quantum mechanical indeterminacy phenomenologically or: what the quantum
physicist can learn from the movement of a chopped onion.
the question concerning digital being is posed, for its origin, which lies
ultimately in Western metaphysics, is by no means clarified in a philosophical
sense. What is digital is usually counterposed to what is analogue. This
amounts to a technical definition. Nowadays, this distinction relates primarily
to the difference in electromagnetic signals of all kinds, whether it be
in telecommunications, electronic music or in computer data processing.
Digital beings are characterized by the fact that they are composed of
binary digits or bits. Signals in telecommunications, for instance, are
transmitted in a digital or binary form through a medium (cables of many
different kinds, the air, space). Basically, an ordered sequence of zeroes
and ones (nothing and something, pure difference) is transmitted which
at the other, recipient's end can be and must be recomposed in such a way
that the appropriate result (a voice, a text, an image, a sound, a TV spot,
a control command, etc.) is brought about. The difference between 0 and
1 may be any arbitrary difference in physical beings such as transmitting
a signal with two different frequencies or two arbitrarily different energetic
states of an electromagnetic system such as the orientation of iron molecules.
Maxwellian electromagnetic force-fields of all kinds (radio waves, electricity,
magnetism, light, molecular bonds, etc. etc.) may be harnessed to generate
a binary difference. The difference as a difference is something
that we humans understand, i.e. we are able to understand (binary) difference
as such and thus to bring forth digital effects. Already in Greek metaphysics,
the category of to\ e/(teron (the other) vis-à-vis
au)to/ (the same, identical), the difference of the one from the
other, plays an important role in the thinking of being and non-being,
especially in Plato's dialectic.
Electromagnetic signals as physical beings (fu/sei o)/nta or beings that of themselves stand in presence including, in this context, also produced things, cultural things), however, in their natural state are not structured or discretely articulated in any form, but continuous. They can be represented mathematically by continuous functions of time (y = f(t)). Aisthaetic beings (Gr. ai)/sqhta, sensuously perceptible beings) are naturally or of themselves (fu/sei) continuous. At first we always perceive a whole (o(/lon) that is not articulated, e.g. we see a car drive past down the street. This is a continuous happening in time. A video camera can record this scene, and the video film can be broadcast on television. The television viewers will still perceive a whole, namely, the scene of a car passing by. Between the live scene and the perceived television sequence there lies the articulated dissolution or taking-apart or decomposition of the scene and its technical reconstitution as a moving image.
So far, so good. This articulated dissolution of what is perceived requires, however, ontological clarification. What is happening, i.e. what must be already given a priori, for digital technology to understand and gain an effective grasp? What does it mean for a being to be whole or one (e(/n)? What does dissolution, decomposition or taking-apart (diai/resij) mean ontologically? What does it mean for a being to move continuously in time, i.e. what is movement, continuity and time? What does the discreteness of digital beings have to do with beings as such? What does number have to do ontologically with beings as such and with movement? And what is the connection between digital dissolution and lo/goj (language, reason, knowledge)?
In digital technology, there must be two different, constant signals, polarizations, bits, states of matter, or the like which are understood as (interpreted as) 0 and 1, as nothing (keno/n) and something (ti/). The categories of something (ti/) and another something (to\ e(/teron) in constancy (a)ei\ o)/n) are presupposed metaphysically. Furthermore, there is also unity (mona/j) and duality (du/aj). How are all these categories indispensable for grasping the digitization of beings ontologically interrelated?
We perceive and understand electromagnetic currents, states, etc. not only as such but as binary difference because these currents, etc. have from the start, i.e. a priori, been interpreted, for instance, by the technological knowledge of the hardware or the communication technology, as such binary differences.
It is impossible to explain, say, the perception of a whole as a temporal process in the brain, for the categories of the whole (o(/lon), of something (ti/) are already 'visible' to the mind's eye in advance, i.e. before any 'data' have been 'registered' by the brain. This a priori dimension - the very general and universal schemata or scaffolding of the categories (cf. also the as-structure with its "pre-structure" (Vor-Struktur) as the "scaffolding" (Gerüst) "from which something becomes understandable as something" (aus dem her etwas als etwas verständlich wird, SZ:151) - must be attributed to the metaphysical (or ontological) power of human vision and has been traditionally the subject of metaphysics today despised by the modern sciences, which have long since staked their pretension to be the 'natural' locus of truth. The sciences investigate their respective subject matters on the basis of an a priori, presupposed understanding of the being of the region of beings into which they do research. Thus the mathematical casting of nature - which made possible modern physics from the seventeenth century on as one of the most momentous events in the history of Western thinking and, in view of its far-reaching consequences, in the history of the world - is not itself a question within physics but rather is presupposed by it. Analytic philosophy of science aids and abets modern mathematical science as its haidmaiden by failing to pose the pertinent ontological questions since, for analytic philosophy, ontology has shrivelled to a matter of classifying what 'exists', where the meaning of 'existence' is taken for granted. The same unquestioning stance pertains to the digital dissolution of beings in progress today which, as we shall see in more detail, is the consummation of the mathematical casting of being. These interrelations with metaphysics wilfully suppressed, denied and dismissed as 'non-verifiable' and 'speculative' guff by modern scientific thinking, and defanged and innoculated by analytic philosophy, must be brought expressly to light in order to see the cast of being on which digital technology is unknowingly, unwittingly based.
Not only Plato (and the Pythagoreans), but above all Aristotle are called upon for assistance, for their thinking is not something past, but, whether we admit and comprehend it or not, maintains its hold on us to this very day. As a starting-point for these considerations, we may take the following passage from Martin Heidegger's Sophistês lectures in Marburg in the Winter Semester of 1924/25.
Dabei ist zu beachten, daß für Aristoteles die primäre Bestimmung der Zahl, sofern sie auf die mona/j als die a)rxh/ zurückgeht, einen noch viel ursprünglicheren Zusammenhang mit der Konstitution des Seienden selbst hat, sofern zur Seinsbestimmung jedes Seienden ebenso gehört, daß es 'ist', wie daß es 'eines' ist; jedes o)/n ist ein e(/n. Damit bekommt der a)riqmo/j im weitesten Sinne - der a)riqmo/j steht hier für das e(/n - für die Struktur des Seienden überhaupt eine grundsätzlichere Bedeutung als ontologische Bestimmung. Zugleich tritt er in einen Zusammenhang mit dem lo/goj, sofern das Seiende in seinen letzten Bestimmungen nur zugänglich wird in einem ausgezeichneten lo/goj, in der no/hsij, während die geometrischen Strukturen allein in der ai)/sqhsij gesehen werden. Die ai)/sqhsij ist das, wo das geometrische Betrachten halt machen muß, sth/setai, einen Stand hat. In der Arithmetik dagegen ist der lo/goj, das noei=n, am Werk, das von jeder qe/sij, von jeder anschaulichen Dimension und Orientierung, absieht.(2)The oneness of each being is indebted to its unambiguous presence within the well-defined contours of its ei)=doj, its look. These are only initial, bare hints from Heidegger, and the passage requires further commentary and deeper probing under the guidance of his phenomenological hermeneutics, to which task we will now turn. Later on, we shall have to take leave of Heidegger's guidance to escape the orbit of what will be called the productionist paradigm of metaphysics.
For Aristotle, the mona/j is the a)rxh/ (principle, starting-point) of arithmetic. It must not be confused with the e(/n, which belongs still to physical beings as an ultimate categorial determination of their being. When some people say that, according to Aristotle, numbers have to be plural, i.e. at least 2, in order to be numbers, this is only sensible when one proceeds from the counting process (cf. Phys. D 12;220a27). If, however, a number is the answer to the question, How many?, then 1 is already a sensible answer and hence a number. The distinction between mona/j and e(/n is more important in demarcating arithmetic from ontology. Proceeding from the mona/j, one comes to two as the first successor in the counting process, and this may be taken as the first counting number. But the mona/j itself must already distinguish itself from something else, from nothing, a nil number, i.e. there must be a difference between 1 and 0 which corresponds to the difference between a unified something (ti/, e(/n) and nothing, emptiness. Only from the principle of unity (monad) can arithmetic, i.e. numbers in the Greek sense, be built up one by one through the iterative counting process. In a further development, and because the base for counting, in principle, is arbitrary, today, all numbers can be represented, manipulated and calculated on a binary basis. The Greeks thought number from the counting process and therefore had no zero, which prevented the assimilation of geometry to arithmetic. To do so would have required the insight into the correspondence between the geometric point and the number 0.(3) Aristotle sees that there is a smallest number, and, proceeding from the geometric line, also that there is no smallest magnitude, but does not resolve the disparity (12;220a30). Even continuity can be captured by a process of limitless approximation by binarily represented numbers, since modern mathematics demonstrates that the continuum consists of the limits of infinite, countable, rational number series. The analytic geometry and differential calculus which Descartes, Newton and Leibniz discovered and developed in the seventeenth century make geometry itself a matter of calculation. We will have to investigate further (cf. 2.6 Bridging the gulf between the discrete and the continuous) the ontological conditions of possibility for bridging the gulf between Greek arithmetic, which was conceived as a discrete counting process, and Greek geometry, whose sensuously imaginable figures are all representable in continuous magnitudes.
2.2. Heidegger's review of Aristotle's thinking on modes of connectedness from discreteness to continuityHeidegger presents the distilling or 'drawing off' of geometric and mathematical structures from physical beings according to Aristotle in his Sophistês lectures (GA19 § 15 Excursus: General orientation on the essence of mathematics according to Aristotle pp. 100ff). The essential, basic act of mathematics for Aristotle is xori/zein and a)fai/resij, separating and abstracting or drawing off from the fu/sei o)/nta which all have a place, topos, locus (to/poj, xw/ra) enclosing and touching them at their outer extremes to which they belong that enables them to come to presence. For instance, a plant belongs to the soil in which it is planted as its proper place to be a plant; an eagle belongs to the mountainous habitat where it builds its eyrie to be an eagle; an actor belongs on a theatre's stage to be an actor. The abstracted geometrical elements and structures already no longer have any place (a)/topoj), but they are posited, positioned (qeto/j) with respect to us and to each other. Geometric entities are no longer in place. The pe/rata are no longer understood as the limits of the physical body, but through the qe/sij they obtain a peculiar autonomy which then can be treated in geometry in this autonomy. This autonomy is heightened even more with numbers (a)riqmoi/), which have neither a place (a)/topoj) nor a position (a)/qetoj). Each number stands on its own (xwrismo/j, discrete), whereas the points of a geometric figure are all identical and are what they are only in relation to other points, i.e. in their position in relation to each other. In other words, they require in addition the determination of pro/j ti, of relation, to be geometrical. Whereas arithmetic entities are formed by sets of numbers in which each number is discrete, geometric figures are not simply composed of points (a line is not simply a collection or heap of points; a surface is not simply a collection of lines; a solid body is not simply a collection of surfaces), but rather they each possess a characteristic complex connected structure which Aristotle sets out progressively in seven steps in his Physics Bk. E Chap. 3. He is concerned with the differing ways in which points and physical beings 'hang together' where continuity, which is closest to the aisthaetic outline of physical beings we perceive with the senses, is ontologically the most complex.
i. a(/ma = coincident; when things are in one placeWhat is ontologically most complex in the way it hangs together, i.e. the continuous geometric figures and physical beings, is most simple for sensuous perception, but is very unwieldy for calculation. And conversely: what is ontologically more simple, i.e. the arithmetic entities in their ordered, countable succession, is not as easily accessible to sensuous perception but can be calculated (logismo/j) without any difficulty. This means that the arithmetic entities and their interrelations can be more easily brought to presence by the lo/goj(or the logismo/j in this case) than geometric entities which, in turn, are closer to sensuous experience, i.e. not so abstract. Herein resides the calculative power of mathematical analysis which reduces the geometric to the arithmetic, the continuous to the discrete, irrational (real) number to rational number, by conceiving real numbers as (Dedekind) cuts or partitions in the (infinite, but countable) sequence of rational numbers. The reduction facilitates calculation in the mathematical language of algebra, and, conversely, the results of the calculation can be translated once again back into the sensuously aisthaetic intuitions of geometry which have a representation in the imagination. With the arithmetization of geometry, the mathematico-logical manipulation of beings thus attains a hitherto unprecedented power.
Heidegger also provides a review in the Sophistês lectures of Cat. 6: 'On Quantity' (poso/n). Quantity is in some cases discrete (diwrisme/non or marked off from itself within itself), and in some cases continuous (sunexe/j or holding itself together within itself). (Cf. Met. V, 13: quantitative means that which can be decomposed into several immanent components. That which is quantitatively countable is an amount; what is measurable is magnitude. An amount is potentially decomposable into discrete components; magnitude is potentially decomposable into continuous components). Continuity "is the ontological condition of possibility for there being something resembling length, me/geqoj" (ist die seinsmäßige Bedingung dafür, dass es so etwas wie Erstreckung, me/geqoj, gibt, GA19:118; cf. extensio in Descartes' metaphysical casting), and motion is only comprehensible when "continuous progress can be made from one point to another." (von einem Punkt zum anderen stetig fortgeschritten werden kann, ibid.)
The numbers and the lo/goi are marked off from each other, i.e. discrete, whereas the geometric figures such as line, surface, solid body, time and places are continuous. Discrete entities are articulated into parts which are not posited, i.e. they do not have any position; the continuum, by contrast, consists of parts which are posited, positioned with respect to each other. Hence figures. Their respective manners of connectedness or their unity therefore differ. The parts of numbers do not have any common o/(roj or limit. The number 10, for instance, has parts 5 and 5 which do not have any common limit; each part is for itself; the parts are marked off from each other, diwrisme/non, each is different, just as with 7 and 3. The mo/ria (parts) cannot be taken together; there is no koino/n (common element) with respect to which each number would be an instance, i.e., it is not possible to generalize the numbers. How, then, is a connectedness possible? Aristotle explains this using the example of the lo/goj: it is meta\ fwnh\ gigno/menoj, spoken with the voice. This speaking (a sensuous experience for both speaker and listener) is articulated into individual syllables as its stoxei=a (elements) which are marked off from each other. There is thus a peculiar unity of a non-continuous, articulated entity in which each part is autonomous, individual. The syllables are autonomous, individual. There is no syllable in general, and also no number in general. The unity of the manifold elements can only lie in the lo/goj or nou=j itself which gathers together and holds together the parts, for there is no merely sensuously perceptible connection, for instance, why certain syllables or numbers should stand next to each other. When the lo/goj appropriates beings in their self-disclosure, it articulates them at the same time into their 'articles'. A diairetic taking-apart takes place which may, in turn, be further articulated into numerical digits. The logic of the lo/goj ends ultimately in the digital de-ciphering of beings in toto, which is equivalent to an en-ciphering of beings. The decryption (disclosure) of beings in their being amounts then to en-ciphering them articulately into digits.
By contrast, one point is like all others. A line has another mode of unity. One can remove something from it and address it in the same way as any other part. The points are all the same. But a line is more than a multitude of points; the points are put in position (qeto/j) and they do not just touch, but hold themselves together (sunexe/j). This is missing in the sequence of numbers which is only determined as e)fech=j (consecutive, sequential) and where no medium in between is necessary. Number is therefore ontologically prior to the points in their continuum. Number is still free of orientation and position and is therefore autonomous and can and must be taken in and comprehended without ai)/sqhsij only by means of the intuition of nou=j. As ontologically simpler and more originary, number is set in an originary connection with the simplest categories such as the something (ti/) when one asks for the structure of beings (to\ o)/n). "This is the reason why Plato's radical ontological determination starts with number." (Darin liegt begründet, daß die radikale ontologische Besinnung Platons bei der Zahl ansetzt. GA19:121) Nevertheless, for Aristotle arithmetic is not the most originary science of beings in their being, for the a)rxh/ of number, i.e. the counting unit (mona/j), must be clarified metaphysically in its connection with the one (e(/n). And this connection provides the key to the ontological interconnection between number and the metaphysical access to beings as o)/n lego/menon in general.
Insofar as physics is mathematical, it relies on the discreteness of numbers because it has to perform calculations on empirical data; however, numbers can also be made to approximate the fu/sei o)/nta arbitrarily closely. That was the great discovery of the mathematical differential/infinitesimal/integral calculus by Newton and Leibniz, for only in this way were physical beings made arbitrarily calculable, i.e. the arbitrarily close approximation of number to the continuum (the digital dissolution of beings to any arbitrary degree of resolution) became possible. For a long time, physics has been marked by a dispute about the fundamental nature of physical beings: wave or particle (atoms)?, continuous or discrete?. This dispute is not decidable within physics itself because the distinction has to be clarified in ontology itself (i.e. coming from being) where the distinction between the continuous and the discrete, i.e. that which holds itself together within itself (the geometric), on the one hand, and that which is marked off from itself from within itself (the arithmetic), on the other (cf. GA19:118), can and must be interpreted as modes of being. Moreover, as we shall see (cf. 2.9 Time and movement in Aristotle's thinking), the phenomenon of movement in time will demand consideration of a twofold presencing in whose light the Heisenberg indeterminacy principle receives a phenomenological interpretation prior to its mathematical cast (cf. 7.1 The Heisenberg indeterminacy principle reinterpreted). The infinitesimal calculus, which enables an approximation (a nearing) of the continuous and the discrete, the geometric and the arithmetic, represents a crucial historical event in the ontology of the mathematical that opened the vista of a mathesis universalis. We shall return to the question of infinitesimals below (cf. 2.8 The calculative assault on movement and time through infinitesimal calculus).
2.3. The crucially important analogy between logos and number for the appropriation of beings: arithmological knowledgeIt is a surprising difference between numbers and points, that each number is autonomous, whereas all points are the same. The analogy between number and logos is also striking and has essential consequences for grasping the being of beings. In the preparatory section of his Sophistês lectures, Heidegger emphasizes the access to the being of beings through the logos for the Greeks. For Plato and Aristotle, Heidegger maintains, a being is o)/n lego/menon, i.e. beings as they are said. At the same time, with the phenomenon of sophistry, it is a matter of beings in the first place being uncovered or covered up and distorted by the logos as speech, even though for Aristotle, the highest form of knowledge, sofi/a, is said to come about through the human mind (temporarily) attaining nou=j, which, he says, is a)/neu lo/gou (without logos). Whereas the lo/goj, as Aristotle discovered, is always a le/gein ti\ kata\ tino/j, and thus is articulated by way of the apophantic As or Qua as a saying-something-about-something, nou=j, by contrast, is a direct looking-at or intuition (Anschauung) of the most general and universal ideas, ei)/dh or sights, i.e. the categories, which cannot be broken down any further and articulated in a diai/resij as saying something about something.
If number and logos are both abstracted ('drawn off') from aisthaetically given, sensuous beings, then in this discrete taking-apart, decomposition or resolution of beings there is simultaneously a distancing from beings which makes it possible for beings to be made present by the logos (and by number) in a different way from the way they show themselves of themselves (aisthaetically). With the logos, another way of making beings present is given. Heidegger writes, for instance, in a striking formulation, "This invasion of the lo/goj, of the logical dimension in this strict Greek sense, into this question concerning the o)/n is motivated by the fact that the o)/n, the being of beings itself, is interpreted primarily as presence and the lo/goj is the way in which I primarily make something present, namely that about which I am speaking." (Dieser Einbruch des lo/goj, des Logischen in diesem streng griechischen Sinn, in diese Fragestellung nach dem o)/n ist dadurch motiviert, daß das o)/n, das Sein des Seienden selbst, primär als Anwesenheit interpretiert ist und der lo/goj die Art ist, in der ich mir etwas, nämlich das, worüber ich spreche, primär vergegenwärtige. GA19:225, italic emphases by Heidegger himself)
As we shall see (2.7 Cartesian rules for an algebra of magnitudes in general as foundation for the modern mathematical sciences), this "invasion of the lo/goj" that articulates beings discretely is exponentiated when paired with the discreteness of the a)riqmo/j which enables also a calculability of beings in their being with historically far-reaching consequences of such arithmological knowledge. The pinnacle is reached when the a)riqmo/j and the lo/goj fuse into abstract algebra in the nineteenth century.
Here, the sense of being as presence and, more particularly, as presence-at-hand (Vorhandenheit) uncovered by Heidegger, and the mutual entanglement of logos and being are at work. The primary sense of being according to Heidegger, ou)si/a or what underlies, i.e. the u(pokei/menon, is what lies at hand for speaking about it in the present. What is of interest here is that the early Heidegger is seeking an access to the phenomenon of truth without the logos. What does he have in mind? This mode of access is not simply Aristotelean nou=j, i.e. the immediate intuiting of the categories, but the two fundamental modes, understanding (Verstehen) and attunedness (Befindlichkeit, Gestimmtheit) in which world opens up to Dasein. To start with, this search can be marked off against Gadamer's:Worte springen wie die Affen von Baum zu Baum, aber in dem dunklen Bereich, wo man wurzelt, entbehrt man ihrer freundlichen Vermittlung. (Robert Musil Der Mann ohne Eigenschaften I Tl. 2 Kap. 40)
Sein, das verstanden werden kann, ist Sprache. Das hermeneutische Phänomen wirft hier gleichsam seine eigene Universalität auf die Seinsverfassung des Verstandenen zurück, indem es dieselbe in einem universellen Sinne als Sprache bestimmt und seinen eigenen Bezug auf das Seiende als Interpretation. [...] Denn sprachlich und damit verständlich ist das menschliche Weltverhältnis schlechthin und von Grund aus.(4)Gadamer's hermeneutic approach makes it manifest that his starting-point is not as originary as Heidegger's, for the latter is essentially concerned with breaking the hegemony of the lo/goj in philosophy after two-and-a-half millennia precisely by situating the originary disclosure of the world prior to the articulate interpretation of the world in language. Heidegger's "hermeneutic as" (SZ:158) is prelinguistic. Dasein has always already discovered the world and interpreted it in dealing with practical things "'without losing a word'" ('ohne dabei ein Wort zu verlieren', SZ:157). When the world comes to language, articulating itself in the lo/goj of the proposition, beings are shown up in saying something about something. This is the phenomenon of "'something as something'" ('etwas als etwas', SZ:159) or the "apophantic as" (apophantisches Als, SZ:158) which itself is derivative of the more originary "hermeneutic as". In uncovering a prelinguistic access to the world in its truth, Heidegger follows the guiding thread of the sense of being as presence and comes upon time as the originary transcendence to the world. Now, instead of presence as the temporal sense of what lies to hand for speaking about it, the phenomenon of time itself in its multidimensionality (enabling also a simultaneous presencing and absencing) comes into the intense focus of thinking. It is thus not a matter of Heidegger's having set his gaze on something resembling a "Ding an sich" (Kant), i.e. something which he properly cannot speak about, which he cannot grasp and conceive, but which he names nevertheless, nor is it the immediate, intuitive, noetic sight of the most universal ideas, but rather it is a matter of a world-opening which lies prior to speaking-about, as demonstrated in the equipment analysis in Being and Time.
Equipment (practical things, pra/gmata) in its being-(good)-for... (Um-zu) is discovered, understood and interpreted in its being prior to any grasping in language by Dasein, and Dasein's taking care of daily life using practical things is interpreted ultimately in its temporality as the everyday sense of Dasein. In Being and Time, Heidegger takes great pains with a "demonstration of the derived nature of the statement" (Nachweis der Abkünftigkeit der Aussage, SZ:160), i.e. of the lo/goj, in order to "make it clear that the 'logic' of the lo/goj is rooted in the existential analytic of Dasein" (deutlich zu machen, daß die 'Logik' des lo/goj in der existenzialen Analytik des Daseins verwurzelt ist, ibid.). He wants to retract the "lo/goj as the sole guiding thread for access to beings proper and for the determination of the being of beings proper" as it "functioned in the decisive beginnings of ancient ontology" (in den entscheidenden Anfängen der antiken Ontologie der lo/goj als einziger Leitfaden für den Zugang zum eigentlich Seienden [...] fungierte, SZ:154). A corollary of this is the future historical possibility that calculable, discrete number as the hegemonic "guiding thread for access to beings proper" in the mathematico-scientific age could also be retracted. The 'one-dimensional' sense of being as standing presence or "standing presence-at-hand" (ständige Vorhandenheit, SZ:96) is unfolded into the full three-dimensionality of temporality.
Time, being-in-time enables an access to being, i.e. it holds it open, without the logos, or prior to the logos. After Heidegger's momentous incursion into Western metaphysics, the temporality of human being (Dasein) can no longer be clarified by following the guiding thread of the logos, but rather, the logos and its hegemony as ontology can only be clarified by starting from the multidimensional, temporal meaning of being and Dasein's temporality. Time, however, is neither linearly continuous nor logically discrete; it therefore cannot be dissolved and grasped digitally, because it does not lie before us as something present from the start. It does not lie before us like a u(pokei/menon to be spoken about; it is not a something (ti/, ou)si/a) lying before us to be spoken of, for a something lying present at hand is only present, which would reduce time proper to the instantaneous now (nu=n) which, tellingly, has the ambiguous ontological characteristics of both discrete presence-at-hand or standing presence, and fleeting continuity or non-being. Time is and, simultaneously, is not. "From the hegemony of this concept of being it becomes clear why Aristotle interprets time itself starting from the present, the 'now'." (Aus der Herrschaft dieses Seinsbegriffs wird deutlich, warum Aristoteles die Zeit selbst aus der Gegenwart, dem 'Jetzt', auslegt. GA19:633) Does this mean that the decomposing taking-apart (diai/resij) of physical beings, including practical things, performed by the logos and mathematics depends essentially on the state of beings as things lying before us as present? Yes, indeed. Whereas Dasein has to be interpreted in the full temporal three-dimensionality of its existence as a cast and casting already-being-with... (entwerfend-geworfenes Schon-sein-bei...) that understands and interprets the world in attunement with it, in order to adequately capture the phenomenon as it shows itself of itself, what is already lying ready before us to be spoken about can be clarified, starting from this fully unfolded interpretation of Dasein, as present merely in a derivative mode of temporality. As we shall see (cf. 2.9 Time and movement in Aristotle's thinking), the phenomenon of time itself cannot be interpreted within the metaphysical framework which has only been able to grasp time ontologically on the tacit assumption that it could be captured as standing presence.
The hegemony of the meaning of being as presence-at-hand (ou)si/a as Vorhandenheit or standing presence) tacitly assumed and established already in ancient Greek ontology carries over to how the lo/goj as proposition itself is understood, namely, as something present-at-hand that can be taken apart into its components, the (syntactical) sequence of words and in particular the sequence of subject and predicate joined by the so-called copula, "is". A simple proposition of the form, S is P, is then taken as the starting-point for all philosophical reflection on the lo/goj, starting with Plato and Aristotle themselves. The proposition, however, is rooted more originally in the "phenomenon of 'something as something'" (Phänomen des 'etwas als etwas', SZ:159) according to which something - a hammer, for example - is understood and interpreted wordlessly as being something, namely, "'too heavy'" ('zu schwer', SZ:157) for the job at hand. Whereas Aristotle at least still saw the paradoxical simultaneous putting-together and taking-apart, i.e. the su/nqesij and diai/resij, characteristic of every lo/goj as proposition, later philosophy formalized this to a relation in a "system of attributions" which "becomes the object of a 'calculating', but not the topic of an ontological interpretation" (in ein System von 'Zuordnungen' aufgelöst, es wird zum Gegenstand eines 'Rechnens', aber nicht zum Thema ontologischer Interpretation, SZ:159). From here it is not far to interpreting the merely formal copula as an equals sign in an equation or as the subset sign in a Boolean algebra of sets. The proposition, S is P, hence becomes interpreted as the statement, 'S is an element of the set of all things having the attribute P'. Such sets and their interrelations can be calculated in a formal algebra that presupposes that the S, P and sets of suchlike are all things present-at-hand open to such calculative manipulation. Finally, in particular, the elements of the lo/goj thus decomposed and formalized are all representable in binary code that can be organized into a calculus. The incompleteness theorems of Gödel within mathematical logic point to an excess of the truth or otherwise of what can be said (predicated) that always remains outside what can be grasped calculably by the lo/goj, thus vitiating the dream of total machine calculability within mathematics (through recursive functions).(5) this process of historical transformation passes through the key figures Diophantos, Vieta, Simon Stevin, Wallis and Descartes. The difficulty obstructing this convergence resides in the circumstance that the Greeks thought the a)riqmo/j as countable, starting with the unit or mona/j. As unit, the unit is indivisible, discrete, so the best Greek mathematics could do was to form proportions of natural, counting numbers, that is, positive fractions, broken integers or so-called rational numbers. From the geometric side, however, the Greeks were aware that somehow there were some numbers missing from the countable integers and fractions, namely, those numbers 'in between' the fractions that could not be brought into the form of a fraction, i.e. a ratio of two whole numbers. They were therefore called irrational numbers or surds or incommensurable because they could in no way be measured by the unit for counting, the mona/j, by way of creating a ratio (lo/goj). The simplest irrational number arises already in considering the diagonal of the unit square, whose length is the square root of two. These irrational numbers are the magnitudes arising from geometric figures which, in turn, are obtained by abstracting the contour outlines of continuous, physical entities. Geometric figures clearly (i.e. for the visual imagination) hold themselves together; they are continuous. How are all the points on the fundamental geometric figures of a line or a plane to be captured numerically if number is conceived as fundamentally countable? This countability, in turn, derives ontologically from the implicit Greek preconception of being as presence-at-hand: a definite number arises from actually counting the things lying present at hand. For Greek thinking, that which lies present at hand is the u(pokei/menon, and such u(pokei/mena in a multitude are countable. As we have seen above, Aristotle thinks the phenomenon of continuity ontologically starting from discrete beings which can touch, be lined up in succession, hang together and, finally, hang tightly together.
The counting unit is indivisible, whereas the unit line is infinitely divisible. Not all the possible magnitudes contained in the unit line can be captured by countable, i.e. rational numbers. The rational numbers have to be complemented by the irrational numbers to attain the entirety of a continuous line with all the possible magnitudes it contains. Although rational numbers can be made to approximate each other as closely as one likes, between any two rational numbers whatever there is an irrational number, i.e. a magnitude that cannot be expressed as a fraction of two integers. How are the countable, rational numbers to be completed to get the real numbers? Real number is an appropriate term because only by means of these real numbers can all the magnitudes of sensually perceptible, real, physical bodies be assigned a number. The task is how physical res can be captured mathematically by number, and not merely by geometry. Only number opens the possibility of calculation, whereas geometry has to rely on intuitive proofs for which the geometrical objects have to be imagined sensuously in an immediate intuition. To be continuous, and thus to capture all physical magnitudes of any kind, number has to become real, uncountable. Uncountability implies that, since the rational numbers are countable, between any two rational proportions of integers, no matter how minimal the difference between them, there are always non-rational numbers, i.e. rational numbers can come infinitely close to one another without ever gaining continuity, i.e. there is always a gap between them that is not rational (i.e. irrational), and in this sense they do not hang tightly together like the geometric line. Richard Dedekind's small but crucial step was to fill in the gaps between the rational numbers by conceiving the real numbers as the limits of infinite, but countable sequences of rational numbers.
2.7. Cartesian rules for an algebra of magnitudes in general as foundation for the modern mathematical sciences
So the problem becomes, how can there be a mathematical calculus of uncountable, real numbers, and what is the ontological (pre-)conception or (pre-)casting of number on which such a calculus could be soundly based? That is the problem of the ontological recasting of mathematics as algebra in the modern age. Number has to become magnitude pure and simple, which is uncountable, but nevertheless calculable. Magnitude is the quantity pertaining to any extension whatsoever of a real, sensuously perceptible being from which sensuous data, and therefore quantifiable data, can be received. Such extension need not be only spatial extension such as the three Euclidean dimensions of length, width and depth, but can be any one of the countless dimensions whatsoever of a perceptible res such as colour or "weight" (gravitas, XIV.16). Thus, Descartes writes in the twelfth of his Regulae ad Directionem Ingenii (Philosophische Schriften Meiner, Hamburg, 1996), "For example, you may suppose whatever you like about colour, but you will not deny that it is extended and consequently has figure" (Verbi gratia, colorem supponas esse quidquid vis, tamen eundem extensum esse non negabis, et per consequens figuratum, XII.6). A figure is geometric, and a geometric figure of whatever kind has magnitudes. The Cartesian ontological casting of beings as res extensa is essential for their reduction to figure and thus, since figure is grasped as a simple manifold of magnitudes, to mathematically calculable magnitude....wenn Licht, Wärme, Kraft, Genuß, Bequemlichkeit Urträume der Menschheit sind, -- dann ist die heutige Forschung nicht nur Wissenschaft, sondern ein Zauber, eine Zeremonie von höchster Herzens- und Hirnkraft, vor der Gott eine Falte seines Mantels nach der anderen öffnet, eine Religion, deren Dogmatik von der harten, mutigen, beweglichen, messerkühlen und -scharfen Denklehre der Mathematik durchdrungen und getragen wird.
Descartes goes on to show in Rule XII.6 that the dimension of colour (of any kind of physical beings), for instance, can be represented simply by different figures which amount to different symbols representing the various colours. And he notes, "The same can be said of all things since it is certain that the infinite multitude of figures suffices to express all the differences of sensible things" (Idemque de omnibus dici potest, cum figurarum infinitum multitudinem omnibus rerum sensibilium differentiis exprimendis sufficere sit certum, XII.6). When the intellect is examining something "that can refer to bodies, this idea must be formed in the imagination as distinctly as possible; to bring this about comfortably, the thing itself which represents this idea must be exhibited to the external senses" (quod referri possit ad corpus, ejus idea, quam poterit distinctissume, in imaginatione est formanda; ad quod commodius praestandum, res ispa, quam haec idea repraesentabit, sensibus externis est exhibenda, XII.11). But if the intellect is to think through and deduce (deducat, XII.11) from a plurality, "everything not requiring attention at present is to be thrown out of the ideas of the things" (rejiciendum est ex rerum ideis quidquid praesentem attentionem non requiret, XII.11). Therefore, "then the things themselves are not to be laid before the external senses, but rather certain abbreviating figures" (non tunc res ipsae sensibus externis erunt proponendae, sed potius compendiosae quaedam illarum figurae, XII.11). These "abbreviating figures" are then elaborated in Rule XVI as "the briefest of signs" (brevissimas notas) which enable the intellect to think through things without being distracted by concrete details. All the dimensions of beings thus become insofar representable in a manifold of quantities represented by symbols. This is, of course, a questionable strategy, as can be learnt already from Hegel: "Due to their simplicity, the simple first figures and numbers are suitable for use without misunderstanding as symbols, but they always remain a heterogenous and paltry expression for thinking. [...] For richer concepts, these means are completely insufficient since their extrinsic composition and the contingency of the linkage in general is inadequate to the nature of the concept..." (Die einfachen ersten Figuren und Zahlen eignen sich ihrer Einfachheit wegen, ohne Mißverständnisse zu Symbolen, die jedoch immer für den Gedanken ein heterogener und kümmerlicher Ausdruck sind, angewendet zu werden. [...] Aber bei reicheren Begriffen werden diese Mittel völlig ungenügend, da deren äußerliche Zusammensetzung und die Zufälligkeit der Verknüpfung überhaupt der Natur des Begriffs unangemessen ist... Enz. II § 259 Anm.).
No matter whether the aid of the imagination is required to represent a state of affairs to the intellect, or whether this can be done through concise symbols, if the state of affairs is not simple and immediately apparent to intuition, it can only be clarified, as Descartes prescribes in Rule XIV, by comparing it with a known state of affairs. Such comparison consists in establishing that "what is sought is in this or that respect similar or identical or equal with some given" (quaesitum esse secundum hoc aut illud simile, vel idem, vel aequale cuidam dato, XIV.2). Equality, however, immediately becomes the standard of comparison between the unknown and the known. Where the comparisons of equals are not "simple and open" (simplices et apertas, XIV.3), but are concealed in "some sort of relations or proportions" (quasdam habitudines sive proportiones, XIV.3), the task of the human intellect lies in "reducing these proportions in such a way that the equality between what is sought and something known becomes clearly visible" (in proportionibus istis eo reducendis, ut aequalitas inter quaesitum, et aliquid quod sit cognitum, clare videatur, XIV.3).
The culmination is then to note that the kind of equality required between the sought and the given, the unknown and the known, is an equality of magnitudes: "It is to be noted finally that nothing can be reduced to this equality if it does not admit a more or less and that all this is to be comprehended under the term 'magnitude' so that [...] we understand that from here on we are involved only with magnitudes in general" (Notandum est deinde, nihil ad istam aequalitatem reduci posse, nisi quod recipit majus et minus, atque illud omne per magnitudinis vocabulum comprehendi, adeo ut [...] hic tantum deinceps circa magnitudines in genere intelligamus nos versari, XIV.4). This holds true no matter whether the intellect is assisted by the imagination or is employed purely (intellectu puro utamur, XIV.5). The aim is to find a relation of equality between something unknown and something known, where both these somethings are nothing but "magnitudes in general". The "relations and proportions" that at first conceal the equality between the unknown and the known must be equations in "magnitudes in general" that can be reformulated so as to finally bring forth the required equality. But this is a description of the general algebraic procedure, no matter whether an image is used to assist the procedure or not. Magnitudes in general are represented in the equations by "brief signs" or symbols, and the equations themselves can be manipulated by the pure intellect to reformulate them in such a way that the unknown, x, is brought into equality with what is given and known. This amounts to solving a set of equations for the unknown, x.
"From here on" we are dealing only with sets of equations in "magnitudes in general" which are to be solved by algebraic methods. These magnitudes are the knowns and unknowns occurring in equations. They are no longer pinned down as continuous geometric quantities or discrete arithmetic ones but are simply the data and solutions to sets of equations of such and such a type. The data given by real beings are all quantitative by virtue of casting the being of beings solely as extension, so that all the many qualitative dimensions of a being, no matter what it and they may be, are reduced to magnitudes that can be inserted into equations as knowns. What is unknown is then discovered by solving the equations for x. The behaviour of real beings must therefore be described in equations, and certain knowledge is to be gained by solving equations of certain kinds. Mathematics itself can then become the motor driving the quest for knowledge through the investigation of kinds of equations with the aim of being able to solve them algebraically for the unknown, x. Whether the magnitude in question is geometrically continuous or arithmetically discrete is no longer crucial, because magnitudes in general can be represented by symbols, and these symbols may be defined simply as the solution to a certain kind of equation within a certain kind of mathematical entity such as a field, ring or group defined solely by a set of logically consistent axioms whose validity relies on immediate intuition. The steps beyond the natural numbers to the rational numbers and on to the real numbers need not stop there. The complex numbers, for instance, can be introduced simply as the solution to certain kinds of equation that do not have solutions among the real numbers, but require the square root of minus one, the imaginary number i. And even these complex or imaginary numbers can still be represented to the imagination as planes, which themselves are imagined as extended. The quest for knowledge (starting with, but soon proceeding beyond, classical mechanics in the natural science of physics) is then guided by applying the mathematical intellect to finding solutions to ever more complex systems of equations in abstract, algebraic symbols standing for magnitudes in general. The future historical trajectory of mathematics for the next few centuries as an abstract symbolic discipline is thus fore-cast by the Cartesian ontological rules, thus laying down the blue-print for the modern age.
If the Greek beginnings of mathematics, in which there is an hiatus between arithmetic and geometry, is papered over in a Cartesian mathematics of magnitudes in general, culminating in abstract algebra, it may be objected that the distinction between digital discreteness and analogue continuity loses its importance and is overcome in the modern age. Accordingly, so the objection goes, analogue computing could, 'in principle', serve just as well as digital computing for the cybernetic cast of the Cartesian modern age. In fact, for certain species of problems concerning especially the dynamics of physical systems that have to be formulated using differential equations, analogue computers have some advantages over digital computers, since the continuous, physical movements of voltages or fluids can be contrived to move continuously and analogously to a given dynamical system. This is correct. However, the antinomies between discrete number and continuous magnitude in mathematics remain (cf. Feferman 1997, Weyl 1918, 2.8.1 Excursus 1: On the antinomy between countable discreteness and the continuum in twentieth-century mathematical foundations) which makes itself felt practically in the convertibility between the two domains. Calculations also have to read by human beings or by digital computers, e.g. as inputs and especially as outputs, and such reading in or out demands a conversion of continuous physical magnitudes (such as lengths, voltages, currents or pressures) into definite numbers (with an accuracy specified by a number of discrete decimal/binary places) which, as definite, are necessarily finite, rational, that is, digital. At the interface, the error in the determination of significant figures by reading off analogue computers is considerably greater than for digitally computed measurements. Likewise, although the results of an analogue calculation may be stored more or less stably, say, as a voltage in a capacitor, or as a physical length, this is of no use for the arithmological human or digital interface which demands definite numbers either as a result or for further digital calculation.
The principal deficiency of analogue computers, however, is that they cannot be (logically) programmed, but must be (physically) constructed. A program is a pre-script, that is, it is logical, specifically, arithmo-logical (cf. 2.3 The crucially important analogy between logos and number for the appropriation of beings: arithmological knowledge). A logical understanding of a segment of the world is pro-grammed 'literally', broken down into bits, into a digital machine for it to carry out the pre-scripted algorithmic calculations. With an analogue computer, by contrast, the computer itself has to be built physically, i.e. its circuits set up, for a specific calculation task. There is no universal analogue computer whereas, by virtue of logical programmability, there is a universal digital (Turing) machine which is first fed with the digital program for the task at hand. A logical understanding is programmed and outsourced to a digital machine in which it can be set into motion to calculate and control movements/changes automatically. Digital calculation, and hence digital beings, 'live' off the intimate affinity between the lo/goj and the a)riqmo/j for human understanding. The human mind must define, delimit, articulate to understand, so that continuous physical magnitudes, as employed in analogue computing, have to maintain a convertibility with digital number. Hence it is incoherent to speak of continuous magnitude being representable as 'numerical code', for coding per se implies digitizable logification. It is therefore also no historical accident that digital computers have won out over analogue computers, and that today hybrid analogue-digital computers are employed for certain specific problems, especially where differential equations of motion arise. An analogue computer is incorporated into a universally programmable digital computer to perform a specific task for which an analogue computer (a suite of electronic circuits that behave physically in analogy to a given dynamic system) is particularly suited.(6) Time is thus thought in the interstellar cold of this natural-scientific ontology as a manifold of now-points or instants, i.e. as presence; both future time and past time are only now-points greater than or less than a given now-point, respectively. Time is measured empirically by gathering the countable data now-points of some very regularly periodic physical process (just as Aristotle's Physics laid down: "Not only do we measure movement through time, but also time through movement because they mutually determine each other." (Ou) mo/non de\ th\n ki/nhsin t%= xro/n% metrou=men, a)lla\ kai\ tv= kine/sei to\n xron/on, dia\ to\ o(ri/zesqai u(p" a)llh/lwn. Phys. D 12;220b15)). Equations of motion in (x, y, z, t) arise according to physical laws of motion whose solution can be sought, depending on which variables are known givens and which unknown.
When the mathematically formulable Newtonian laws of classical physics are modified to take into account that there is no absolute time variable, t, but rather that there are differences in time between two inertial frames of reference (the 'proper time' with the symbol tau, t) which are determined mathematically by the Lorentz transformations involving the speed of light, c, the movement of bodies (particles) in such a (Minkowski) space-time is still formulable in four-dimensional equations in which the resemblance to the classical Newtonian laws of motion is still clearly recognizable.(7) Calculation with both classical Newtonian and relativistic equations of motion requires the use of infinitesimal calculus because the velocity of a body is the derivative, and its acceleration is the second-order derivative of a space 3-vector (with respect to time, t) or a 4-vector (with respect to the time-difference, t), respectively. Rates of change of continuous mathematical variables of whatever kind necessitate a calculus with infinitesimal magnitudes to gain a calculative hold on the phenomenon of movement (strictly: locomotion, i.e. only one kind of movement) through real, continuous variables such as space and time co-ordinates.
Space-time - no matter whether Newtonian-Galilean, Minkowski-relativistic or Riemann-relativistic (including gravitational mass points) - is the context for the motions or, more precisely, locomotions of physical bodies which may be celestial bodies, including stars, planets, galaxies, black holes, supernovae, pulsars, etc., bodies moving on Earth such as cannon balls, ballistic missiles, ships, etc., or those peculiar invisible particles of quantum physics whose motions are governed by complex differential equations. As Descartes' Rules already prescribed, however, extension is not restricted to spatial dimensions, but covers anything admitting of "more or less", including time, colour, weight, stress, pressure, reproductive potency (biology), emotional tension (psychology), propensity to consume (economics), ad infinitum. It depends solely on scientific ingenuity whether any phenomenon at all can be reduced, or led back, to the movement of a magnitude. Such quantification demands a mathematics to calculate such movement through the appropriate equations. It makes no difference whether the magnitudes are exact or inexact, or the equations involved can be solved uniquely, approximately or only within certain ranges of probability. Mathematical statistics as a calculus of probability distributions is the way, in the modern mathematical age, of making those phenomena that do not move with necessity, but only with regularity (Aristotle's category of e)pi to\ polu/), calculable nevertheless.
Because of the universal applicability of quantitative mathematical methods to all regions of phenomena, it was crucial for mathematics to put the infinitesimal calculus on a firm foundation. This was begun by Augustin Cauchy in the nineteenth century and finally accomplished by Karl Weierstrass with the rigorous, epsilon-delta definition of limit, which obviated having to introduce infinitesimals as mathematical magnitudes smaller than any real number. Any number on the real continuum can then be defined as the limit of a countable, infinite sequence of rational numbers. Continuity and differentiation (and its inverse operation: integration) could then be rigorously formulated within the real numbers, perhaps with the aid of the imaginary number i, and the historically momentous nineteenth century program of the arithmetization of geometry, or the convergence of the discrete and the continuous, consummated.
All mathematico-scientific treatment of movement of whatever kind requires at least a quantifiable concept of time, which may be conceived, or rather: imagined, as a simple, continuous, 'linear' variable of now-points usually taken to be non-reversible, but not necessarily so.(8)Why time should be non-reversible remains undecidable in a purely mathematical conception of time. No matter whether an absolute or relativistic time is assumed, this time is regarded as scientifically 'objective', as opposed to the so-called 'subjective' time of psychological, cultural, historical, poetic, etc. experience. But objective time is the conception of time employed by a certain kind of thinking in order to make movement of all kinds calculable and, in many cases, predictable. That is, the concept of objective time is such only for a subject, viz. human being, underlying this kind of calculative will to power over movement. The ontological casting of the phenomenon of time quantitatively as amenable to mathematical calculation is a determinate historical conception of time that determines, i.e. truncates, also the possibilities of the human experience of time and hence also of the human experience of movement.
2.8.1 Excursus 1: On the antinomy between countable discreteness and the continuum in twentieth-century mathematical foundations (Solomon Feferman and Hermann Weyl)
(July 2009)The antinomy between the discrete and the continuum returns at the beginning of the twentieth century with the crisis in the foundations of mathematics involving questions concerning how the continuum of real numbers necessary for mathematical analysis can be derived logically from primitive elements, plus, perhaps, the arithmetical basis of the natural numbers taken as given: "... it is when we come to the real numbers that we get into serious problems about the logical foundations of mathematics..." (Solomon Feferman 'The Significance of Hermann Weyl's Das Kontinuum' second of three lectures for the conference Proof Theory: Historical and Philosophical Significance held at the University of Roskilde in Denmark, 31 October - 01 November 1997, accessed July 2009. All further quotations in this Excursus are also from this paper.). The foundations of mathematics become logically problematic at the beginning of the twentieth century, most strikingly demonstrated by Russell's paradox, related to the ancient Greek liar's paradox, which Russell communicated to Gottlob Frege with depressing effects on the latter logician.
As Feferman clearly and succinctly lays out in his 1997 lecture, Hermann Weyl did not accept Russell's attempted resolution and put forward his own attempt in his 1918 book Das Kontinuum. The antinomies unearthed in mathematical logic concern the limits of what is predicable, sayable at all with mathematical exactness of a mathematical entity. Russell's antinomy shows that it is not possible to take any predicate at all (Russell spoke of "properties") and then posit the underlying ('subject') set of mathematical entities to which that predicate/property applies. The property was then said to be "non-predicative" for it had no "extension" in an admissible set conceived as a collection of mathematical entities (and in this sense a res extensa). In other words, the saying was unsayable, the proposed predicate impredicable, for it referred to a logically inadmissible mathematical entity. What is sayable has to refer to (mathematical) entities whose existence is established by an appropriate definition or proof invoking only already existing mathematical entities. A mathematical predicate is said of a mathematical subject which cannot be simply posited by positing the set of all entities satisfying the predicate (the so-called Axiom of Separation), but whose existence has to be secured in terms of a construction from entities whose own existence is already assured.
Russell hence proposed a stepwise procedure starting with the simplest entities of type 0 and then proceeding to successively define higher types in terms of sets of elements of the same or a lower type. One could not presume the existence of a totality (such as the set of all sets) containing the entity-subject to be defined by a predicate/property (such as the set of all sets which are not members of themselves). That way lies patent nonsense. Russell therefore built up mathematics on the basis of a so-called Ramified Theory of Types (RTT) that respects the stepwise, countable procedure in building up authentically predicable mathematical entities, i.e. entities that can be said, defined without saying demonstrable nonsense. Because of countability (which is a natural feature of anything sayable, predicable, which, in turn, can always be broken down into bits) it is straightforward to logically define the natural numbers simply as equivalence classes of equinumerosity of finite sets. From the natural numbers, one can then proceed to logically define the integers and the rationals, both of which are also countable.
How then to define real numbers, which exceed any finite, countable definition in terms of natural or even rational numbers? They can be defined either as an infinite, countable, convergent series of rational numbers (Cauchy series) or as Dedekind sections, which can be regarded as infinite, countable, upper-bounded sets of rational numbers (namely, such a set which does not have a maximum element). If the rationals are cut, or bisected, in this way, the cut is precisely at an irrational real number. A single real, or a finite set of reals, or even a countably infinite set of reals (such as a real sequence) can still be conceived, i.e. built up logically, on the basis of countable sets of rational numbers, and so remain within the bounds of countability and thus of sayability, predicability.
But, as Feferman outlines, when one comes to the least upper bound axiom for the reals, which is indispensable for classical mathematical analysis, reference has to be made to the total set of all the upper bounds for a bounded set of reals, which is uncountable and exceeds all the levels through which definitions of the reals have been built up. From the uncountable set of all real upper bounds, a least one has to be picked, and this proves to be impredicable, unsayable, illogical. Uncountability thus proves itself to be the limit for the sayable in a mathematical domain based only on basic logic and the natural numbers. The reals themselves can just barely be reached through countable sets of rationals, but that is also the boundary of logical rationality, i.e. of what can be said mathematically. Feferman concludes, "[a]nd since that [the least upper bound axiom] seems to be a basic essential principle of analysis, RTT proves to be unworkable mathematically".
He then goes on to discuss Weyl's proposed axiomatics which makes do with being able to say anything about natural numbers and (countable) sets of natural numbers, but not about the (uncountable) totality of sets of sets of natural numbers. (Note that this is not the problem of Gödel incompleteness, for this latter concerns not what is sayable, predicable as a set but whether what can be said truly within an axiomatically generated, consistent system rich enough to generate the natural numbers is also provable within that system. Nevertheless, countability plays an essential role in Gödel's proofs because statements/predicates within a theory have to be counted.) Weyl arrives at the same impasse that Russell arrives at with the least upper bound axiom but, instead of attempting to form a least upper bound of a bounded, uncountably infinite set of reals, he forms only the upper bound for a bounded, countably infinite sequence of reals. This is admissible because it remains within the pale of countability, and therefore of sayability and logicality.
With this weaker, but logically admissible, least upper bound axiom, mathematical analysis happily can still be done, but only up to a point because, as Feferman puts it, a "continuous function ... (say on an interval) is determined by its values at rational numbers". Hence, one can fill in the gaps in continuity of a real function from the rationals and therefore one does not have to assume the existence of the uncountable set of reals making up a real interval. The interval may just as well remain rational, countable, sayable, and the uncountable real continuum as a totality is not derivable within the Weylian axiomatics. Since continuous real functions can be brought within the pale of logicality, differentiation and integration also pose no problems and "all reasonable 19th century analysis can be reconstructed, or redeveloped, on the basis of Weyl's system".
But it is not possible to logically build up discontinuous real functions in this way, so the sophisticated functional analysis required for twentieth century mathematical physics falls by the wayside. Presumably, the perplexing quantum indeterminacy struck upon early in the century hangs together with the strange illogicality of the uncountable continuum and also with the supposed continuity of movement and time. Is the phenomenon of time already beyond the reach of what can be said logically? Is time itself discontinuous? Is time outside the domain of the mathematical and exact altogether? These are not merely rhetorical questions, but go to the heart of this study and of still unresolved antinomies in both mathematical logic and quantum physics. (See also Excursus 2 and Excursus 3)
To sum up and also to anticipate later chapters (cf. also my 'Digital Being, the Real Continuum, the Rational and the Irrational' 2010): An irrational, real number can be regarded as an infinite, countable sequence of rational numbers approaching a non-rational limit. Thus, an irrational, real number can only be approached by an infinite counting process that gets as close as you like to it without ever reaching this limit. This implies that an irrational real number can only be conceived as a counting movement toward that can never be made present as a logical, computable ratio of natural counting numbers. An irrational real number is forever absent from the infinite series of rationals approaching it in a counting movement. The irrationality of an irrational real number could therefore be said to consist in its being never present, but forever arriving, forever heralded by the endless row of rational numbers announcing its arrival. The irrationals fulfil the illogical condition of the Aristotelean ontology of movement in general as a twofold of presence and absence (cf. 2.5 The essentially 'illogical' nature of time and 2.9 Time and movement in Aristotle's thinking). They are illogical because they can never be brought to a standing presence by the rationals. Otherwise they can only be symbolized by algebraic symbols (cf. 2.7 Cartesian rules for an algebra of magnitudes in general as foundation for the modern mathematical sciences) symbolizing numbers that are forever absent and beyond the grasp of a calling to presence by the logos in a definite rational number amenable to arithmetic calculation.
Moreover, this movement of counting infinitely through a rational sequence toward an irrational limit takes place within the continuum of real numbers, so that each step from one rational number to the next must pass through an infinity of irrational real numbers. The movement of rational counting itself requires the medium of the real continuum, which is largely irrational. The continuum of real numbers can be imagined geometrically as an endless continuous line. It is geometrical figure that contours real, physical bodies, so the name 'real' for the real numbers is well-chosen. On the other hand, however, only rational numbers can actually be calculated to obtain a definite arithmetic number that is a kind of logos as the result of a calculating logismo/j.
What can we infer from this antinomy between the real, irrational continuum and countable rational discreteness for the being of digital beings (cf. Chapter 3)? A digital being is, in the first place, a finite sequence of binary code, consisting perhaps of billions and billions of bits, that is interpreted and calculated by the appropriate hardware in sequences of nested algorithms to bring about a foreseen effect. As binary code, i.e. a string of zeroes and ones, a digital being is nothing other than a finite rational number, whereas even a single irrational real number is a countably infinite string of bits. (If, following Cantor, Aleph is the symbol for the countable infinity of the natural numbers, the smallest infinite number, then the infinity of the real continuum of numbers is 2 to the exponent of Aleph and the real continuum, in binary representation, is the set of all countably infinite strings of bits.) Therefore even a single irrational real number never can be inscribed logically-digitally. And yet, this binary code, interpreted as commands to be processed by a digital processor, brings forth change and movement in the real world of real, physical beings. A digital being can only represent the real world in terms of binary bits, which are logical, rational, computable numbers that always must miss the irrational continuum of the real.
For example, a computer-controlled robot on a production line can bring the robot's arm into a precisely precalculated position, which is always a rational number or an n-tuple thereof. The robot's arm, however, will always be in a real, physical position, no matter how accurate the rational position calculated by the computer is. There is therefore always an indeterminacy in the computer-calculated position, a certain quivering between a rational position and an infinity of irrational, but real positions. An irrational, real position can never be calculated by a computer, but only approximated, only approached, forever just beyond a final, rational grasp. This signals the ontological limit to the calculability of physical reality for mathematical science. It is not an experimental result, but is obtained from simple, self-evident-but-overlooked, phenomenological, ontological considerations. We must conclude: physical reality is irrational.
What does this imply for the understanding of being as standing presence? The standing presence of being is a temporal determination that goes hand in hand with the understanding of time as composed of a continuum of now-instants. According to the ontology of standing presence, a physical body assumes a definite position at a definite instant of time. In mathematical physics since the beginning of the modern age, the position and motion of physical bodies become calculable, but only by developing a mathematics of the continuum of real numbers that allows also the calculation of velocity and acceleration as infinitesimal differentiations of position with respect to the real, continuous variable, t. An irrational, real instant of time or an irrational, real position, however, can never by precisely calculated, but only approached by rational approximation. Insofar, a phenomenological interpretation of the calculability of the real position of physical bodies by means of the infinitesimal calculus shows that there is no definite position of a physical body at time, t, but only ever an indeterminate quivering of it between a here-and-now and an incalculable infinity of irrational there-and-thens.
Since, as we have seen in 2.7 Cartesian rules for an algebra of magnitudes in general as foundation for the modern mathematical sciences, the mathematical access to being is generalized to all properties insofar as they are represented quantitatively by magnitudes, changes of all kinds in physical beings can be conceived as movements of a variable with respect to the one-dimensional, real, continuous variable, t, that is always essentially both rational and irrational, standing and quivering, present and absent. The state of a real physical being, however, can only be calculated from real, rational data as a countable rational number. Hence the state of any real physical being is always an indeterminate quivering around a rationally calculable state. Physical reality, even on a banal macroscopic level, therefore always exceeds what can be logically, mathematically, rationally, definitely calculated.(9)It starts in the first book A with a critical review of his predecessors' thinking on the being of movement, ki/nhsij, including that of Parmenides with his mono-archic determination e(/n to\ o)/n, "being is one", which leads to a denial of the possibility of movement altogether.
On pronouncing that "it must not remain hidden what movement is" (dei= mh\ lanqa/nein ti/ e)sti ki/nhsij. Phys. G 1;200b13), Aristotle proceeds to introduce the ontological concepts that will allow him to overcome the shortcomings of his predecessors, namely, above all, the famous triad du/namij, e)ne/rgeia and e)ntele/xeia, a triad as hackneyed as any other from ancient Greece in our snotty unphilosophical times. Although we are entirely familiar with the phenomenon of movement, Aristotle claims that it remains hidden to us. This is the classic situation of philosophical thinking: it starts with what is most familiar, and thus in some sense known, in order then to show that we have always already skipped over the simplest of questions and appeased the understanding with only apparently adequate notions that take the phenomenon in question for granted.
In the following I will provide a condensed re-run of Aristotle's stepwise unfolding of an ontological concept of movement.
Movement concerns all beings in the world, not just beings in some kind of 'nature'. In the Greek understanding of being, that which is present is, and what is present most of all is the ei)=doj, look or sight that a being presents of itself. The ei)=doj is e(/n, one, i.e. a well-defined, single look or Gestalt that can also be addressed by the lo/goj through the manifold of simple categories that define (o(ri/zein), predicate the being in how it is present in its predicament. Movement is the phenomenon of change (metabolh/), and that with respect to four categories: a being can change with respect to what it is (to/de ti, ou)si/a), how it is (poio/n), how much it is (poso/n), and where it is (pou, kata\ to/pon) associated with the phenomena of becoming/decay, mutation, waxing/waning and locomotion, respectively.
Significantly, Aristotle does not consider anywhere, as a kind of movement sui generis, the change that takes place through the exchange (metabolh/, a)llagh/) of one thing for another, as in exchange in the market-place, which would have brought in the category of pro/j ti, relation, and another kind of movement, namely, the social movement of interchange.(10) The ambiguity residing in that crucial Aristotelean term, metabolh/, which can mean both 'change' and 'exchange', has had fateful consequences for Western history. Replacing one light bulb by a new one is a banal example of movement as exchange which can still be thought as a composite movement composed of the movements of the old and the new light bulb. But the social exchange among human beings in which goods exchange or in which mutual recognition takes place can by no means be thought through merely by composing individual movements, because the starting-points of the movement are multiple and also interlinked in a mirroring process (as captured, for instance, in the process of recognition in Hegel's Phänomenologie des Geists). The metabolh/ of greeting each other on the street, for instance, is an interchange whose ontological structure is already more intricate than the productivist movement of a du/namij being realized one-sidedly through its e)ne/rgeia.
The peculiarity of the phenomenon of movement is that it cannot be pinned down to the present. Anything in movement has a twofold (dixw=j Phys. G 1;201a3 (10a)) presence: first of all it shows itself in the look of its ei)=doj, but secondly, it also has a lack (ste/rhsij) that points to something absent which it could also be, i.e. which could also be brought into presence. For instance, a full moon has the lack that it could also be a new moon, or vice versa. In what it is, it is also in a certain way, i.e. potentially or 'absently', what it is not, a mh\ o)/n. Or a piece of timber presents itself in its ei)=doj as timber and also as lacking what it could also be, namely, a table, for instance. What/how/how much/where something could be through the appropriate movement is its du/namij, i.e. its potential, potency or power to be something else, which is more than a mere formal or so-called 'logical' possibility. The thing itself has an inherent tendency to become other than it is; it is not yet finished. Aristotle conceives the lack in the twofold presence of a being in movement through the pair of concepts, du/namij and e)ntele/xeia. A being with a potential, a duna/mei o)/n, has the power to become something else, but as it is in its presence, it is still a)telh/j, unfinished. It could only have itself in its finished presence in achieving e)ntele/xeia, i.e. through its having-itself-in-its-end.
Thus does Aristotle come to his first definition of the being of movement. It is the presence of the potential being as such, stretching itself toward its finished presence, and thus a peculiar twofold presence of both presence and absence in which the potential being is on its way to becoming other than it is, in a finished state in which the movement will have ceased and come into its end. In achieving its presence as a potential being, the du/namij is already fully present, i.e. in its e)ntele/xeia, insofar as it is duna/mei o)/n, but it has not yet attained finished presence as something else in its realized potential. In movement, the potential being is still exercising its power of change. "The finished presence of the potential being insofar as it is such is movement." (h( tou= duna/mei o)/n e)ntele/xeia, $(= toiou=ton, ki/nhsij e)stin. Phys. G 1;201a10f). In movement, the being's power to be what it can be is at work, i.e. it is e)ne/rgeia. Therefore, Aristotle can say that movement is the e)ne/rgeia of a du/namij in its e)ntele/xeia. Movement itself is a phenomenon that cannot be defined by a single category; it has, at least, a twofold presence and therefore must be addressed by a double concept, i.e. by a pair of ontological concepts, du/namij and e)ntele/xeia as lack (ste/rhsij), whose unified twofold presence is a third phenomenon, namely, the at-work-ness of the potential under way or in transition to finished presence.
Now, if the being does not have the source of its movement within itself, which would make it an ensouled (e)/myuxon), living being, it suffers itself to be moved by something else. A being with the potential to be moved has a du/namij paqhtikh/, whereas a being that is potentially a mover has a du/namij poihtikh/. A piece of timber has the passive potential, or power, to suffer itself to be transmuted into a table, and the know-how of carpentry has the active power to move or transmute the timber into a table. Despite this twofold, passive-and-active, aspect of movement, the movement at work, its e)ne/rgeia, is still just one movement, and not two.
Moreover, movement is a continuous (sunexe/j, Phys. G 1;200b19) phenomenon which means that it is connected (e)xo/menon) and also that it holds itself together within itself (sune/xein). The continuum is that which can be divided limitlessly (a)/peiron diaireto/n, 200b21), i.e. for which there is no discrete limit where the division has to stop. The indefinite, double or twofold determination of movement as both du/namij and e)ntele/xeia at once would seem to have to do with its continuous, limitlessly divisible nature. The presence of the du/namij cannot be separated from the likewise present absence or lack of the e)ntele/xeia as the perfect, finished present toward which the du/namij in its e)ne/rgeia is stretched. Instead of a well-defined, unambiguous presence of one (e(/n) that could be captured by a single category, we have an ambiguous, inseparable presence of both a power and the not-yet-finished end-presence of its being-at-work. Even more than that, with the advent of e)ne/rgeia, there is a triad of elements whose unity constitutes the full ontological structure of movement of all four Aristotelean kinds.
With this triad, Aristotle has all the elements in his hand to think through also the ontology of the phenomenon of time, albeit he goes a completely different path in his chapters on time in Phys. D Chaps. 10 14.(11) There he notes that "it is obvious that time is not without movement and metabolism/change" (fanero\n o(/ti ou)k e)/stin a)/neu kinh/sewj kai\ metablolh=j xro/noj. D 11 219a1). The gateway to the phenomenon of time is thus through movement: Something present has the potential, the power to be something else, which it can become through the appropriate movement which itself comes to presence when the potential achieves its finished presence as a potential, namely, in being at work as movement itself toward its end. What was (past) a potential power at rest is now (presence) a power at work toward (future) a realization of the potential in a perfect presence. The three ontological elements of movement thus map onto the three dimensions or 'ecstasies' of time itself which, two-and-a-half millennia later, and foreshadowed by Husserl's phenomenology, will be explicated as the temporality of Dasein in Sein und Zeit, whereas the Aristotelean conception of quantifiable time, now designated as the "vulgar conception of time" (vulgäres Zeitverständnis, SZ:428 §82a), will be shown to be derivative of a more primordial conception of the phenomenon of time (cf. Sein und Zeit Division 2, Chap. 6). When a power is at work, all three elements of movement are present, albeit that two of them, namely, the power as potential and the power realized in a finished presence, are present as absence, i.e. as no longer and not yet. This ontology of time is therefore thought on the basis of the paradigm of production, a particular kind of movement. A piece of timber, for instance, has the potential to be a table. This potential becomes present as such when the timber is worked upon by the carpenter on its way to attaining a perfected presence in a finished table. The piece of timber as in movement is thus stretched in time between what it was potentially and what it will be finally, and only in this transition as a simultaneity of presence and absence is it in movement. Being itself is thought in Greek ontology as a pro-duction, a Her-Stellung, namely, as a coming from an origin, a whence (a)rxh/, ge/noj, ti\ h)=n) into the perfected presence of its sight (i)de/a, ei)=doj) most succinctly summed up in Aristotle's famous formula for the beingness (ou)si/a) of a being: to\ ti/ h)=n ei)=nai (the what-it-always-was-ness).
Aristotle eschews the possibility residing in the triad of concepts he has fashioned to grasp the ontology of movement, and famously determines time instead quantitatively as the number (a)riqmo/j, 219b2) or measure (me/tron, 221a1) of movement: "This namely is time, the number of movement with respect to earlier and later. Time is therefore not movement but movement insofar as it has a number." (tou=to ga/r e)stin o( xro/noj, a)riqmoj kinh/sewj kata\ to pro/teron kai\ u(/steron. Ou)k a)/ra ki/nhsij o( xro/noj, a)ll" $(= a)riqmo\n e)/xei h( ki/nhsij. 219b1ff).(12) And "time is the measure of movement" (o(/ xro/noj me/tron kinh/sewj, 221a1). The now (to\ nu=n) divides the earlier from the later like a point (stigmh/, 219b18) divides a line (grammh/) into two parts (220a21). The succession of nows counted off as 'now', and 'now', and 'now' is the progress of time coming to presence and simultaneously disappearing from presence. Aristotle raises the aporia that only the now is, so that time consists predominantly of that which is not, namely, the no-longer and the not-yet. As a quantity lifted off the phenomenon of movement, "we measure" (metrou=men, 220b15) time; it is a number, a measure, a magnitude (me/geqoj, 220b27), and, like movement itself, it is continuous. Insofar as it is simply a number, time is unmoving, i.e. outside time, so it is crucial that the counting of nows in the progress of a movement refers to the transitional character of the nows that they are underway from...to, i.e. always both present and absent.
As a continuous magnitude, there is no smallest time, because any continuous magnitude can be divided further, but as a number (a)riqmo/j, 219b2), there is a smallest one, which Aristotle takes to be two (220a28) because that is the first number one comes to in the act of counting, starting with the one (mona/j). Time is counted by saying 'now' at least twice in succession, thus marking an earlier and later. This raises the aporia in the nature of numbers as either countable and discrete or as endlessly divisible and continuous, an aporia which, as we have seen (cf. 2.6 Bridging the gulf between the discrete and the continuous), was solved in mathematics as late as the nineteenth century with the concept of mathematical limit which allowed the infinitesimally small to be coherently calculated without assuming the infinitesimals as infinitely small magnitudes smaller than any real number. Infinitesimals can be dealt with as the limits of countable, infinite sequences of rational numbers, thus bringing countability and continuity together.
But why should time be quantitative at all?(13) Time is something lifted off (a)fai/resij) movement itself in its transitional character and, as such, is an abstraction. Saying 'now', or a succession of 'nows', is an abstraction from any particular quality of the movement concerned, capturing only the phenomenal moment of transition from what was to what is to what will be. The only difference between successive 'nows' is earlier and later, which makes of the counting of now-moments passing through, the abstracting counting of time itself. Hegel determined quantity as the abstraction from all quality,(14) and the counting process of successive 'nows' is indeed an abstraction from all quality of movement apart from its transitional, never-to-be-pinned-down character 'between', underway, or as both presence and absence. A kind of ordinal counting as a steady drumbeat of successive nows can therefore be phenomenally justified, and the successive nows can be added up to attain a succession of (ordinal) counting numbers going on indefinitely, which is the counting of time that can be made mechanical and arbitrarily refined in a clock (beyond the rough counting of days, months, years, which are all regular movements of celestial bodies). The difference between any two counted now-moments can be measured, and since they are read off movement, which is continuous, the measured magnitude of time itself is also continuous. Why the passage of time should be uniform at all is a question taken up at a later stage of our investigation (cf. 5.5 Time in a capitalist economy).
We conclude this section by noting that the quantitative ontology of time has its origin already with Aristotle. The ontology of time offered in Heidegger's Sein und Zeit implicitly breaks with this quantitative ontology but remains within an ontology of time still determined by the paradigmatic movement of production. Now it is not a piece of timber that is produced into a table through the realization of a potential, but Dasein itself that casts its self into the future in a kind of self-production: "Preparing its potential for being, Dasein comes to itself." (Das Dasein kommt, sein Seinkönnen gewärtigend, auf sich zu. GA24:375) Is there a possibility of an alternative ontology of time residing in the paradigm of social interchange, according to which each human being finds its self as it comes about as a who-stand in the intricate, haphazard interplay with others? We shall return to this question in 5.5 Time in a capitalist economy and 5.7 Recovery of the three-dimensional, complexly interwoven social time of who-interplay (cf. also 7 Appendix: A demathematizing phenomenological view of quantum mechanical indeterminacy).
3.1. The appropriation of the truth of beings, digital interpretation of world-movement and its outsourcing through executable, cybernetic machine-codeIn order to clarify the essence of digital beings a step further, they have to be viewed from digital technology which up until now has been left out of consideration. The binary code of a digital being is writing, script, i.e. it is the inscription of a lo/goj into a medium where this lo/goj can also contain numbers, i.e. a)riqmoi/, and thus can have mathematical character in the narrower sense. This logos is that of a techno-logical know-how, which is a special case of the lo/goj as conceived since the Greeks: "In knowing and speaking, the truth of beings, their disclosedness, is appropriated." (Zugeeignet wird im Erkennen und Sprechen die Wahrheit des Seienden, seine Unverborgenheit. GA19:276 emphasis in the original; cf. 274, 391) Technology is essentially a knowledge which provides insight into beings with a view to their manipulation. Productive technology or te/xnh, i.e. knowing poi/hsij, is a knowledge of how an envisaged product (a change or movement of any envisaged kind, which may be regarded simply as an effect or a result) can be brought forth.
Here a distinction must be drawn between digital beings which are in some way or other read by humans, and digital beings which are employed to automatically control some process or other. Productive know-how can be written down. Written-down knowledge was first of all read by humans who appropriated and applied the knowledge for their own purposes, e.g. in artisanal production. With digital technology, however, knowledge is not only written down in a written script legible to humans, but in a written script which can be read by a machine as a sequence of machine commands bringing forth envisaged results in a certain, determinate context. The written script itself can be input into a machine to control it. Written script thus becomes a digital program, or literally, a pre-writing or pre-script, which controls a machine of one kind or another and is 'productive' in the sense of bringing forth an effect which is always some sort of change (metabolh/).
Written script as binary code, i.e. as a finite sequence of discrete binary numbers (for any written script at all can be represented in binary code), is 'read' sequentially (e)fech=j) by the machine, i.e. each digital character or each string of digital characters taken together (i.e. syllables in the Greek sense of sullabei=n, aor. inf. act. 'taken together') serves to control the machine's movements by means of commands that the machine (its 'chip') has been preprogrammed to 'understand' and 'interpret'.. The hardware and software mesh together like a su/mbolon, a 'symbol' in the Greek sense, as in two pieces of code that only make sense when fitted together. They fit together to form an 'automatic computing machine', or Turing machine, for a certain calculatory task. The hardware itself is the computer, a universal Turing machine, into which at first a "description number" (Turing 1936) is input that today is called software, which is then able to operate on and compute data (a number) that are fed in. Turing's ingenious insight (Hodges 2007) was that the instructions for a definite computing task (the software) are just a single digital number that works on another digital number (the data).
An elementary example of such control is when a binary-coded, digital text is 'read' by a digital device such as a word processor, mobile telephone or PC, etc. in order to represent or reproduce the text on a screen through an ordered sequence of pixels. The pre-script in this case is not merely the text itself in a digital form (the data-number), but the word processing program and the control characters embedded in the text which together compute a number that, translated back to the physical world, enables the text to be reproduced on a screen by means of control instructions. The program pre-script used to control a machine is always a 'logically' fixed knowledge insofar as the lo/goj appropriates beings in their truth with a view to some practical end (in this example, an electromagnetic state of matter interpreted as an ordered sequence of pixels and legible to the eye as text). The essential and immensely powerful characteristic of digital technology is that human knowledge can be outsourced by the pre-script of a program into a machine where it then automatically brings about effects at any place whatsoever. Already the idea of a Universal Turing Machine (Turing 1936) provides for outsourcing the algorithm for a computing procedure into the tape-memory of a computing machine. The knowledge is a theoretical pre-understanding of a certain matter or state of affairs which, as a digital program, enables certain predefined procedures to be automated. In principle, all human tools are the outsourcing of a knowledge or know-how. A tool as simple and banal as a potato peeler, for instance, is the outsourced knowledge of how to peel a potato effectively embodied in a practical thing designed for the specific purpose. A better potato peeler is the embodiment of a better, more efficient potato-peeling know-how. The potato peeler is not simply a tool for an operative execution of human know-how but rather as such, in its very fashioning and making, already embodies, materializes partially a restricted kind of practical culinary know-how.
Outsourced know-how, however, comes into its own when it is automated, e.g. when the know-how of how to produce a table is outsourced via a digital program into a automatic, numerically controlled lathe. Contemporary debates over artificial intelligence and expert systems turn upon the extent to which, and which kinds of, practical human understanding can be digitally, logically encoded and thus outsourced. Digital technology opens up hitherto inconceivable possibilities for outsourcing (segments of) practical world-understanding in such a way that movements of all kinds (e.g. the motion of a door, the movement producing the result of a calculation or a signal that a predefined state has been achieved) can be automatically brought about. Computer programs inscribe a partial practical understanding of world, say, into the hard disk of a network server, and make the interpretation of this understanding processable and calculable by a microprocessor, thus producing functional effects (such as the 'production' of a search result by a digital search 'engine'). The digital capture and taking-apart of the totality of beings thus goes qualitatively beyond mechanical technology, which is still oriented toward physical (loco)motion, into the dimension of the automated control of systems of movement of all kinds.
Since the onset of modernity, in which beings were cast as res extensa for the first time, the theoretical access to beings in their being has been enabled through measurability. The theoretical appropriation of beings is then a disclosing of beings by quantitative measurement, both practical (e.g. empirical data collection) and theoretical (e.g. postulating algebraic variables for certain physical dimensions). The way a given matter behaves is then graspable and knowable theoretically through quantitative relations (equations), and this knowledge can then be programmed into computing machines of all kinds which further calculate what is measured on beings in accordance with a theory. For instance, digital photography is enabled firstly by casting colour itself ontologically as a purely quantitative multi-dimension (i.e. a triple of positive integers plus other numerical parameters to form a colour vector). The further calculation then serves either a deeper knowledge of the matter (e.g. digital chromatic rendering) and/or the (possibly automatic) control of a process already set in motion in which the measured or further calculated matter or state of affairs is fed back into the process as a control variable (e.g. to produce a colour print).
Whereas the written, legible logos preserves knowledge - i.e. in this context, primarily technical knowledge -, with executable digital character sequences, knowledge is converted into an functional form which allows it to bring forth effects and to control processes automatically. The logos in the form of digital code is thus fed back into beings in order to manipulate them in a kind of self-poiesis. Digital beings legible for humans comprise not only text-like files, but all code sequences such as images, sounds, moving images which, when they are re-presented by the appropriate hardware, have effects on the senses and can be taken in by sensuous perception and understood as a meaningful whole. Machine code, on the other hand, controls processes in pre-understood and pre-calculated ways. To do this, the process itself must have been already understood and taken apart in a mathematically calculable way which itself builds on various natural and technological sciences such as physics and electrical engineering. The programmer transforms this understanding into machine-readable, sequential digital code (for every programming language must be ultimately translated into digital machine code in the narrow sense which consists exclusively of binary bits to be processed stepwise by the digital processor or 'chip' as executable commands) which then brings forth calculable control effects in a definite, foreseen context. Thus, cybernetic-technical knowledge becomes automated and tendentially makes itself independent vis-à-vis humans for, although each program can still be read and understood individually, the possible implementations of automatic control are well-nigh unlimited and thus lead to intricate, intermeshed, non-transparent control complexes that may even feed back automatically into each other in feedback loops - including in unforeseen ways.
Control processes that are no longer co-ordinated with the particular context foreseen, automatically bring forth nonsensical or even detrimental effects. An understanding programmed in digital code can thus possibly turn into a severe misunderstanding with serious consequences. If each digital program can be conceived of as the implementation of a partial understanding of the world, then the possibility of arbitrary replication of binary code means that the digitized cybernetic knowledge transformed into software is available and can be called up anywhere, including in wholly unintended contexts.
The interpretation of the world through executable machine code takes place factually and mechanically (i.e. without understanding) in the interpretative processing of what is given by the world (data) and this interpretation is already latent in the pre-script of the program itself that just 'mechanically' processes the data. Viewed thus, a computer program pre-script is not only a productive technical know-how producing functional effects, but, more deeply and prior to that, a pre-interpretation of (a restricted segment) of the world written down by us which is ready to receive data at any time in order to calculatively interpret the world, on the basis of the data fed in, in a certain predefined direction and to control some system or other on the basis of this interpretation. Human being, for which the world opens up in understanding, can today outsource to a computer its interpretation of the ontically understood world in segments into binarily programmed, functionally effective pre-interpretations of the world, where the understanding of world itself already has to be compatible with a digital decomposition (e.g. time has to be conceived quantitatively as a continuum of timeless now-points). Such a world-understanding as a whole is oriented toward setting up and controlling the various kinds of movements of beings in their totality.5.6 The global power play measured by money-value and its movement). Cyberspace itself has its own peculiar spatiality; it is not merely 'virtual' but has its own orientation and dimensionality (cf. 4.2 Dasein's spatial being-in-the-world: approximation and orientation), and in this cybernetic space, the digital beings can be arranged, moved and reproduced arbitrarily at will. Cybernetic (from kuberna=n, 'to govern') space is called thus because it enables total control through digital know-how. In a certain way, digital beings, insofar as they are viewed merely as ordered sequences of binary code, are nothing other than written 'texts' stored in the electromagnetic medium which can be called up arbitrarily at will, including by that automated will preprogrammed into computer programs. Because the electromagnetic medium is homogenous, and digital beings are nothing other than an impression or imprint in this medium, any topologically continuous network of such electromagnetic medium, such as the internet, potentially facilitates total control through total traceability, for each and every digital being leaves its calculable 'footprint' in the electromagnetic medium.
Such arbitrariness of place stems from the circumstance that, viewed ontologically, logos and number are both attained by being 'lifted' or 'drawn off' from physical beings. The placelessness of the logos thus assumes a new meaning: not only is arithmological knowledge attained by an abstraction that 'lifts' measurements from beings, but this knowledge now assumes the garb of binary code in a technically ubiquitous form. Binary code as a pure form impressed in an electromagnetic, ubiquitously present medium is entirely compatible with all kinds of formalistic thinking that abstracts from the particular situation. These include especially the formalistic bureaucratic and legal thinking that the state employs to impose its rule 'neutrally' over its subject populace. Knowledge is then not only universal in the sense of a universal comprehensibility and applicability but also materially universal in the form of universally accessible binary code that can be embodied arbitrarily as executable code in the homogenous electromagnetic medium of the appropriate digital devices for the control of movements of all kinds.
A medium is something through which other beings can move. The technically produced electromagnetic network technically enables the arbitrary movement of digital beings through the medium of the network. Every place in the network can be specified by co-ordinates. Since the electromagnetic medium is homogenous (every place is thus equivalent to any other place), each place in the network can be specified by purely numerical co-ordinates. These co-ordinate places are therefore not places in the Aristotelean sense to which a digital being essentially belongs and to which it owes its presence, nor even geometric positions, but rather, paradoxically, merely positionless, placeless, numeric n-tuples enabling calculation. Branches of mathermatics called combinatorics and graph theory even arise to enable the calculative control of networks. Networking means only that all co-ordinates are connected with each other (Aristotle's e)xo/menon cf. 2.2 Heidegger's review of Aristotle's thinking on modes of connectedness from discreteness to continuity) directly or indirectly in such a way that digital beings can move without obstruction through the homogenous medium from one arbitrary co-ordinate place in the network to any other arbitrary network address. The restrictions to this movement are of a merely technical nature which, in turn, can be overcome technically (or, conversely, even imposed technically for security reasons).
In a computer program, technical knowledge itself translating a partial understanding and interpretation of some aspect of the world is made into something lying present at hand and to hand, i.e. it is then available as something that can be called up ubiquitously, and it is a being which is good for something (mode of being as being-(good)-for...). Whereas the 'logical' or logos-like call-up of beings takes place through language calling beings to presence by addressing them, with the digitally decomposed beings this presencing is different, for here, binary code is called up through the electromagnetic medium, i.e. is de-distanced or approximated(15), in order to be processed further, e.g. read by a human, or to unfold automatically its programmed effects. One could say that for the construction of many kinds of digital beings, physical beings serve as models, for binary code represents technical knowledge of the world and of practical ways of comportment in the world in some respect or other. Physical beings are brought to presence in knowledge through the numbers and language 'lifted' from them in a way different from their presencing of themselves unmediatedly for aisthaetic perception in a situation. The knowing re-presentation of physical beings in digital code depends on both the geometric abstraction from physical beings and the discrete arithmetic abstraction that is able to algorithmically approximate continuity to any desired degree of accuracy.
When, as we have seen, the knowing, disclosing appropriation of beings through numbers and language, i.e. arithmological knowledge, is inscribed in a computer program, physical beings too then become cybernetically manipulable and that not only merely by a technically skilled human hand, but by automatic machines controlled by binary machine code, where such machines can assume also the most subtle and inconspicuous forms of appearance such as biochemical nano-machines. As cybernetic programming, arithmological knowledge intervenes 'in writing' in the world of things. Arithmological knowledge not only enables a technically productive manipulation of beings, but arithmological script as cybernetic program code transforms this arithmological knowledge automatically into effects. Such automatic cybernetic systems represent a hybrid between fu/sij in the sense of beings which bear the governing source of their own movement within themselves, on the one hand, and a technique under the control of a human hand in which the governing source of movement lies in another being (the producer, the programmer), on the other, for these automatic systems have something fu/sij-like in their nature, where fu/sij is understood as self-poiesis.
Tellingly, Aristotle conceived fu/sij precisely as self-poiesis, so the cybernetic, auto-poietic systems confronting us today are the consummation of his ontological dream which is now revealing its ambivalence as a nightmarish dream. An auto-poietic being in the Aristotelean sense is one that has the principle (a)rxh/, starting-point, source) of its movement and change within itself. We may as well call these auto-poietic systems and things robots and note that we have long since been living in the robotic age, the epoch unwittingly cast by digital ontology. In automatic cybernetic systems, the governing source of movement no longer resides in a living, breathing human operator, but has been outsourced knowingly (i.e. through knowledge) into material beings insofar making it seem that these systems themselves had souls and were in this sense alive, animated (anima = soul). Such outsourcing introduces a split between the knowing designer (electrical engineers, programmers, etc.) of the cybernetic system, and the users, who need know nothing about how the system works, but only its operating instructions, thus deepening the gulf between technically skilled labour and unskilled labour. Unskilled workers have not even forgotten something they once understood in principle or in technical detail, but inhabit the cybernetic world as if in a fog in which beings are discernible only in fuzzy outline.
The phenomenon of digital automation also reflects back, through the totalizing tendency of the digital cast of being, onto the self-conception of human being itself: a science of neurophysiology arises which conceives of human thinking itself as an intricate, auto-poietic computational program, residing in the brain, which reacts to sensory impulse-data given by the outside world. This is a kind of forgetting of an entirely different order: truth is understood then only as effective knowledge, and human thinking is conceived as the effectivity of its functionality, i.e. through the interconnections between cause and effect, stimulus and response, data input from the environment and brain-calculated reaction. The thinking human brain is then considered to be simply extremely good in calculating given inputs, but in principle (i.e. ontologically) as the same as a digital computing machine. In this kind of effective scientific thinking, the ontological difference itself has been forgotten, i.e. consigned to oblivion.
3.5. The onto-theological nexus in abstract thinking, cybernetic control and arithmological access to movement and timeTo come back briefly to the Greek origins, there is no denying that, under the influence of the Pythagoreans, Plato accorded a special place to the abstraction of number on the way to the ideas, which are the ontological abstractions enabling beings as such to come to appearance, i.e. to show their looks, their sights. Learning geometry and arithmetic demands a kind of abstract thinking necessary also for grasping the abstract ideas at the heart of ontological thinking, and so these disciplines may be regarded as preliminary finger exercises in philosophical thinking. Just as the Pythagoreans accorded divine and mystical status to numbers, so too does Plato regard the ideas as divine and located in a special, transcendent place beyond the heavens. Aristotle also sees philosophical thinking as a divine (qei=on), happy activity enabling the philosopher to catch a glimpse of the divine precisely through being able to see the sights beings offer of themselves in the ei)/dh. This is the key to understanding why metaphysics can be understood both as ontology and as theology, a double trait, the view of which has been clouded by Christian theological metaphysics.
Be that as it may, with regard to digital ontology we could ask what has become of the theological aspect of metaphysics and answer that the (Cartesian mathematical) ideas enabling a productive power and cybernetic control over the movements of (both non-human and human) beings in the world are the sober Cartesian ideas setting down the rules for modern mathematical sciences which, however, precipate in material beings themselves insofar as (pieces of) human understanding of the world, borne tacitly by the implicit digital-ontological thinking that has made the dissolution of beings in the world (ontologically) conceivable, can be coded (piecemeal) into executable binary computer code. Human subjectivity in the modern age has insofar assumed god-like cybernetic powers. But this engenders only the illusion that we human beings are in control.
It is not simply a question of complexity that, say, because of the countless aspects, we cannot see through what computing machines of all kinds perform and hence become entangled in an intransparent, automated, cybernetic web, but already, prior to that, there is the primal onto-arithmological casting of access to the world which today enables the outsourcing of productive world-interpretations in a digital form. These autonomized systems now turn upon us, challenge us. And even more, the arithmological way of thinking is an access to disclosing beings as such that also obscures the phenomena. It is important to recover from historical oblivion that the ontological origins of the powerful onto-arithmological casting of the world lie in Greek metaphysics that implicitly understood being as constant, standing, defined, and therefore unambiguous presence that underlies beings' as such themselves being addressed as 'one' (o)/n = e(/n) and as a well-defined look (ei)=doj). As we have seen (cf. 2.9 Time and movement in Aristotle's thinking), the categories appropriate to grasping ontologically the phenomenon of movement (ki/nhsij) are not just one, but at least two, and thus lie on the other side of the famous diagrams (dia/gramma) of Plato and the Pythagoreans in which the elements on the left belonging to ei)=doj face their opposites, such as finite-infinite (pe/raj - a)/peiron), resting-moved (h(remou=n - kinou/menon), reason-opinion (nou=j - do/ca), one-many (e(/n - polla/).(16)
The achievement of metaphysical thinking has been to grasp the phenomenon of movement in terms of both presence and absence (ei)=doj and ste/rhsij) in such a way that what is present (to\ duna/mei o)/n) governs the pro-ductive coming-to-presence of what is absent. This is the Western will to power over movement of all kinds, an all-pervasive megalomania inspired also by 'good causes'. Access to the world through the lo/goj depends on beings' being grasped in a well-defined, discrete way as o)\n lego/menon, and the discrete lo/goj can be broken down into countable, finite, calculable number as binary code that articulates numerically a piece of world-understanding in executable digital pre-script or pro-gram that mirrors our world-understanding in automated processes/movements (mathematized as Turing machines). Such logical pre-script is outside of time; it is timeless. Why? Because time is conceived simply as the real variable, t, consisting of pure now-points which are either present or absent, but not both. The unity of time in its ambiguity as both presence and absence simultaneously eludes pure number which, as the Greeks knew, is outside time. And yet, modern physics discusses the question of the possibility of the reversibility of time purely in terms of dynamical equations ("of Newton, Maxwell, Einstein, Schrödinger, Dirac, and others", Penrose 1989, 1999 p. 454) in which time, t, occurs merely as a variable, so that, of course, they can be read symmetrically either forward or backward.
It is therefore an historically momentous obscuring of the phenomena of time and movement to conceive time as a mathematical variable. If, however, human being itself is, in truth, exposed to three-dimensionally stretched, ecstatic time, then the productive power enabled by metaphysical thinking that culminates in today's digital technology, is a narrow-minded access to the world that makes certain phenomena inconceivable, i.e. invisible to the mind's eye. It thus fails to allow room to move for those movements, including the movement toward death and the movements in interplay with free others, that are beyond the reach of the Western will to epistemic power over movement including, in its latest historical garb, as automated cybernetic systems of all kinds.
The electromagnetic medium is precisely an a)nai/sqhton which accepts all possible impressions and can be written over again at will, re-inscribed by means of electronic signals. The impressions, however, are digital, i.e. sequential binary code, i.e. minimal electromagnetic differences, which we understand as 0 or 1 and which can be represented in various sensuous ways with various contents and differing functions. The bits are invisible in themselves, but they can be transformed into ai)/sqhta by the appropriate hardware and software which are then accessible to the human senses. Itself neither air, water, earth nor fire, the electromagnetic e)kmagei=on enables beings to "appear" (fai/nesqai 51b).
In which sense, however, can we speak of the electromagnetic medium as a space? The electromagnetic medium is a stampable mass which is able to take in digital beings. Digital beings, however, can also move arbitrarily through this homogenous medium and find an arbitrary, or placeless, place in it. Insofar, the electromagnetic medium is, like the xw/ra, a space for accepting digitally, i.e. arithmologically decomposed or dissected beings. The electromagnetic medium as a dimension that can be passed through insofar deserves the name cyberspace which now has to be investigated more closely. The treatment of space in Heidegger's Being and Time will serve us as a guiding thread.
The essential approximation and orientation of Dasein means that it is 'always already' away from its bodily place of sojourn and that Dasein as oriented-approximating is always already reaching out spatially toward faraway places and that this is the condition of possibility for its being able to be there also factually (bodily, or medially through speech, writing, voice, image). We approximate bodily by reaching for, looking at, going to, etc. Hearing speech, however, approximates - through the presencing inherent in speech - also what is to-hand or that with which we have dealings in the world. Heidegger even adduces the example of the electromagnetic medium, radio: "All kinds of increasing of speed which today we go along with more or less under coercion push toward the overcoming of distance [Entferntheit, in contrast to Ent-fernung, approximation, nearing, de-distancing, the elimination of distance; ME]. With the 'radio', for instance, Dasein today performs an approximating of 'world' by way of an extension and destruction of the everyday world surrounding us which is not yet assessable in its sense for Dasein." (Alle Arten der Steigerung der Geschwindigkeit, die wir heute mehr oder minder gezwungen mitmachen, drängen auf Überwindung der Entferntheit. Mit dem 'Rundfunk' zum Beispiel vollzieht das Dasein heute eine in ihrem Daseinssinn noch nicht übersehbare Ent-fernung der 'Welt' auf dem Wege einer Erweiterung und Zerstörung der alltäglichen Umwelt. SZ:105)
This passage provides an important clue for thinking through the multimedia in their spatiality, especially since it also poses the question of the "sense for Dasein" of the electromagnetic media in general, for it does not make any difference in this connection whether one is speaking of radio, television or the internet. The "everyday world surrounding us" is not only extended but also destroyed by the telemedia; there is thus no change or extension of the everyday world without loss, which, however, should not mislead us into making pessimistic pronouncements on civilization or about the destructive nature of technological progress. Hearing a report on the radio is an approximating of the region itself from which the report comes. The media allow other regions of the world over there to presence here through approximation. Later, too, with the example of the "ear-piece of the telephone", the "inconspicuousness of what is at first to-hand" (Hörers am Telefon ... die Unauffälligkeit des zunächst Zuhandenen, SZ:107) comes into play. Dasein is always already far off beyond what is close at a physically close distance. The acquaintance with whom I am talking on the telephone is closer to me than the telephone's ear-piece which I am holding physically and bodily in my hand and to my ear. Accordingly with the internet too: the entire hardware and software which is used as medium is "inconspicuous", far away, absent, but enables, through electromagnetic approximation, the encounter with beings and other Dasein which are then close to and present for Dasein.
The approximation of regions, however, has various modes; there is, for instance, a difference between seeing/hearing a live report from Moscow on the internet or on television, and reading about it in the newspaper, or reading about Moscow in Tolstoy's War and Peace (which last is a literary casting, and even founding, of the city of Moscow itself, and not merely a description of Moscow). These are different ways for presencing the city of Moscow. Live reports on television are accorded a high ranking only insofar as they are nurtured by the sense of being as presence in the now (simultaneity) and the priority of the sense of sight. Since time is conceived proceeding from the standing presence of now, the future as what is yet to come and the past as what is no longer now are experienced as a 'less' in being, e.g. as passé or guesswork. The 'immediate', 'simultaneous' presence of a live television report (or better still, an image and best of all a moving TV image) now is nevertheless highly mediated (through the electromagnetic medium). The medium itself, however, is inconspicuous and in the main even an a)nai/sqhton, unless there is a disturbance, such as a flickering of the image, which draws attention to the medium itself. A live TV news report suggests im-mediacy, i.e. an absence of medium, and also that truth resides in what you can see 'with your own eyes', but in truth, a live TV news report is an impoverished presencing of the happenings on which it is reporting.
When one walks on the street, the medium for walking, the street "slides underneath, so to speak, certain parts of the body, the soles of the feet." (schiebt sie sich gleichsam an bestimmten Leibteilen, den Fußsohlen, entlang, SZ:107), i.e. this medium can at least be experienced bodily whereas movement on the internet is experienceable in a bodily way only through clicking on the pointing device (mouse) or shifting one's eyes slightly. This is an approximation without movement or, in other words, one moves through cyberspace with a minimum of bodily involvement. Dasein's approximating does not depend on the physical, bodily movement, but can also be performed without an involvement of the body. "The spatiality of Dasein is therefore not determined by specifying the point at which a body-thing is presently occurrent." (Die Räumlichkeit des Daseins wird daher nicht bestimmt durch Angabe der Stelle, an der ein Körperding vorhanden ist. SZ:107) This implies that the spatiality of the internet, albeit mediated by a mathematico-calculative reduction of space effectively to Cartesian co-ordinates, is a genuine spatiality conforming to Dasein and is not merely virtual. Dasein orients itself in this space and is able to purposefully approximate digital beings through this space. Even more than that: the internet as a navigable cyberspace is only possible at all because Dasein is spatial a priori.
4.3. Abstraction from bodily experience in cyberspace through reduction of place to numeric co-ordinatesIf digital technology 'advanced' so far as to be able to decompose the body itself into electromagnetic waves (and not merely take measurements on the body by 'lifting' numbers from it) and to reconstitute the body at will (through a conversion of energy back into matter), then, to this extent, there would no longer be any bodily experience of space at all, but there would still be an experience of space in the sense of Dasein. Then, the finger movements of clicking on the pointing device, which serves to orient and approximate in the electromagnetic medium, would also be done away with. The history of the technical overcoming of distances is simultaneously a history of the smoothing out and elimination of the bodily experience of space. Even with the transition from the horse to the automobile, the bodily experience of space through approximation regressed, for there is a difference between riding on a horse and gliding through a region sitting comfortably in a motorized limousine. On the internet, spatial orientation is provided by URLs (= DNS = a number) and signposts (with numerical links). Approximation is done by clicking a pointing device. The pointing device points to what is to be approximated. Insofar, cyberspace is a very simple space, but nevertheless a space to which both the essential existentials of orientation and approximation specified in Being and Time have to be attributed.
In Being and Time, the place where equipment belongs is given through the totality of applicability in use (Bewandtnis), which is the understood interconnection in which the various useful things stand in relation to each other. Equipment must be in its proper place for it to be to-hand and so that it can be put to use. Each piece of equipment thus belongs somewhere in its place. This is quite different from the way in which Aristotle thinks the belongingness to place of physical beings. We also do not cease to be in the mode of taking-care-of (daily life) when we approximate things in a different way in a digital, electronic medium. When, say, we call up a digital being which then flickers on the screen and can seem to us to be very near or very far, this seeming is not merely virtual or 'subjective', but rather: "Only in such 'seeming', however, is the world in each particular situation properly to-hand." (In solchem 'Vorkommen' aber ist die jeweilige Welt erst eigentlich zuhanden. SZ:106, italics in the original) This means that digital beings and the electromagnetic media can also be interpreted from being-in-the-world and not merely from the standpoint of the arithmological casting of being. This also implies inter alia that the electromagnetic medium enables a mode of Dasein's being together with other Dasein. Insofar it is erroneous to speak of a merely virtual being-together in the network, for being-together means fundamentally a sharing of the truth of being by Dasein and other Dasein and not merely a bodily adjacency at one place in space. Communication by no means requires a bodily togetherness of human beings, nor even a simultaneity of presence, whether bodily or otherwise. Communication can take place across centuries and epochs through legible signs in various media.
What can be designated as a technicization of approximation is the point where te/xnh comes into its own with regard to spatiality. Te/xnh poihtikh/ always rests upon a mode of disclosing or decrypting beings and therefore also on an understanding of being which is mostly implicit and thus forgotten as such. It is always a knowledge enabling a know-how, and can and must be implemented in technical devices. In particular, the various media such as paper, the printed word, etc. are enabled by technical knowledge such as printing technology. The digital electromagnetic medium is the consummation of all technical media insofar as it not only appropriates beings in arbitrary far-off places through the logos, in 'lifting' the logos from beings, but also appropriates them through numbers which then also enables further calculation. The beings situated there are given a digital (i.e. basically arithmetic) representation through calculation, whether it be in words, sound, images, video, which can then be sent at will to any place through the electromagnetic medium. Thus, digital, electromagnetic approximation arises which of course presupposes the knowledge of digital technology as well as the mathematical casting of the totality of beings. I.e., situated a priori or 'before' technical knowledge is the (invariably implicit) ontological understanding of the arithmological decomposition and appropriation of beings which has come down to us from Aristotle via Descartes.
There are thus two steps: first of all, the digital, calculative appropriation of beings through which they attain a purely numerical representation in digital code, and secondly, the digital medium through which the digital beings can pass through and 'measure through' as their own di-mension. Because digital approximation takes place through the electromagnetic medium without bodily experience of space, this kind of spatial experience is somewhat ghostly. Dasein spirits bodilessly through the electromagnetic medium without having to leave its place bodily. This signifies in a certain way a collapse of all places into one place which insofar destroys the possibility of farness. But that has always been the case with technology; it destroys an old world by opening up a new one. The special feature of the digital, electromagnetic medium is that it is a mathematical space which can also be represented numerically, thus opening possibilities of calculation and cybernetic control. Since, however, numbers are not only placeless but also without position, the movement of Dasein in cyberspace is reduced to a game of numbers even though the user interface presents itself to Dasein in a sensuous form, say, with 3-D graphic elements, etc. The interface with Dasein must adapt itself to the sensuous, bodily givens of Dasein, which is, however, only an illusion. Behind the interface there is merely a numerical representation of the beings shown along with the network which is physically spread over the entire globe without the geographical scattering being sensuously experienceable as such, and without the user having to understand anything at all about digital code. Nevertheless, Dasein knows that it is approximating beings from all over the world and thus appropriating them. By virtue of the sensuous graphic interface, Dasein can immerse itself in a simulated reality generated by digital code as if cyberspace were a second world for leading a second life. Much has been made of this 'virtual reality' of cyberspace without, however, its ontological underpinnings in the digital dissolution of being having been adequately clarified (M. Heim 1993, 1998, D.R. Koepsell 2000).
The two steps named are supplemented by a third which, however, goes far beyond the first two. This third step, as already explicated, is the further cybernetic calculation of the beings appropriated in digital form in computing machines of all kinds, such as PCs, movement sensors, robots, implanted microprocessor chips. I.e. it is not simply a matter of presenting the appropriated being merely as linguistic or image information (which, of course, also presupposes a certain amount of further processing of the digitally captured beings), but, furthermore, the measurement data obtained are processed further in a digital program (which always represents a certain, fixed pre-understanding of the data) in such a way that control functions are triggered in a cybernetic system. For instance, numerical data on traffic flow on various roads are automatically gathered through electronic sensors by telematics services, and calculated and processed in such a way that the driver of an automobile can be offered a graphic representation of a congestion-free route on the screen of the car's navigation system. This example shows how the spatiality of the digital-cybernetic network intermeshes with and feeds back into the spatiality of bodily being-in-the-world. The will to power over movement and time thus extends also to a will to power over space on a global scale.
A necessary precondition for breaking down networks into a matrix calculus is the study of networks through graph theory, combinatorics and topology. Topology as a branch of mathematics clearly shows its geometric origins, and it deals especially with the connectedness and non-connectedness of geometric objects (therefore covering also problems of graph theory) easily representable sensuously to the imagination, but very hard to calculate. The topographical objects of geometry therefore had to be reduced to a kind of calculus by abstract algebra in which not merely numbers play the key role, but symbols representing the placeless and positionless elements of abstractly defined mathematical objects such as groups. The elements of a group are abstract symbols representing magnitudes in general, and therefore can be calculated. Whether a given geometric object is connected or not is converted into a problem in abstract algebra involving chains of groups. The geometry of a space thus becomes algebraically calculable (more powerful than arithmetically calculable, because more general) which, in turn, is a precondition for it becoming amenable to digitization and specifically digital calculation.
If place in the global network is made mathematically calculable, the electromagnetic network is placeless, and positional only insofar as the co-ordinate numbers or symbols preserve an order (ta/cij). It is not a genuine geometric structure, or rather: all geometric structures can be represented algebraically and thus become representable and manipulable by computing machines. Hence, the global electromagnetic network itself can be represented as a mathematical, i.e. digital, structure which accordingly can be controlled in a mathematical, calculative way. The technically constructed world of cyberspace is thus a mathematically comprehensible space in which beings appropriated by mathematical knowledge circulate. But the reduction of physical beings to geometric figure and further to algebraic magnitudes accomplished by modern mathematics is not a one-way street: the calculative manipulation of digital entities in the global network also has a translation back into a sensuous form. This is the so-called graphic interface that makes the handling of computers and the 'sojourn' in cyberspace itself more natural for Dasein. Dasein can therefore experience cyberspace from the non-technical 'inside' as an independent spatial dimension in which it can orient itself and also approximate digital beings, and which also maintains easily negotiable interfaces with the surrounding sensuous physical space of the world.
4.8. Difference between Aristotelean/Platonic and digital ontology and the latter's specifically totalizing nature - Merely an oppressive over-presence of digital beings?How is the being of beings thought in a digital ontology differently from Aristotelean and also Platonic metaphysics? A digital ontology views beings neither from a supersensuous topos where the ideas reside, nor only from the categories (without which there would be no world-understanding at all), but ultimately from the calculating mathematical dimension inhabited by abstract algebraic symbols whence physical beings appear only insofar as they are representable as algebraic symbols, thus becoming also measurable and digitally decomposable. The lo/goj has become not only logically and mathematically calculable, but also, as we have seen (cf. 3.1 The appropriation of the truth of beings, digital interpretation of world-movement and its outsourcing through executable, cybernetic machine-code), outsourced into self-poietic things. In the everyday world, it may very well happen that digital beings gain an overwhelming precedence over continuous, physical, 'analogue' beings, which means that dealings with the medial dimension of the electromagnetic medium would attain the upper hand in the life-world vis-à-vis other possibilities of existing. For instance, the practice of reading news reports digitally could gain the upper hand over reports which are simply printed on paper and circulated. The reading of newspapers actually printed on paper in the literal sense could thus die out since, in the digital world, the messages and news may continue to circulate only in digital form. Or digital products such as computer games may make youths completely insensible to what is going on outside the digital cyberspace dimension. Their life-world may then be totally absorbed into the digital dimension. Or movies, today increasingly reliant on digital technologies, may almost totally smother the viability of live theatre.
If, however, through digital ontology it can be seen that digital beings still represent something abstracted from the sensuously experienceable world, then the digital beings will appear as the technical constructions which they are in truth, i.e. in the complete uncovering of their being. Despite all the digital technology, humans remain bodily, mortal beings that experience the world sensuously with its dust, dirt, blood, sweat, wine, meat, light, its fragrances and colours, etc. and can also sometimes bash the table forcefully, take a walk through woods, recite a soliloquy on the stage, etc. And even digital beings have to take account of the bodiliness of Dasein, e.g. that messages and images have to be legible to the physical eyes, and a computer or a mobile telephone has to be operated by hand. Cyberspace can also present its digital beings to Dasein in a strikingly natural, sensuous way. Despite all the abstraction that makes calculation possible, digital technology can be translated back to humans in their bodiliness as sensuously experiencing beings. Is this sufficient to appease qualms about the invasion of robots from cyberspace?
The question is not just whether all these sensuous bodily aspects are only admitted as existing when they appear from the digital dimension, but whether the digital way of thinking totalizes to become the natural mode of human thinking, along with all the convenience and cybernetic control that goes along with having digital devices of all kinds 'at one's fingertips'. No doubt, the omnipresence of digital beings can become oppressive and absorb or push aside the natural life-world. Like any other casting of being, digital ontology makes an absolute, totalizing claim which, however, cannot be relativized simply by referring to natural ontic givens (dust, fragrances, walks, live theatre, etc.) and life-nourishing practices in the 'old' physical world in a competition with the seductions of dwelling in cyberspace. A relativization of the digitally decomposed world is only possible through an ontological destruction that goes to the root of the digital way of thinking as it has been cast throughout the centuries and millennia of Western history, as we have outlined. It cannot be a matter of repeating the 'call' of other castings of being and keeping them alive and vigorous alongside this historically 'latest' casting of being. For instance, it cannot be a matter of reviving Christian 'spiritual' 'values' to compete alongside the seductive convenience of a digitized 'materialism', because the very core ontological concepts of 'spirit', 'matter', 'value' are themselves in question and have to be recast in another historical time which are our own times. The conflict among historical castings of being arising in this way opens up again the forward-looking gigantomaxi/a peri\ th=j ou)si/aj (Plato), the question of being concerning how beings as such are to shape up and appear in their truth.
Western history has been and, insofar as it still has a future, remains the struggle (especially against the powerful complacency of established ways of thinking) to cast being alternatively and thus to fore-cast future historical ways of thinking-and-living in the world, starting always from what has already been cast as an historical world based subterraneanly and obliviously on a definite cast of ontological thinking. The digital casting of being totalizes to cast all that is - i.e. 'reality' as a whole, the 'universe' or the world - as digitizable, computable 'information'. An alternative casting is not just another way of understanding and experiencing the world, thus shaping it historically, but, more deeply, has to prove itself in the struggle as more adequate to the phenomena and thus to human being itself as an historical way of dwelling on Earth (and perhaps elsewhere?). This leads to Heidegger's insight that the 'place' where the question of being is posed is the clearing of Dasein, and that Dasein is fundamentally open to understanding being as a casting of beings in their totality. The 'natural ontic givens' and traces of other world-experiences referred to above which do not properly fit into the mould of a digital casting of world therefore serve merely as a reminder that the question concerning who we are has to be posed more fundamentally than how it is implicitly answered by the digital casting of world, which is only able to grasp, i.e. to see, humans from the possibilities and potentials of digital technology, right up to biotechnology with its genetic code and its genetic understanding of 'life' as well as neurophysiology with its conception of human being as a whole in mechanistic terms, where the preferred paradigmatic machine for conceiving human being today is the computer. The digital casting of being makes it seem that certain questions concerning human being are impossible, senseless questions. Therein lies the grave historical danger of being absorbed by the Cartesian-mathematico-digital way of thinking.
The totalizing nature of digital ontology is therefore not merely a matter of cyberspace - along with other, now digitized media such as radio, television, telephone, portable music players, etc. - 'invading' our life-world as the ubiquitous medium. In these shiny forms of appearance, digital ontology does indeed already dominate the surface of everyday life in technologically advanced countries. It is only a small step now from having a mobile telephone permanently glued to one's ear to having a chip connected to the internet implanted in one's brain. Digital technology itself is only the consummating tip of the mathematico-Cartesian iceberg. Underneath it is the epoch-making mathematical casting of being, which is deeper and truly totalizing. Why? Because the Cartesian casting today underlies all science, and science, with its empirical, quantifying mode of access to the world, has become the locus of truth in the modern age. Truth for us has become universal, generalizable, 'objective' truth established by scientific method. All else has become mere 'subjective' opinion, at most a colourful embellishment not 'properly' grounded in scientific methodology. Or quantitative scientific methodology has been applied to phenomena that do not at all fit the mould of precalculative reason, notably, the entire gamut of phenomena associated with social interplay including economic interplay (thus, for instance, we thoughtlessly put ontological faith into sophisticated computer models of the economy to precalculate its movements).
Truth has become that which is established by scientific method according to experimental observation of the facts. If the experimental data agree with the theoretical model, the scientific theory is held to be true until further notice. This seems self-evident, and thus it is believed. But what guarantees the truth of scientific method itself? It specifies that scientific truth resides in the correctness of theoretically predicted observations corresponding to the experimental facts. On its own terms, therefore, scientific method itself cannot be true because its own correctness cannot be verified. For the issue of the truth of scientific method, correctness will not do as criterion, let alone the 'success' and pracitcal 'effectivity' of science. Scientific method depends on a prior (Cartesian) casting of the access to the world whose truth or otherwise can only be assessed by questioning how such a mathematically cast world shapes up, for this casting determines from the outset as what and as who beings as a whole take shape, come to a stand in understanding and show themselves in the world. The untruth of scientific method lies in its riding roughshod over the phenomena in its inexorable striving to ascertain the correctness or otherwise of the empirically testable scientific theory with the facts, without ever questioning that the beings interrogated have always already been precast in the science's foundational concepts.
Our ways of thinking about the world and our own being have long since become totally infiltrated and infected by modern scientific ways of thinking, indeed, so much so, that we can scarcely even imagine another way of thinking and are all too quick to reject other modes of thinking as pre-modern, unscientific, merely poetic and subjective, ideological, or similar. (Or, conversely, one champions the poetic and artistic over the 'coldness' of calculating, scientific rationality, which is the same as its opposite, just with a minus sign, but not an alternative. Or one seeks a counterweight in ethics and morality.) Computer science is only one science among others, but all sciences today process empirical data using mathematical formulae, and therefore rely more or less on computers to automatically carry out the required, preprogrammed calculations.
But that is not what is decisive. The striking hallmark of digital ontology in today's world is also not so much the ubiquity of digital media but, more essentially, that through digital technology our world-understanding can be outsourced piecemeal to computing machines (automatons, robots) for the sake of automated cybernetic control over movement of all kinds, especially the free movement of human beings. This outsourcing is often seen one-sidedly as allowing us human beings to make our lives more comfortable and convenient, as relieving us from unnecessary toil, and this perspective has its justification, but the other side is that we become entangled more and more in cybernetic systems that function inexorably and inflexibly according to the logic, i.e. the world-understanding, that has been programmed into them. The possibilities this opens up for state political control are only one aspect, albeit one not to be underestimated, for the state's will to power, including its will to power through caring for its populace, is insatiable.
5.1. Two exemplary industries at the forefront of the digitization of beings: telecommunications and bankingIn particular, it is instructive to see how the disclosure achieved implicitly by the digital casting of beings as such has its effects in the world of capital, i.e. in the economy. There are two exemplary industries mightily affected by the new dimension of cyberspace and digitization in general: the telecommunications industry, and the banks or finance industry, and this for reasons which have to do with the essence of digitization itself. The telecommunications companies are subject to the compulsion to techno-logically bring forth the unified, all-comprehensive (o(/lon) dimension of the electromagnetic medium, to open it up and to make it available so that the digital beings can move freely, without borders and limits. This dimension must encompass the entire globe if such a telecommunications capital is to survive in the long run. The digital entities must be able to move through cyberspace anywhere on the globe without resistance; this is the final sense, the teleology of networking. At present, huge transnational capitals in the telecommunications industry are still working on this. Under the coercion and discipline of competition they are corresponding to a metaphysical destiny (the digital encoding of the totality of beings) without inkling at all that they are doing so. "They do not know it, but they do it." (Marx) The essence can remain unknown - and as a rule it remains unknown - whereas the phenomenal forms of appearance, such as an opportunity for revenue growth and competitive pressure, thoroughly correspond to it nevertheless.
Another industry significantly affected by digitization is the banking or finance industry. The banks are also forced to completely explore and exploit the homogenous, unified dimension of the electromagnetic medium, and they can and must do this because the 'commodity' which they trade in, namely, money, on the one hand, as universal equivalent of commodity wealth, is universal and, on the other, it can be stamped into an arbitrary material as a number. Coins are already stamped material; today, it is enough for a (state sanctioned) number to be stamped electronically into the electromagnetic medium and that this number, which is owned by someone or other, can be transferred from one owner to the other. This is already money and in this form it can move freely also as capital.
Capital needs the dimension of value(17) which is also determined in an abstract, quantitative way. Only in this quantitative dimension of being which admits of a more and less is money what it is, and only in this quantitative, monetary dimension can and must capital calculate. All economic phenomena can be grasped quantitatively, i.e. measured, more often than not in monetary terms, and this circumstance forms the essential precondition for all economic phenomena potentially and necessarily being taken into the grip of digital technology, whether it be, say, through macroeconomic simulation models based on econometric data, or through decisive statistics such a inflation rates, unemployment rates, etc. which provide guides for the attempt to steer the economy. The essence of capital is a movement of objectified value which is also essentially quantitative, and the competition among capitals has to correspond to this essence and thus also obey the movements of the value-numbers. From here comes the compulsion for banks to merge. Either they do so, or they go under, or are swallowed by a larger capital, for cost-savings are achieved mainly by facilitating the relentlessly increasing masses of monetary transactions which can be processed most efficiently, i.e. automatically, by computers at nodes in the electromagnetic network. Since the business of banks to a large extent consists of monetary transactions and monetary movements, such movements can be registered electronically in a digital form and thus executed in such a way that enormous costs can be saved. Bank customers too must be enticed, partly through monetary incentives, into learning to deal with digital technology and thus into contributing to their banks' cost-saving drives.
Of course, the revolutionizing effects of the digital casting of being (which is a mathematical way of thinking the being of beings as a whole that is also materialized as digital code embedded in electromagnetic media, thus doubling beings into physical beings and their digital, virtual counterparts) are not restricted to only two exemplary branches of industry, but these two industries are particularly suited for illustrative purposes because they themselves are so digitally abstract, i.e. in the one case, they do their business with the 'formless', homogenous electromagnetic medium itself, and in the other, they do their business directly with money which is determined purely quantitatively in its form and can be materialized simply as a stamped number, including in the electromagnetic medium. In electronic commerce, too, it is primarily and decisively the monetary side (order processing, accounting, inventory control, etc.) which is exclusively carried out in the digital dimension and automated as far as possible, whereas the goods ('unfortunately') still have to be produced, packaged, despatched, etc. physically, of course with the assistance of a massive employment of digital technology in automated logistics. In the finance industry, the commodity itself is a money-form or a money-near form, and this then does not require any physical supplementation (transportation, etc.), but rather the trade can be carried out completely within the digital-arithmetic dimension with all the accompanying advantages of cybernetic automation.5.5 Time in a capitalist economy). The digital casting of being is ambivalent; it opens up existential possibilities for us on the one hand, through digital "conveniencies of living" (Adam Smith), and on the other, it makes us into mere cogs in developments which roll in over us. We mesh in like cogs and run along, somewhat breathlessly, behind 'developments'.
In all the hype (idle chatter) about globalization today, the essence, of course, is not seen at all. People are ontologically completely blind and forgetful in this regard and are satisfied with sociological explanations. It is still unfathomable, incalculable and unforeseeable how the digital casting of being will further unfold, say, in the next fifty years. What is most questionable, however, is that the origin of digital technology as a mode of being is not a question at all. We have lost sight of the indispensable role of philosophical knowledge which, in Hegel's words, consists in "investigating what is normally regarded to be well-known". "But such well-known phenomena are usually the most unknown." "The business of philosophy consists only in bringing expressly to consciousness that which, with regard to thinking, has been valid for human beings from ancient times. Philosophy thus does not set up anything new; what we have brought out through our reflection is already the immediate prejudice of each individual."(18) Instead of telling the story of the string of events through which globalization has been enabled by a string certain key (digital) technologies and other events (the ontic narrative so amenable to normal understanding), the deeper, philosophical task is to uncover how the digitization of the world is enabled by a certain, historical way of thinking the being of beings.(19) Both are associated, but in different ways, with the Aristotelean-Cartesian, ontoarithmological casting of the totality of beings and its consummation in the dissection (or taking-apart or decomposition) of beings into logical bits which, as placeless, calculated beings can be inscribed arbitrarily in the dimension of the electromagnetic network whence they can be called up to present themselves anywhere, anytime. In particular, in money, the value of beings, their valuableness, is embodied quantitatively in a reified way, i.e. in a separate thing (res). Money as the "universal equivalent" facilitating universal exchange is an arithmetic (and therefore a)/qetoj, a)/topoj) abstraction which can be calculated, and insofar it corresponds to the digital casting of being and thus also to the world-encompassing, unified, techno-arithmologically produced cyberspace; it is absorbed in this medium and the circuits of capitals can assume in virtual reality, too, their own, independent life, just as Marx analyzes in the fetishism section of Das Kapital and elsewhere. This is an autonomization of capital vis-à-vis human existence and it is enabled in a consummate form through the digital casting of being, for both capital and digital beings are in their essence arithmo-logical, i.e. they both have a numerical, calculative, calculating nature.
So one might be tempted to think of capital and capitalist economy like a machine that can be controlled calculatively, just like a technological cybernetic system. In this case, the essence of capital would correspond entirely to the pro-ductive, calculative, controlling essence of technology that sets up the totality of beings as a standing reserve for endless circuits of production, and the political polemic against the machine-like 'capitalist system' would have some justification. To clarify this, and without yet having answered the question posed in the subheading, we have to dig deeper into the essence of capital and tease out two quite different meanings of 'calculating'. To this end, we first consider what it means for beings to be valuable.
The exchange of use-values is the exercise of a certain power inhering in the use-values concerned to exchange or interchange for each other. The exchange-values possessed by the exchangers constitute starting-points (a)rxai/) governing an exchange (metabolh/, which can signify both 'change' and 'exchange') and insofar fulfil a modified ontological definition of power as laid down by Aristotle in his Metaphysics, where only a single starting-point is considered. The exchange of use-values for the sake of enhancing living (which Marx calls "simple commodity circulation", einfache Warenzirkulation, MEW23:164) is therefore already a power play in the strict ontological sense of the term, 'power'. Money arises (both ontically and ontologically) as the universal equivalent facilitating the exchange of everything offered for exchange. As a thing (res), it therefore itself embodies the power of exchange for everything with a price, i.e. it is the reified crystallization point for exchange-value which is nothing other than an ongoing power play of the exchange of everything valuable. Since use-values are generally (i.e. apart from purely natural products that 'come forth' entirely without labour) the result of the exercise of human labouring powers, i.e. of human abilities and excellences, the power play of exchange is fundamentally an interchange of human labouring powers, so power is implicated even on the most 'innocent' level of exchange.
Whereas the power play of the exchange of use-values can be viewed as the metabolism of the goods of living to enhance living, i.e. exchanging one use-value for money in order to acquire another, desired use-value, or C - M - C', the play of exchange can also be inverted in order to make more money from money, or M - C -M', where M' > M. This is the simplest formula for capital: advancing a principal sum in order to have it return augmented from its circular movement. A production process P may or may not be incorporated in this simple circling of exchange-value as capital, thus distinguishing between industrial and commercial or finance capital. The power play played by exchange-value advanced as capital is subject first and foremost to the simple rule that M' >= M. Otherwise, if M' < M, the power play would consume itself until eventually nothing more were left to advance. The power play of capital as a movement from M to M' requires at least two, and in general many, exchanges, each of which is a power play between the exchangers, most notably between the capitalist (large corporation, small firm, or whatever) and the hirers of labour power, the workers, who comprise all those (including even the top managers and executives) contributing to this movement.2.9 Time and movement in Aristotle's thinking), which in this case is the turnover time of capital (cf. my Critique of competitive freedom and the bourgeois-democratic state 1984/2010 Appendix §§42ff), the measure of a circular movement (kuklofori/a, Phys. D 14;223b19) from M to M'. The success or otherwise of the circuit of capital can thus be measured by the simple finite-difference formula dM/dt = (M'-M)/(t'-t), where t and t' are the points in time at which a capital sum is advanced and returns. Such a formula measuring the result of the capitalist value-play relies, of course, on the reduction of the phenomenon of time to a linear variable consisting of now-points and also on the reduction of the phenomenon of value to a quantitatively determined money-value wherein the power play underlying value becomes invisible. The differential calculus developed by Cartesian (Newtonian/Leibnizian) mathematics in the modern age for physical movement therefore applies also to the social movement of value as capital, albeit without necessarily requiring infinitesimals but only a calculus of finite differences.
In decisive and essential contrast to the movement of physical bodies described by Newtonian (or even Einsteinian) laws of motion, however, there is no formula to compute the difference M' - M, because this difference is merely the outcome of a value power play in which exchange-values are actually exchanged. There is no intrinsic potential exchange-value inhering in a use-value that could pre-determine its quantitative exchangeable value simply because exchange-value itself only comes about or happens in a power play on the market among at least two, and usually many players. Such is the power play played by capital in its plurality whose ontology represents a rupture with traditional metaphysics because it can cope only with mono-archic movement, not with the poly-archic, 'playful' movement of social interchange. Capital is therefore calculating in that it reckons with a surplus value at the end of its circuit, but it cannot precalculate this surplus with calculative certainty, for the gainful interplay on the markets is essentially risky and uncertain.
Moreover, the time required for the movement of an exchange transaction also has no ground in a law of social movement according to which it could be calculated, nor is this time interval uniform. Commodities offered for sale on the market are at rest (h)remei=n, Phys. D 12;221b28) with respect to their value-transformation and only jolt into movement upon being sold. They are nevertheless at rest only within the overall movement of capital, so that this their being-at-rest is only a limiting case of their movement as value, just as, analogously, a piece of timber at rest on the carpenter's bench is still within the overall productive movement of being made into a table. The movement of a single capital involves many individual transactions and therefore many individual value transformations, each of which takes its own time, so that the overall movement of one turnover of capital depends on many, even myriad value transformations being achieved before the advanced money-capital returns. This circumstance implies already that the circular movement of even a single capital comprises a series of jerky movements of value transformation plus the movement of production itself, which may be organized technically, through logistics and supply-chain management, to run smoothly. Especially at the interfaces where commodity-value has to be transformed into money-value, the movement of value comes to rest for a time which may be brief or extended depending upon market conditions.
The circular movement of a single capital is hence both incalculable and uneven. The reproduction of an entire capitalist economy involves the intricate intermeshing of many individual circuits of capital. The turnover of the total social capital is therefore even more complicated and intricate than that of a single capital, so that the counted number or time associated with this total social movement is both incalculable and non-uniform, since the underlying movement of total social capital itself is both incalculable and uneven. This contrasts with Aristotle's determination of the measure of time as an "even circular motion" (kuklofori/a o(malh/j, Phys. D 14;223b19). The regular period of even circular motion makes counting easier and its number, viz. time, easier to deal with calculatively, and a public measure of time in a standard periodic movement facilitates the co-ordination of movements not only of capitals but among the economic players in general. A uniform measure of time, such as the year, can be imposed on the movements of value as capital, but this is only the abstract subsumption of many complicated, uneven movements under a convenient standard.
If the turnover of the total social capital is the basic, underlying movement of a capitalist economy, the measure of this turnover also provides the basic measure of time in such a society whose rhythm is determined by the circular, augmentative movement of capital. As we have seen, this underlying social movement is uneven, which implies that time in such societies is also uneven (not like the more regular movement of, say, an agricultural society in tune with the movements of the seasons). Furthermore, the measure of the success of a turnover of capital is not only the amount of surplus value it throws off on its return as money capital, but also the turnover time taken for this circular movement, i.e. the faster the turnover, the more profitable the capital. Since capital is this augmentative movement from money to more money, it achieves greater augmentation by shortening as far as possible its turnover time, thus reducing the denominator in dM/dt = (M'-M)/(t'-t) and increasing it overall. If the turnover time of total social capital is an underlying, basic measure of time in a capitalist society, the tendency of capital to shorten the turnover movement means that time in such a society becomes shorter and shorter. That is, a capitalist society tends to continually accelerate time, even though such acceleration is not precalculable (but at most postcalculable), depending as it does already on the simple, but nevertheless incalculable transformation of commodity-value into money-value (sale of the finished product on the market) and money-value into commodity-value (e.g. if supply on the market is short).
Since the augmentative movement of value as capital is the fundamental underlying movement of a capitalist economy, and this movement draws in all the players striving to earn income in the gainful game, and participation in the competitive gainful game is a vital aspect of social life in such a society, any obstruction to the gainful game threatens the very movement of society, and time stutters and in some places can even come to a standstill. Essential reasons for obstructions to the underlying valorization movement of capital reside in disproportionalities among economic sectors in the reproduction of total social capital and overaccumulation (cf. my 'Anglophone Justice Theory, the Gainful Game and the Political Power Play' Section 8. 'Anomalies in the gainful game and the political power play' and Critique of competitive freedom and the bourgeois-democratic state §§65ff). Some industries and some regions may lose out in the competitive struggle and be transformed into industrial wastelands where time has come to a halt. To the competitive players themselves on the surface of society, however, hindrances to economic movement can have many phenomenal guises and may even be caused contingently when, for instance, some (entrepreneurial) players misjudge risk. A capitalist society therefore has a vital interest in keeping the gainful game moving at any cost. Once the state proclaims by fiat that its paper money is legal tender within its territory, it also has a way of steering, or interfering with, the gainful game through central bank monetary policy, and may or may not be partially successful in preventing the gainful game, and time itself, from faltering and coming to a standstill.
Money is the mediator in this dimension of quantified value, the medium which enables universal access to acquiring what is valuable. It is inconsequential in this context whether what is valuable is a thing, a human service (i.e. the exercise of labour power of some kind or other), a piece of nature or - in a derived way and, so to speak, of second order - money itself (interest). Money is then, as the representative of wealth in general, the universal key, by means of exchange, to, the universal social power over all that is valuable. Money is reified social power par excellence. All possibilities of existing in the world are enabled by valuable things and therefore entertain, directly or indirectly, relations with money as mediator. In any appropriation of valuable things in the broadest sense by human being, a more or less is disclosed from the perspective of the gainful game through the proportions in which they are exchanged on the market. Therefore money, as universal mediator of exchange and the abstractly universal representative of what is valuable, is abstracted from any quality and is thus only quantitatively distinguished within itself. Hence it can assume the form of pure countable number (a)riqmo/j) which, in turn, of course, can also be digitized. As universal mediator for the exchange of what is valuable, money itself is valuable, i.e. a social power, and therefore access to it must be regulated. Money must be acquired according to certain rules for acquiring it (property rights, especially those pertaining to contracts of all kinds that regulate the interplay). The rules for exchange and acquisition constitute the framework of the gainful game in which human beings must participate insofar as they exploit their options for existing well.
The way of viewing everything from the perspective of money, which allows beings to be in a certain mode of quantitative valuableness, means that money is something resembling the material precipitate of a homogenous, universal, quantitative dimension in which value discloses itself, also quantitatively. Even when a satellite is sent into space or make observations of distant galaxies, etc., or research is done into the sub-atomic world, these activities are guided and constrained also by flows of money, i.e. money mediates, through enabling and constraining, the dimension within which we also measure and fathom our existential possibilities and activities, our movements as social beings, as well as their limits. It is a medium for the movement of human living itself. Money and value, too, distinguish us from animality insofar as we are exposed to fathomless value and thus also to limitless greed and desire. Animals do not have desires, but only limited drives which can be satiated by their environment. They do not look at anything as valuable; they are not exposed to the apophantic as which shows up beings as such and in such a way that they can be addressed by the lo/goj, including in the category of quantity (poso/n). Only we humans can be voracious and greedy, and voracity as a mode of human being's comporting itself is a possible way of responding and corresponding to the gathering of promising possibilities of gain (of course, including also the 'negative' or detrimental possibility of losing, of failing to achieve success). Money has assumed historically differing garbs such as gold, silver, state paper money and today, strictly guarded numbers stored in the electromagnetic medium. Digital money is the pure consummation of money in its purely quantitative value-being for it hardly requires matter at all, only the electromagnetic alignment of a couple of molecules. These numbers and their flow (cash flow) encompass, and insofar steer, all possibilities of human existence either directly or indirectly. Without a flow of money mediating the gainful power play, human being itself cannot move in its existence, something that becomes painfully experienceable in every severe economic downturn in phenomena such as the so-called 'credit crunch'.
We could also formulate the fundamental condition of a capitalist world (i.e. of a world that is sustained in its movement through the power play of gain) in the following way: Nihil est sine valore - Nothing is without value, in resonance with Leibniz' principium grande: Nihil est sine ratione - Nothing is without ground or reason. But value itself is not a ground, coming about as it does groundlessly in the interplay only as a promise. This means that everything has its value in the sense that all beings are open, disclosed, first of all as use-values, but also as exchange-values suitable as pieces in the gainful game. Like every ontological world-opening, this casting of world as a value game is itself ungrounded, i.e. there is no founding ground or reason why all beings should be caught up in the gainful power play of values and also no reason or ground for value, measured in money as price, having a definite quantitative magnitude. The principle, nihil est sine valore, posits of itself - fathomlessly, from the very depths of being - a mode of disclosing beings as a whole and setting them into motion in the gainful game whilst enticing human beings as the players in this game. If the use-value of a commodity, depending as it does on the constantly shifting ways of living in which human beings customarily live, is without ground and thus also quantitatively indeterminate as an exchange-value, then it is also the case that nothing is without risk, for the values are determined only through the interplay among many players in which they are exposed to validation by others. The circling of value as capital has to pass through several or many value-validations on various markets before the success of its circuit is 'home and hosed'. Risk is that which cannot be brought under the control of an a)rxh/, but just happens, comes along contingently (to\ sumbebhko/j).
The entrepreneurial risk familiar to capitalism itself derives from the fundamental groundlessness of values as they come about in the power play of exchange-values of all kinds. Above this groundless abyss and before the horizon of the being of exchange-value, all the players in a capitalist economy play, above all, however, the entrepreneurs themselves who, as the lead players and initiators of a circuit of capital, are exposed to the essential contingency of value. Having and spending money is indeed a deeply rooted historical custom enabling our existential options. Money, however, is not simply a technical instrument but the ontic, material precipitate of an ontological dimension, namely, the dimension of value, which is an historical way in which world opens up for human being. Money as capital is the autonomized movement of the augmentation of money with its own simple, finite-differential formula for success or failure denoting the accumulation or destruction of value as capital. Since money and capital as embodiments of value are infected with the groundlessness of an interplay of powers, there are no laws of motion for the economy analogous to the laws of motion for physical beings investigated by physics.
To bring the fair face into perspective requires, as an ontological condition of possibility, reappropriating the time that has been quantified as a mere mathematical (and hence timeless) variable t in the formula given above for the success of a circuit of capital, for the time in which we play the gainful game is our own, finite life-time of our own finite life-movements. Firstly, as elaborated in Sein und Zeit, the mathematical variable t has to become thought and experienced as the three-dimensional ecstatic, finite time of human being itself that casts its self into the open dimension of the future by retrieving who it has been and fashioning its ownmost, singular possibility of existing. Such self-casting, however, is close to being misunderstood as auto-production. Therefore, in a further twist, this three-dimensionally stretched time, which accords with the life-movement of human being itself stretching toward its future, has to become thought and experienced as the social time of social movement itself, which is not just the measurable movement of total social capital, but the immeasurable, complexly interwoven movements of social interplay in which each individual haphazardly comes to stand (or fall) as somewho, to gain or lose its self, in the power play of social recognition and social validation of its powers and abilities.
A further form of fetishism inheres in interest-bearing capital which seemingly 'of itself' brings forth precalculably an interest yield: M becomes M' = M + i over time t. 'In truth', however, i.e. in the full disclosure of its essence, the interest yield is a form of appearance of the augmentation of value in analogy to the quantitative growth of reified value through its productive circuits as capital which sets human abilities into motion. Everything thus depends on clarifying the ontological origin of value, i.e. on entering and thoughtfully deciphering the value dimension itself as a dimension seemingly with a life of its own.
Marx's concept of fetishism includes as its foil the thought of the loss of (socialized, collective) human control, for the value dimension in his thinking is traced back ultimately to labour (which is thus implicitly proclaimed to be that which ultimately is solely valuable and quantitatively determining for exchange relations) and it is shown that, because of the commodity form predominating in capitalist societies, consciously organized social labour as such does not serve as the underlying basis, i.e. as the subject, of human economic activity, but rather that this subject role is usurped by objectified labour itself as capital. This Marxian retracing to the essence is located in modernity's metaphysics of subjectivity in the form that the (sociated, collectivized) human is postulated and cast as the ultimate subject of the totality of beings. Only within such a metaphysics can capital be designated as something resembling an (alienated) 'human product' or even as alienated human freedom. For how could humankind be free as a socialized collectivity whilst quashing the interplay of individual human powers in rivalrous, gainful play with one another?
Nor can capital be addressed at all as a kind of technology, a 'machine' that could be mastered by (collective, socialized) humanity, but rather it can at most be brought into a connection with Platonic te/xnh kthtikh/ (an art of acquiring) in contradistinction to te/xnh poihtikh/ (cf. Sophistês 219c and GA19:272ff). Capital is the wager to augment value through the interplay of exchange, and not a fore-knowing, precalculable movement of bringing forth a value-product, for value itself cannot be pro-duced. The fetishism of capital therefore has nothing at all to do with technology and technical products or any autonomization of suchlike, nor with surface phenomena such as consumerism (but solely with the appearance that value, as the power to exchange, seems to inhere in certain things of themselves). That would be a complete misrecognition (all too common) of the phenomenon of economic fetishism as first brought forcefully into view by Marx. Rather, the origin of fetishism has to be sought in the ontological dimension of value and the interplay in which value comes about in a power play, and the phenomenon of technology can be brought in only indirectly, via this value-dimension, for instance, as a factor for enhancing the chances of value-augmentation in the competitive power play.(20) Value itself, the disclosure of beings as valuable, is not a human product, not a human machination, even though humans necessarily participate in the value dimension in the same sense as they 'participate' in the ideas, for it is 'natural' for human being to be open to understanding and 'practising' use-value and therefore also use-value-for-others, i.e. exchange-value.
Early on in the analysis of the essence of capital in Das Kapital, capital is determined as an "automatic subject" (automatisches Subjekt, MEW23:169) in the sense of the self-valorization of value. Capital is not anything resembling a cybernetic subject controlling a total-social economic reproduction process. Rather, the subject-character of capital must be sought via the concept and dimension of value. Capital is then subject only in the sense of the underlying (u(pokei/menon, subjectum), incessant and therefore "automatic" movement of value through the value-forms of money, commodity and back to money in the striving for an augmentation of value. This (formal) movement takes place according to the rule of play that the rate of change of capital be positive, i.e. dM/dt=(M'-M)/(t'-t) > 0. This rule of value interplay asserts itself inexorably in the long run. Value interplay itself is, in nuce, the game of mutual recognition of powers, starting with individual human powers or abilities, but including also derived powers inhering in property and money as exchange-values. Value can therefore be thought of in the first place as the dimension of mutual social recognition, a simple phenomenon lying at the basis of all human sociation: we estimate and esteem each other's powers, abilities (even, and especially, when we are indifferent to or detract from each other's powers and abilities).
As we have seen, capitalist economic activity is undertaken under the principle (a)rxh/) that from advanced capital (money M), more capital (money M') is supposed to flow back, which is a kind of rule of play, constraining boundary condition or condition of existence for capital: M' - M > 0. This principle is by no means complex but rather, extremely simple, and its origins do not lie in the nineteenth century but already in ancient times. Aristotle already thinks about the endless striving for the augmentation of riches by way of chrematistics. The principle of the valorization of money, its self-augmentation when viewed from the standpoint of advanced money-capital bending back onto itself to determine a difference, comes to us from a far-off origin (as a sending from destiny), just as modern technology comes from te/xnhas a poietically knowing mode of disclosure that was taken as the foundational, paradigmatic phenomenon for Western (productionist) metaphysics that has subterraneanly shaped Western history. Marx says that since Aristotle not a single step has been taken forward in clarifying the concept of value, the key to deciphering the ontology of our social being. That sounds similar to Kant's parallel remark regarding Aristotelean logic.
"Destiny" here does not mean anything like a fate in the sense of an alien power that decides our fate, but rather the historical disclosure of a world sent, or eventuating, from being (hiddenness, nothingness) which we can never completely fathom nor control. This disclosure comprises the various historical ways in which we can encounter phenomena and address them and also be addressed by them. Certain simple 'ideas' form the ontological 'scaffolding' on which an historical world hangs. A world shapes up for us as historical beings existing in time, and as time, and the shape this historical world assumes is determined first of all by our deepest and simplest shared ways of thinking, the ones most unquestioned and apparently unquestionable in any given historical epoch which seem absolutely self-evident (such as 'objective' scientific truth seems to us today). The ground-categories of an epoch are also those with which our thinking identifies and hence those with which our very identities, i.e. who we are, are bound up. A shift in historical destiny is always a matter of a disquieting, conflictual transformation in the way in which world discloses itself to us and also of letting go of how we have understood ourselves hitherto in our self-standing. Destiny is not a being, an instance, but must be understood as a sending and receiving of deep, simple ontological 'messages', a giving and taking to which human beings belong as recipients. It should be apparent that only the strongest, and not the pusillanimous, are able to receive the message. The strongest here are not the most unbending and steadfast, but the most receptive and ontologically 'sensitive' who are able to suffer the concussions associated with a seismic non-identity between thinking and established world.
To return to the question of complexity: Marx investigates the complexity of capitalist economies only in the second volume of Das Kapital, The Circulation Process of Capital, where the opaqueness and intricate complicatedness of the economic whole is dealt with, which leave open many possibilities in the overall process of reproduction of the economy for frictions to arise in the intermeshing and intertwining of the many individual capitals, etc. But already the simple value-form itself (which presupposes a plurality, a sociating of commodities) is essentially contingent and incalculably unpredictable, since what or how much a commodity is worth is determined only in the exchange interplay itself on the market. The analysis of the essence of the commodity, money and capital (their socio-ontology) is carried out already in the first two chapters of the first volume of Das Kapital. In the determination of the essence (the socio-ontological principle) of capital as self-augmenting value, it becomes apparent that this circling principle confronts humans as an alien, alienated power, not because of the complicatedness of the economic system, but because of the fathomlessness of value as the ongoing outcome of an interplay of powers and ultimately of the gainful game itself, to which, however, human being in its desirous striving for the goods of living, and evermore thereof, belongs.(21) in connection with the essence of technology and its eery will to will, according to which all beings are set up sense-lessly for endless production, do have a kinship relation with the circuits of capital, and this not even necessarily in another phenomenal guise, insofar as it is the capitals, that is, the enterprises themselves, which forcefully promote the development of technology and control technicized production in order to stay ahead in the competitive, gainful game. Not only with technical progress, but also with capital accumulation there is an endless progress for the sake of progress at work where the two tendencies intermesh with each other, for staying ahead technologically is one major way in which an individual capital, through increased productive efficiency, enhances its chances of survival in the competitive power play of exchange-values. But the kinship in essence between technology and capital, between precalculative, productive setting-up and calculating, risky valorization of value - ultimately by virtue of the difference (diafora/) between one (e(/n) and many (polla/) and the consequent incalculability of movement (ki/nhsij) as interchange (metabolh/) (cf. 3.5 The onto-theological nexus in abstract thinking, cybernetic control and arithmological access to movement and time) - breaks down and an estrangement between the two emerges insofar as the will to productive control so exquisitely consummated in automated digital cybernetics is the very opposite of the willingness to risk engagement in the incalculable play for gain with its many, competing players. Insofar, in answer to the question posed above (cf. 5.3 Does the essence of capital correspond to the essence of technology?), we have to say that the essence of capital (whose ontological structure is that of poly-archic interplay) does not correspond but runs awry to the essence of technology (whose ontological structure is that of mono-archic production) which, especially since the advent of the Cartesian mathematical casting of world, seemed to be close to fulfilling the dream of total, calculable, even materialized, automated control, the ultimate consummation of the will to power as pro-ductive power governing the movement that brings beings to stand in presence. The essence of capital, by contrast, is playful.
What, then, do the essential contingency and incalculability of the value-form have to do with Heidegger's characterization of the consummation of subjectivity as the "securing" of the will to will. He writes, for instance, "Humans of themselves align their essence with security in the midst of beings, against them and for them. They seek security amidst beings through a complete ordering of all beings in the sense of contriving a planned securing of standing reserves, which is how setting up in the correctness of security is to be performed." ("Der Mensch stellt von sich aus sein Wesen auf Sicherheit inmitten des Seienden gegen und für dieses. Die Sicherung im Seienden sucht er durch eine vollständige Ordnung alles Seienden im Sinne einer planmäßigen Bestandsicherung zu bewerkstelligen, auf welche Weise sich die Einrichtung im Richtigen der Sicherheit vollziehen soll."(22)) First it should be noted that security is the antithesis to the groundlessness of freedom, so the passage points to the headlong rush of human being from the possibility of freedom into a subjugation to calculable security. Moreover, Heidegger's gaze is directed at totalizing cybernetics which, however, has to be uncovered as a self-delusive illusion insofar as cybernetics only calculates and can only calculate with beings pro-ductively, for it is impotent vis-à-vis value as a mode of being which is unfathomably incalculable and beyond any cybernetic control, but is nonetheless an historical way of world-disclosure for a plurality of human beings at mutually estimating play with one another. The dimension of value in its economic sense is never a subject in Heidegger's writings, and we have to learn this in an 'unsettling' re-reading of Marxian texts inspired by Heidegger: Inherent in the interplay promising gain is an essential limitation to productive technology in general, and cybernetic technology in particular, for technology is essentially not able to steer and control the augmentation of value for which there is no sure-fire, calculable, winning strategy.
To see this requires going beyond the horizon of Heidegger's thinking. It is noteworthy that in the above quote, Heidegger speaks of "der Mensch", which is here translated naturally as "humans". What is singular in German translates naturally into plural in English. This is because the phenomenon of value of its nature involves a plurality of commodities and of exchangers of commodities. Whereas use-value always signifies a usefulness for a human user, the second-order exchange-value always signifies a usefulness for another human user. Now there are at least two free starting-points demanding ontological consideration. Something useful has to be offered by a seller, and someone else has to bid for what is offered, and only in this ongoing interplay of offers and bids does the exchange-value come about, or happen, as an abstract, quantitative exchange proportion (since anything at all can be exchanged for one another, all quality is abstracted from in generalized exchange). Exchange-value is therefore essentially quantitative (poso/n) and relational, pro/j ti, and not a substance, an ou)si/a. Contra Marx, whose thinking was still held fast by the Cartesian casting, there is no "value-substance" (Wertsubstanz, MEW23:49). The human world is characterized by a plurality of human beings engaged with each other in exchanges and interplays of all kinds. Economic interchange is only one kind of human interplay.(23)All human interplay, however, is fathomless because each human starting-point in the interplay among at least two is free, i.e. essentially abyssal, fathomless, groundless. It is this essential groundlessness in the plurality of human interplay that vitiates any dream (or Heideggerian nightmare) of total cybernetic control through digital technology, even though the possibilities of surveillance and welfare from 'above' of human beings and their intertwined movements opened up by automated digital technology conforms entirely to the state's caring will to political power. In itself, the economic game of striving for gain is a groundless, incalculable interplay subject only to the simple principle or rule of play that it be gainful rather than loss-making.
The productivity of the labour of circulation can also be boosted through the employment of digital technology, although here it is important not to confuse the productivity of the labour of circulation with that of the labour of production. Labour of circulation includes all the 'backroom' operations of invoicing, accounting, etc. along with sales, marketing and advertising efforts. Since accounting is performed in quantitative monetary dimensions, and is thus of its nature arithmo-logical, it is particularly amenable to digitization by means of appropriate accounting software that enables automated cybernetic processing of all sorts of transactions such as purchase of raw materials, payroll processing. invoicing of customers, etc. etc. It must be kept firmly in mind that the labour of circulation is not a pro-ductive bringing-forth, but rather a retro-spective mopping-up or a pro-spective smoothing-the-way for value-form transactions which i) have to be processed and recorded for bookkeeping purposes after they have occurred, i.e. retrospectively, or ii) instigated through rhetorical prodding in the market-place. Labour of circulation includes also retailing, which can be made more efficient in diverse ways by employing digital technologies, right up to setting up retailing outlets on the internet that can sell goods with all the efficiency advantages offered by trading in a purely digital, near-zero-cost environment in which the goods can be presented digitally and also sold via sophisticated cybernetic sales transaction and logistics systems. The movement of money-capital in the broadest sense (that is, not merely monetary movements, but also commerce, circulation, turnover and revenue/earnings transfers, financing, credit-lines, etc.) quickly makes itself at home in the electromagnetic, digital medium as a useful, cost-saving aid that also enhances oversight of the total process.
Digital technology can also be deployed for another kind of labour of circulation, namely, marketing and market research, especially where the markets themselves are digitized, as on the internet. Since all movements in the electromagnetic medium leave a data trace, these data can be mined to distil regularities in consumer behaviour which, in turn, can aid in conceiving marketing strategies. Again, such marketing strategies are not productive in the strict sense, but merely game strategies of enticement in the gainful capitalist game which nonetheless can turn out to be highly profitable. Prediction of, say, consumer behaviour (e.g. researching the market for a new fashion collection) or future demand for a certain raw material is done by collecting data on what has already happened or on what a sample of consumers say it intends to do, their 'tastes', etc. These data are then sifted using sophisticated statistical methods, and then extrapolated in some way into the future. Digital technology is today indispensable for such statistical processing which, nevertheless, in such prospective applications, does not have cybernetic, pro-ductive power, but only surmising, probabilistic power. Scenarios for future markets can be very useful for capitalist companies in planning their game strategies.
So, is there anything new in the digitization of the capitalist economy? Digitization means the breaking-down of beings into a digital (binary) representation that enhances their calculability. This is a legacy of the Cartesian rule that beings be approached through what can be abstracted from them by way of magnitude, which presupposes the Cartesian casting of physical beings as a whole as res extensa culminating, as we have seen (2.7 Cartesian rules for an algebra of magnitudes in general as foundation for the modern mathematical sciences), in modern abstract algebra and its coupling with functional analysis. This is the ultimate basis of the modern mathematical sciences, both physical and social. What is new in the digital age is that our mathematico-scientific understanding has become representable piecemeal in chunks of automatically executable digital code that can be inscribed in electromagnetic media. This enables the doubling of familiar, physical beings into their digital, virtual counterparts already mentioned which, through sensual, graphic interfaces, behave in much the same way as their 'analogue' 'originals'. E-banking is just one familiar and now ubiquitous example, along with its inevitably associated e-fraud, which is anything other than merely virtual fraud. Robots (cybernetic devices) of all kinds controlled by executable digital machine code are the decisive step in materializing human mathematico-scientific understanding of movements of all kinds - including physical locomotion, the classic conception of a robot - to create automatons. Cyberspace, in particular, is a dimension in which all sorts of movements (e.g. pop-up advertisements, billing and logistics processes) are automatically triggered in a pre-programmed way through executable binary code embedded in matter. We have quickly become adept at digitizing and thus automating chunks of our practical understanding of the world, so much so that it already seems 'natural'.
The dimension of digitized beings also offers many new opportunities for playing the gainful game that are the doubles of their physical counterparts, e.g. e-casinos and e-markets such as online travel agencies and online stock exchanges. We have little trouble dealing with these virtual digitized entities because they duplicate the physical ones and rely on the same practical understanding. It is nevertheless eery that we encounter our own practical world-understanding, now automated and materialized in another medium that has quickly become familiar, ubiquitous and global. We are oblivious that, without an Aristotle who was the consummation of ancient Greek ontology, a digitally calculable world whose calculability has today been materialized in automatic machine code embedded in its own medium could never have eventuated. The digitization of the capitalist economy is an adjunct of this materialized, outsourced, practical world-understanding, but what capital is and the gainful game in which we are entangled confront us with questions concerning the ontology of interplay, which is an alternative paradigm to that of production, of which the digital casting of the world and digitized cybernetics are perhaps the consummate historical consequences.
The digitization of the logos is a special case of the digitization of beings in general, and is most natural because the logos itself is already a discrete articulation that can be easily broken down further into binary code or bits. Therefore letter correspondence and the postal system are quickly digitized as e-mail correspondence on the internet. But the spoken logos, too, and images of the world's happenings can also be digitized and made vehicles of communication with the aim of sharing an understanding of what is constantly going on in the world.
Written communication does not have to be one-to-one. It can be a general, public sharing of the world's moving facticity first enabled historically by the printing press and the newspaper which employed paper as the medium for the articulated signifiers of a language. Digitization leads to an explosion of news because now factual world happenings can be shared worldwide in writing, voice and image easily and at zero cost. What has happened (historical fact) can also be shared in the same way. Digitized media are global as a matter of course because the worldwide circulation of messages in the electromagnetic medium knows no technical bounds geographically nor with regard to the type of message. All the various media (news'papers', photo journals, radio, television, video) are now one digital medium distinguished only by the source of dissemination. Hence, no doubt, we are suddenly living in a world of global digital communication.
These communications of news of the world's factical movements, however, are shared by countless individuals, each with a different perspective on the world, with a different basic world-understanding against which news events are assessed, evaluated. The basic evaluation is whether the news is good or bad, i.e. whether the factical movement of the world is deemed to be for the good of or to the detriment of humankind in general or particular (a particular country, region, industry, etc.). Insofar news is always political, always controversial and conflictual, concerning as it does the differing particular interests and more universal views of different groups of people. The divides in how news is understood do not depend on the news itself, but on the underlying understanding of the world, and ultimately and crucially, on the individual understanding of the deepest concepts of human being itself such as freedom and justice. The controversy over such issues as such is not a matter of the communication of news, and it is shared and fought out only by relatively few. Otherwise, the controversies continually raging over issues of freedom and justice take place only between divergent positions representing particular configurations within a deeper-lying problematic concerning the question of human being itself. In other words, these controversial issues lead us back ultimately to philosophy, whose movement in time is slowest of all and does not depend on the instantaneous ease with which messages today can be globally communicated.
Political controversies and conflicts of all kinds are waged between differing positions (parties, organizations, segments of the population, etc.) that depend on the dissemination of news messages. Hence there is a continual power struggle to get one's message disseminated and placed favourably, and most news messages have some political import, so that control of the disseminating media becomes a decisive and divisive political factor, for it is important to occupy the news audience's minds with the 'right' messages for the sake of legitimating a particular political constellation, especially a government's rule, or a particular political tendency or struggle. The understanding of world news events is only in part a matter of fact, and more deeply a matter of deeper conceptions of human being itself, which is always also a conception of the world, of how it shapes up for understanding in fundamental categories and concepts.
On both a deeper and a more superficial level, therefore, there is always an ongoing struggle to disseminate one's message and to get it across. The truth of the world at all levels is a power struggle. Getting a more superficial message across depends on the audience's preconceptions and prejudices, on what it is inclined to take in, on what it can understand easily, on what is pleasing or even flattering to it. The dissemination of an average message is therefore a rhetorical power struggle employing all the available techniques of rhetorical persuasion to flatter and thus win over the senses, hearts and minds of recipients. The power struggle over deeper messages is more difficult insofar as such messages are neither news nor views and are therefore not comprehensible in general, but demand for their reception a smaller audience's developed ability to comprehend. Such deeper-lying, but nonetheless crucial questions are therefore pushed into niches or pushed aside altogether in the global communication of messages.
The ease and cheapness with which messages can be communicated through the global network itself causes a problem of the superfluity of messages, of information of all kinds which materially are simply an in-formed electromagnetic medium. We become over-informed without necessarily improving our understanding one whit, for the latter can only take place outside of cyberspace in quiet study. Digital messages can become a kind of plague. We are flooded with messages to the point of over-saturation and of being overtaxed by endless reports on factual movements in the world, to say nothing of advertising messages we would rather do without.
Modern communication theory was founded in the wake of the emergence of electromagnetic communication media such as telegraphy, telephone and radio. With a mathematically brilliant, seminal paper published in 1948 entitled 'A Mathematical Theory of Communication', Claude E. Shannon, an engineer and mathematician, is generally regarded as the father of modern communication theory. It is no accident that precisely a mathematical theory was accorded the honourable status as founding theory and that this mathematical theory deals first of all with digitizable discrete communication between sender and receiver, which are the appropriate physical beings for transmitting and receiving discretely generated, digitizable messages. Human beings as communicators are initially put to one side so that communication is conceived as machine communication. As an engineer, Shannon was interested solely in the efficiency of getting a message generated at a source through a medium to a receiver where the message is to be reconstituted with as few errors as possible. The amount of information transmitted in unit time was of prime concern, regardless of the message's content, let alone its interpretation and meaning. Abstracting from qualitative content left a quantity of information, whose amount was measured by the rate at which information was generated by the source and the channel's or medium's capacity to convey information in bits per second as the appropriate mathematical measure of the entropy of a source generating symbols (e.g. letters in a natural language, image pixels) with certain probabilities.
In digital code, of which a message considered as a digital being is composed, the information content is appropriately measured by the number of bits required to encode it, each bit representing a power to base 2 of binary code. When a bit code is extended by one binary place, the amount of information encodable together with the extra bit doubles. Therefore, the appropriate measure of information content is the logarithm to the base two, which gives the number of bits required for encoding and thus the amount of 'freedom' in 'choosing' a message, i.e. the amount of entropic digital difference among possible messages. A bit-coded message has to be conveyed through a channel whose capacity is likewise measure in bits (per second). Since the stochastic process probabilistically generating a message in a natural language is not completely random, but constrained by the probabilities of certain frequencies and sequences of letters or characters in the language, there is also statistical redundancy in the message and therefore a potential for saving channel capacity (bandwidth) by transmitting just enough bits to reliably and correctly reconstruct the message as a sequence of bits at the receiver's end.
Shannon proceeded first by assuming a transmitting source discretely generating a finite number of symbols that first had to be encoded by a transducer to produce a signal to be transmitted electromagnetically through the channel which, once received, first had to be decoded back into a legible symbol. Discreteness is the appropriate place to start for considering the transmission of a message in an articulated, finite logos of some kind since, as we have seen in Chapter 2.3 on arithmological knowledge, the finite logos has an affinity in essence with calculability. The theory of discrete, finite message generation and transmission was then extended to consider noise interfering with the transmission of the encoded signal, thus giving rise to errors for which the transmission procedure had to make allowances. Finally, the theory was extended to cover continuously generated messages such as (radio) voice or (TV) moving image by the usual approximation and limiting procedures familiar from differential and integral calculus. Shannon was therefore interested in productive power, i.e. the power to bring about a change, namely the correctly delivered message, at the receiver's end. This abstract mathematical perspective is the appropriate one for setting up a reliable, effective process of message transmission, entirely regardless of the content of the messages transmitted which may equally well be highly semantically charged, like James Joyce's Finnegans Wake, or trite.
In subsequent work, Shannon's collaborator, Warren Weaver, broadened the communication model beyond a purely mathematical engineering problem to take into account the social dimension of communicating messages. The effectiveness of message transmission therefore no longer had a merely technical measure, but a (non-mathematizable) social one, namely, "the success with which the meaning conveyed to the receiver leads to the desired conduct on his part". Now the criterion of effective communication is no longer simply a message correctly decoded, but a message whose meaning is correctly interpreted and executed in the sense that the recipient's behaviour accords with the sender's desires. This is now a question of social power (cf. my Social Ontology Chapter 10 and my 'Social Power and Government' 2010) that, at least tacitly, brings into play the interplay of acknowledgement and estimation among social players in communication without, however, ever clarifying the ontology of social power.
One communication theorist, Harold Lasswell, characterizes the seminal Shannon-Weaver theory as regarding communication as a matter of "who says what in which channel to whom and with what effects?"(cf. collaboratory accessed July 2010. ) This characterization makes it apparent that modern communication theory proceeds unwittingly and unquestioningly from a problematic of social power whose origin lies in Plato's and Aristotle's treatments of rhetoric (cf. my 'Assessing How Heidegger Thinks Power Through the History of Being' 2004 Section 3. 'Rhetoric as a test case for power over the other'), without ever unearthing this source. Subsequent elaborations of the Shannon-Weaver theory introduce other communication 'factors' such as feedback (N. Wiener 1948), two-way communication between 'communicators', gatekeepers filtering or censoring information-transmission between source and recipient (a kind of social noise distinct from technical noise interference, although the two can be coupled, e.g. in politically motivated transmitter jamming), the social status of 'transmitter' and 'recipient', social contexts, etc. The core problematic of communicative power, however, is tacitly adopted and left unchallenged and unclarified by such later theoretical 'models', which usually have merely schematic form.
6.3. The intermeshing of the movement of digital beings in the global network and the movement of value as capitalThe efficiency of the cybernetic digital network is welcome to the movement of capital because, as we have seen in the preceding chapter, both productivity increases and the acceleration of turnover time boost the self-augmentation of value by enhancing the chances of coming out on top in the competitive struggle for gain. Capital therefore slips into the global digital network like a hand into a glove. The speed with which messages can be communicated accelerates the circulation labour and shortens the circulation phase of capital. Moreover, the near-zero reproduction costs of digital code lead to dissemination throughout the global network and massive cost reductions for many productive and circulation functions of capital. A banal example: invoices can be communicated, i.e. billed, by sending a digital being (a digital-electronic invoice-file) through the network both quickly and at zero cost. Both tendencies lend themselves to maximizing the value-augmentation-formula for capital's gainful movement: dM/dt = (M'-M)/(t'-t) (cf. 5.5 Time in a capitalist economy).
Consumers too, who are just as much enticed by and caught up in the gainful game as capital itself, benefit from the near-zero cost of all sorts of digital 'messages' in the global electromagnetic network, where such messages comprise written texts of all kinds, music, photo, film, etc. Although the original 'production' costs for a digital being may be considerable (programming, writing, recording, photographing, filming, etc.), the reproduction and distribution costs are next to zero, requiring only the cost factor of the electromagnetic medium itself. This provokes the question whether the deeper-lying telos of the global digital network is for the sake of sustaining, expanding and accelerating the movement of value as capital and, more generally, whether it is bouyed by the striving of all economic players for gain, which can take the form simply of saving money.
The will to power in the guise of total digital cybernetics therefore dovetails neatly with the striving for gain, especially with capital's incessant striving to bloat value with surplus value through the course of its circular movement. It is therefore justified to speak of an inversion of human purposes behind our backs, for what seems to be simply a desire for ease of communication globally and for cheaper, more convenient products and services, turns out to further something unintended, but perhaps inkled, viz. the will to (cybernetic, pro-ductive) power over all movement and the incessant acceleration of the rivalrous power play of all kinds of powers (labour powers, personal skills and abilities, productive powers of means of production, the power of money as capital, the power of land and sea as factors of production) for the sake of monetary gain.
The discourses of the media communicate, and must communicate, in terms of people's average understanding and cater to their tastes and what they want to hear. Average understanding is broken down into many different segments, some of which are inconsistent with one another, or downright contradictory, whilst nevertheless co-existing, each catered to by a segment of the media. But underlying them all as the 'natural' foundation of average understanding is the unquestioned, hegemonic mathematico-scientific worldview, if only because it is immensely effective in shaping our world on all levels. Furthermore, scientific method consists in a theoretical modelling of 'the facts' which, again, as mathematically souped-up common-sense, appeals to it. Truth is measured by undeniable effectivity. The esoteric discourse of mathematical physics, especially in connection with deeper questions concerning the cosmos, is unquestionably given credibility in the media because of its experimental basis in 'the facts' (measurable) and the incontestable effectiveness of such discourse in life-shaping phenomena such as cars, aeroplanes, power plants, atomic weapons, etc., etc., whereas the esoteric discourse of philosophical thinking is regarded as speculation in the pejorative sense or as a matter of personal taste and values. Even an apparently critical questioning of science in media discourse is unable to unearth anything like the simple ontological presuppositions of the modern casting of world, because such an unearthing requires a kind of questioning alien to any segment of people's average understanding, including especially the complacent, know-all world-views of the highly educated and scientifically trained comfortably established in our age.
Our communication consists for the most part in keeping in touch and keeping abreast in terms of that which we always already know, namely, the diaphanous categories and basic concepts within which an historical world shapes up. Only the facts, the ontic occurrences change endlessly within an unchanging, settled, historical constellation of world-understanding particularized into countless configurations as individual, differing, and often opposed, world-views. The fundamental historical casting of the world is taken for granted and as such remains invisible. To question the unquestionable amounts to leaving the community of common-sense in which communication about what is already all too firmly understood, whether digitally enabled or not, continually takes place.
An historical world shapes up only in an interplay, strife and struggle between hiddenness and disclosure in which the foundational categories of a world are forged and recast.(23a) Only insofar as we belong to this interplay and struggle can we take cognizance not only of what the case is, or adopt a political stance toward the state of the world and its injustices, but also engage in questioning the categories enabling, in the first place and from the ground up, the world to be understood as a world. Taking cognizance of beings in their respective modes of being is our destiny as human beings. Mostly we understand beings in their being implicitly, and without such an implicit understanding, we would not understand the world at all. Information as news is one way in which human being takes cognizance of the factual states of affairs in the world and their movements within an implicit, tacit categorial understanding. Taking explicit cognizance is philosophy.
As we have seen, such philosophical knowledge turns out to be a knowledge of whence (a)rxh/, ge/noj, ti\ h)=n) and what (ei)=doj) enabling knowing insight into how beings come to presence (du/namij, e)ne/rgeia, e)ntele/xeia cf. 2.9 Time and movement in Aristotle's thinking).Western ontological knowledge is a pro-ductive understanding of being that in its precipitates today confronts us as an oppressive wealth of productive scientific and technological knowledge that can now also be made digitally-cybernetically effective in automatons of all conceivable kinds. The question, however, concerns another kind of knowledge that is, in a certain sense, a not-knowing: What insight can we gain into that which eludes a knowing, in the sense of a productive grasp? What possibilities still lie latent in the first Greek metaphysical beginning that were addressed there already in passing, but had to be excluded or pushed to one side for the sake of productive knowledge? We have pointed to one such possibility above (cf. 5.7 Recovery of the three-dimensional, complexly interwoven social time of who-interplay), namely, an alternative to the conceptualization of movement enabling control from a single (e(/n) a)rxh/, which had to be demarcated vis-à-vis its polyvalent opposite (polla/), thus provoking the alternative question: What happens to the ontological structure of movement when more than one a)rxh/ comes into play? Movement as the transition from something to something (e)/k tinoj ei)/j ti) then becomes interplay among two or many sources of power, including in particular human beings as sources of power. Interplay is not the causal power play among whats resulting in productive control, but a non-precalculable, and in this sense uncontrollable play among whos. This already represents a rupture with the will to power pure and simple underlying productivist metaphysics, under whose very success today all is "sicklied o'er" with the digital cast of being.
With the plurality of interplay, the Anaximandrian-Aristotelean question concerning justice(24) (the pro/j ti among human beings, and perhaps among all beings) is posed anew in an alternative ontological landscape. Being kata\ sumbebhko/j (contingency, or that which comes along without ground) - which as the opposite of being kaq" au)to/ (being in itself) had to be excised from metaphysics - then invites renewed interest as a feature of interplay in its essential unpredictability and incalculability. Those features of phenomena that fail to offer an unambiguous, determinate, definite look and withdraw partially into the hiddenness the Greeks call lh/qh from knowing, controlling insight then incite questioning attention. With such questions, we have already taken a step beyond digital ontology.
We must not pass up the opportunity to draw a corollary from the discussion of Aristotelean movement and time (cf. 2.9 Time and movement in Aristotle's thinking) with regard to the obfuscation of the phenomena practised by modern physics. The Cartesian cast of modern knowledge prescribes that all phenomena must be approached by way of measurement to determine quantities that are entered into equations which, in turn, can be manipulated mathematically according to a mathematically formulated theory. This is accepted today unquestionably as the paradigm of scientific method. If the focus is on quantities and their measurement already from the outset, so that there is nothing to consider beforehand, then the ontological structure of the phenomena themselves, i.e. their modes of presence in the world, is obscured.
This obscuring, or a certain vacillation, is indicated already by the doubling of terminology for the Heisenberg principle, which is called both the indeterminacy principle and uncertainty principle or, in the original German, Unbestimmtheitsrelation and Unschärferelation. Uncertainty, however, refers to, and is mostly understood as, a lack of sharpness in principle in the observed measurements of phenomena of motion at the sub-atomic level, namely, the motion of entities such as electrons, protons, neutrons, photons and many other sub-atomic entities whose existence has been inferred from experiments based on theories of the physics of very small entities imperceptible to the unaided senses. The uncertainty principle is then understood as a limitation in principle to the accuracy of experimental measurement observations of the motions of sub-atomic entities due to the unavoidable interference to the physical system observed caused by the physical process of observation itself. The macroscopic clumsiness of the experimental apparatus needed to make sub-atomic motion visible to the human senses is said to introduce myriad hidden, uncontrollable variables, a viewpoint based on still unshaken conceptions of causality from classical mechanics. Thus, for example, to determine observationally the position of an electron, a photon is 'shot' at it, which itself disturbs the electron's position and thus causes it to be observed somewhere else than where it would have been if it had not been observed. On this conception, if the system is left unobserved and therefore un-interfered-with, it evolves over time according to deterministic laws of motion, and the uncertainty principle becomes almost common sense.
Werner Heisenberg's deeper insight is that it is in principle already in theory, and not just in experimental practice, impossible to determine accurately, say, both the position and speed, or both the position and momentum, of sub-atomic entities in motion (and they are always in motion), quite independently of whether they are experimentally observed or not. Hence we read in the article on the Heisenberg "uncertainty principle" in Encyclopaedia Britannica, "that the position and the velocity of an object cannot both be measured exactly, at the same time, even in theory. The very concepts of exact position and exact velocity together, in fact, have no meaning in nature".(26) In the article on "physical science", section "quantum mechanics", Stephen G. Brush points out, "Heisenberg's principle is often called the uncertainty principle, but this is somewhat misleading. It tends to suggest incorrectly that the electron really has a definite position and velocity and that they simply have not been determined." To have any meaning within modern physics, such a statement of uncertainty or indeterminacy must have a quantifiable, i.e. mathematical, probabilistic formulation in the theory itself, which is then checked against experimental evidence that, in the case of quantum mechanics, apparently(27) has confirmed results produced by theoretical physicists, including illustrious names such as Planck, Einstein, Bohr, de Broglie, Heisenberg, Schrödinger, Dirac, et al.
What the empiricist methodology of science overlooks, however, is that the theory has always already seen and precast, in its fundamental concepts won by thinking on the phenomena involved, more than could ever be checked through empirical observation. By the time science has thought up an hypothesis and set up its experimental harness to gather the hard factual data, the ontological horse has long since bolted. These precast theoretical concepts prescribe already where the empirical evidence has to be looked for to confirm or refute theoretical predictions concerning the behaviour in motion of physical systems which, in this case, happens to be on the sub-atomic level that is observable only through sophisticated macroscopic apparatuses that allow the physicist to bodily participate in an experimental observation construed according to the theoretical pre-casting.
To satisfy the prescripts of mathematical science, Heisenberg's indeterminacy can and must be expressed 'rigorously' in an equation for a sub-atomic entity such as an electron: (Dx)(Dp)>=(h/4p), where x is the electron's position, p is its momentum, which depends on both its mass and velocity, h is Planck's constant, and D stands for the standard deviations of the probability distributions for x and p, respectively. Thus, Dx is interpreted as the standard deviation of the probability distribution for the spatial position of an electron e, i.e. the quantified chance of e 'being' at position x over an entire infinite range of possible positions that exhaust the possibilities for where e could be, where 'be' is tacitly assumed to imply a determinate, unambiguous 'here'-answer. The interpretation of indeterminacy as a probability distribution is forced by modern physics' having to quantify all physical phenomena, including the phenomenon of indeterminacy, for probabilities are still mathematically calculable and insofar ascertainable. Even in the many-worlds interpretation of quantum mechanics, the many indeterminate positions of e are finally reduced by probability calculation to a real, observable position. This is the indeterminacy in e's position, i.e. its position as a moving entity cannot be pinned down together with its momentum, and the lack of accuracy can never be reduced toward zero under any circumstances, because the product of the indeterminacy of its position and the indeterminacy of its momentum of motion is always at least as great as a positive, albeit extremely small, real number, h/4p, of an order of magnitude of minus 34 to base ten in units of joule-seconds, or energy times time (the action of an energy and thus a motion).
In mathematical language, the indeterminacy of position and momentum taken together manifests itself in the non-commutability of these two dynamic variables describing a dynamical system, or in the vernacular, indeterminacy means position and momentum cannot be nailed down together. The units of the Planck constant, energy times time, are already a hint that movement over time here comes into play to account for indeterminacy of position and momentum together. The two parameters could just as well be taken to be position and velocity, for these, too, suffice to specify a dynamical system. The ineradicable indeterminacy of position and velocity taken together points to the impossibility of instantaneous velocity and to Zeno's ancient paradox of the flying arrow.
One may object that the uneliminable indeterminacy in x and p (or, equivalently, x and velocity, v) taken together, which dynamical variables specify the state of a physical system, applies only to actually obtained experimental observations of these two parameters due to interference with the dynamical system by the measurement process itself, and not to the system's state prior to or independently of observational measurement. This viewpoint would seem to be supported by Paul Dirac himself, one of the founders of mathematical quantum mechanics, when he writes,
According to classical ideas one could specify a state by giving numerical values to all the coordinates and velocities of the various component parts of the system at some instant of time, the whole motion being then completely determined. Now the argument of pp. 3 and 4 [regarding disturbances to the system by the process of observation] shows that we cannot observe a small system with that amount of detail which classical theory supposes. The limitation in the power of observation puts the limitation on the number of data that can be assigned to a state. Thus a state of an atomic system must be specified by fewer or more indefinite data than a complete set of numerical values for all the coordinates and velocities at some instant of time.(28)The indeterminacy would thus arise from a lack of observational data, from a want of 'experimental facts', which is a common-sense viewpoint. But Dirac then goes on to introduce the "principle of superposition" that postulates a superposition of dynamical states necessarily prior to any observation - necessarily, because any observation (according to physics' implicit ontological decree) can only ever determine a 'real' determinate state, and not an 'unreal' or 'imaginary' superposition of states (hence also the imaginary or complex numbers employed in the mathematics of the famous wave function that describes system states). The principle of superposition in quantum mechanics therefore refers to the indeterminacy or 'hovering' of states of a dynamical system independently of observation. From the indeterminacy, or rather observed variation, of the results of observation in sub-atomic experiments, results which nevertheless exhibit regularity, the theoretical principle of complex superposition of dynamical states was inferred, or rather postulated (like any other law of motion - which in itself is unobservable), and this indeterminacy of experimental results was then explained by saying that the experimental apparatus employed in the experimental process, which is itself a physical dynamical system, interferes with the original system, causing the wave function of a single quantum 'particle' to 'collapse'. But an explanation is something different from a postulated principle of quantum mechanical theory, viz. the principle of complex superposition. The explanation only explains something regarding a so-called 'quantum leap' from an indefinite superposition of states to a definite observed state.
(29) to interpret the situation as a disjuncture between an unobserved, closed dynamical system describable determinately by mathematically expressed physical laws such as the Schrödinger equation, on the one hand, and the irreducible indeterminacy of experimental observations, on the other. Such an interpretation confuses determinacy with (effective causal) determinism (cf. below). Even the Schrödinger equation is formulated in quantum mechanics as a complex superposition of (usually) infinitely many dynamical states, and in the superposition lies already the quivering indeterminacy, so that the deterministic evolution over time described by the Schrödinger equation is beset from the outset in its 'innards' by indeterminacy in the very superposed entity (a quantum described as a complex wave function) whose evolution is being determined by the equation, for this superposed entity is described by "complex probability amplitudes which weight our linear superpositions", as Roger Penrose notes in his The Emperor's New Mind (1989, 1999 p. 332). The squared moduli of these complex probability amplitudes (a real number) then serve as the "probabilities describing actual alternatives" (ibid.) when a real, determinate observation here-and-now is experimentally forced upon the wave function by interrogative scientific method.
Notwithstanding this, Penrose writes on the one hand, "Quantum-mechanically, every single position that the [single quantum] particle might have is an 'alternative' available to it. [...] The collection of complex weightings describes the quantum state of the particle. [...The] state of an individual particle [...] described by its wavefunction [...] involves our regarding individual particles being spread out spatially [...] The quantum state of a single (spinless) particle is defined by a complex number (amplitude) for each possible position that the particle might occupy." (p. 314, 325, 356) But, on the other, he asserts, "[the wave-function] y is governed by the deterministic Schrödinger evolution. [...] It is the procedure R [wave-function reduction/collapse ME], and only R, that introduces uncertainties and probabilities into quantum theory" (p. 323). Effective causal determinism in time (as prescribed by an equation), however, is not the contrary of indeterminacy (of the superposed wave function), since these two go hand in hand; nor is Heisenberg's 'Unbestimmtheitsrelation' appropriately translated into English as 'uncertainty principle'. The "non-determinism of quantum theory" (p. 383) must be distinguished from the strange complex indeterminacy of the superposed quantum wave function which formulates unwittingly a deeper truth about physical entities in general, namely, that a physical being (i.e. capable of movement) is, in a certain way, also where it is not, even though it does so within the framework of one-dimensional time composed of present instants, as if the alternative positions for a quantum entity all had to 'be' lined up next to each other at the same time-point. By virtue of the complex-number character of the "probability amplitudes" in the wave function, one might equally well say that a wave-function entity is nowhere real. ("[A]ccording to quantum mechanics, in general the notions such as 'here' and 'now' could have only indefinite or potential meaning." Joy Christian 'Why the Quantum Must Yield to Gravity' 2001.) One could perhaps even go one step further to assert that a wave-function quantum-particle is not a unified something, i.e. neither e(/n nor ti/, but an imaginary superposition of infinite potentialities that only becomes real and definite, thus excluding infinite possibilities of being present, upon experimental interrogation, or the intervention of gravity (Penrose 1989, 1999 pp.475f).
What is the nature of Penrose's wave-function collapse/reduction operator, R, that compels the indeterminacy of superposition of many, or even uncountably infinite, potential phase-states through a probabilistic quantum leap into unique definiteness? Penrose writes, for instance, presumably confusing discreteness with definiteness, "discrete states of an atom, for example are those with definite energy, momentum, and total angular momentum [which, however, assume values within a real, non-discrete continuum ME]. A general state which 'spreads' is a superposition of such discrete states. It is the action of R, at some stage, that requires the atom actually to 'be' in one of these discrete states". (p.521) Here, R is sounding awfully like the lo/goj that gathers wavering indeterminacy into the well-defined definiteness of being such that something becomes visible as such to the human mind. The difference is that the physicist, Penrose, imagines some kind of real, 'material' process in R, thus overlooking that even 'matter' is an idea. Penrose's proposed 'one-graviton' criterion for the onset of wave-function collapse (pp. 475ff) suggests that the world falls into definite place under the effects of gravity, whereas the gathering of the lo/goj is simply the eventuation through which world shapes up, discretely and finitely, as world. Is it quantum gravity that grounds beings as such, or is it being itself that "presences within itself as grounding" (Sein west in sich als gründendes. M. Heidegger Der Satz vom Grund Neske, Pfullingen 1986 p. 90.)? Is the "anthropic principle" Penrose adduces, "which asserts that the nature of the universe that we find ourselves in is strongly constrained by the requirement that sentient beings like ourselves must actually be present to observe it" (p. 524; cf. pp. 560ff), merely the latest 'scientific' edition of the famous Parmenidean belonging-together of being and awareness (Frag. 3)? And when Penrose adduces "insight" (p. 541ff and passim) in order to show the "non-verbality of thought" (p. 548ff) that sees more than any algorithm could ever achieve computationally, does this not amount, unbeknown to him, to a latter-day resuscitation of Aristotelean nou=j?
In the complex-imaginary superposition of dynamical states resides also the incommutability of the operators for position and momentum, or position and velocity, of a dynamical system. With observational indeterminacy, which in turn induces the postulation of a theoretical indeterminacy (complex wave-function superposition of infinite possibilities for the phase state of a quantum entity), modern mathematical physics has come up against an (unbeknowns to it: temporontological) obstacle in its striving to govern motions of all kinds by mathematically formulated laws and has had to retreat from totally precalculable determinacy to probabilistic precalculability, which is still quantifiable and calculable. This is still removed, however, from the insight that movements can have wholly unforeseen outcomes or that they can be free and that physical beings per se, as changeable, are not solely present at an instant....daß unter der siegreich gebliebenen Wirklichkeit unzählige Möglichkeiten liegen, die auch hätten wirklich werden können.
The so-called 'uncertainty principle' says that position and momentum, or position and rate of change of position, cannot be seen together or 'at once'. There is a blurring or quivering. Position is the place assumed here and now, whereas rate of change of position, or motion, is not instantaneous, but involves both here-and-there and now-and-then, i.e. an ecstatic stretch of both time and space together, albeit possibly very small. Mechanics set out to lay down the physical laws of this motion, proceeding from the starting-point or a)rxh/ of here-and-now. There-and-then thus become controlled, predictable, precalculable from here-and-now. In classical mechanics, the laws of motion are deterministic, which means that, given the dynamical situation at one instant, t, the dynamical situation at a later instant t' can be calculated by applying the laws of motion to form the appropriate dynamical equations and solving them. This precalculation involves the differential calculus with infinitesimals through which an instant in time can be approached. In quantum mechanics the laws of motion become probabilistic because, surreptitiously, or rather unwittingly, an instantaneous here-and-now is no longer assumed in postulating the non-commutability of position and rate of change of position or, what is the same thing, the superposition of dynamical states. A (perhaps infinite) multitude of heres is admitted, whilst hanging on to a single now-point.
Non-commutability means that it makes a difference in which sequential temporal order position and velocity are measured. The quivering of the physical entity in space-time (which is something different from the disturbance to the dynamical system caused by any observational measuring process) introduces an indeterminacy in the position and rate of change of position, the parameters defining a physical entity in motion, taken together. The space-time co-ordinates of a physical entity become the probability amplitudes of the physical entity's having a given position at a given fixed instant or now-point, t, these probability amplitudes (defining the entity's potential to be at a certain point in space and definite point in time as a superposition of infinite possibilities) being derived from the physical entity's wave function which "is a complex function of the position eigenvalue x' [... that] can be used to reconstruct the state ket |A>," itself an integral over the infinite-dimensional ket-space of position spanned by the eigenkets of position(30). Fitzpatrick notes that for "a simple system with one classical degree of freedom, which corresponds to the Cartesian coordinate x [... a] state ket |A> (which represents a general state of the system) can be expressed as a linear superposition of the eigenkets of the position operator." (ibid.) Even for such a simple system, the complex wave function says that the physical entity is spread out everywhere in the x dimension.
This description still falsifies the situation because time itself is not composed of now-points, t, but, as we shall see in more detail below (7.2 The necessity of introducing three-dimensional, ecstatic time), is itself three-dimensionally stretched. Only the counting of time introduces now-points which, however, also introduce the antinomy between time conceived as continuous and time conceived as countably discrete (see 7.3 The phenomena of movement and indeterminacy in relation to continuity, discreteness and limit).
For normal everyday purposes in situations with macroscopic objects in motion, the indeterminacy in position and momentum or, equivalently, in position and speed of a moving object is said to be so small that it cannot be detected at all by any possible experimental arrangement, i.e. it cannot be measured, and thus 'scientifically' observed and is therefore beyond the bounds of what physical science can know according to its method. The indeterminacy pertains in theory nevertheless, however, independently of the practice of experimental observation. The laws of classical Newtonian mechanics, which make no theoretical allowance for indeterminacy, therefore apply practically to normal macroscopic situations as opposed to sub-microscopic, sub-atomic situations. The theoretical error is said to be negligible in practice, and no quantitative correction has to be made in terms of quantum mechanical considerations. But that does not mean that the same indeterminacy considerations cease to apply in principle, only that it cannot be experimentally detected and confirmed. The dynamical situation for macroscopic systems is thus treated theoretically as the extrapolation of the dynamical situation for microscopic systems (or macroscopic systems in terms of sub-atomic entities) which itself is accessible to sophisticated experimental verification and falsification. Quantum mechanics, it is claimed, provides a truer theory of physical reality, and normal, everyday, macroscopic physical beings are made up of quantum entities. If the quantum-mechanical access to normal, everyday, macroscopic physical beings in principle allows for quantum-mechanical indeterminacy, the issue then becomes whether this indeterminacy is adequately conceived, quite apart from any neglible or non-neglible errors in calculating motion. Insofar there is a deviation from strict empiricist methodology, for no experimental observations could ever be presented that would either confirm or deny quantum-mechanical indeterminacy of position and rate of change of position for macroscopic systems, the numerical scale of the indeterminacy being beyond any possible practical measurement that could be made.
The term 'quantum' refers inter alia to the peculiar duality associated with sub-atomic entities which have been shown experimentally to exhibit the characteristics of both particles and waves. Light waves, for example, are quantized as photons or tiny packets of light that exhibit particle characteristics. This was shown by Einstein. De Broglie proposed an equation, subsequently confirmed experimentally, linking the particle characteristics and wave characteristics of all sub-atomic entities, namely, l = h/p, where l is wavelength (of a wave) and p is momentum (of a particle). This means that a sub-atomic entity, whatever it may be, is also a wave (a spatial vibration) with wavelength l exhibiting wave phenomena such as interference, and by virtue of this dual nature it cannot be precisely localized to a determinate point-position, even in principle, i.e. even in theory. Its position is 'spread out' over a wave length so that the chance of finding it at a determinate point, when it exhibits particle characteristics, is given by regarding its associated wave as a probability distribution for its position, one accepted interpretation of the famous Schrödinger equation. The 'spreading-out' of the position of a sub-atomic entity when considered as a wave is therefore not a spatial spreading-out or 'spreading-thin', but a quivering indeterminacy that eludes visualization (Heisenberg(31) was the one who warned against the desire to visualize quantum-mechancial states of affairs). The wave itself is a quivering indeterminacy of position and momentum taken together.
Probability distributions are still mathematical entities occurring in equations that can be algebraically manipulated and calculated, and probability provides the bridge in the duality, or vacillation, between considering the same sub-atomic entity either as a wave or as a particle (or sub-atomic particle as a wave!) as suits the context, whether it be theoretical or experimental. In experimental practice involving the taking of determinate, observable measurements on macroscopic experimental apparatuses, the sub-atomic entity conceived strictly as a wave is thought to 'collapse' into a determinate state appropriate for a particle so that it becomes a 'real' res. This apparently observed, i.e. measured, collapse of the wave function is most perplexing and to the present day gives rise to controversy within mathematical quantum mechanics as to how it is to be interpreted physically. In any case it is to be noted that the indeterminacy relation between position and momentum, insofar as it is conceived to go hand in hand with the dual nature of sub-atomic entities as both particle and wave, is postulated as a spatial indeterminacy; a temporal indeterminacy and what this could mean do not appear on the quantum physicist's list of perplexities.
But what does all this say about physical entities in motion prior to (Cartesian) quantification and mathematization? What is motion? This question is invariably skipped over in taking the phenomenon itself for granted. As we have seen from the review of Aristotle's thinking on movement (2.9 Time and movement in Aristotle's thinking), physics is the study of those beings that (can) move, kinou/mena. Beings at rest are physical only insofar as they are also able to move, i.e. rest for Aristotelean physics is a limiting case of movement (but what is a limit?). Hence numbers, for instance, are not physical, for they are outside movement altogether (to make numbers move, they have to be conceived as variables with respect to a variable for time, t, a crucial step in the development of differential analysis, or some sort of movement, such as a counting process, has to be introduced to make the numbers flow). In modern physics, movement is thought first and foremost as motion, i.e. locomotion or change of place, which is mathematized as functions on four-dimensional space-time in which place has become position, and position is expressible as a Cartesian co-ordinate (x, y, z). All other types of movement, in order to be mathematized, must likewise be converted into change of a magnitude whose rate of change can be calculated, thus reducing the scope of the Aristotelean panorama of the phenomena, but with the gain of being able to mathematize them all in algebraic, usually differential equations.Schließlich wird derselbe Fehler gemacht, wenn man, wie es in der Quantentheorie geschieht, die Zeit als einen reellen Parameter beschreibt. Zeit ist mit Uhren meßbar. Der Zeitpunkt ist eine Fiktion. Er könnte wiederum nur durch einen irreversiblen Vorgang, und durch diesen nur mit endlicher Ungenauigkeit bestimmt werden. Durch diese ungelösten Fragen weist die Quantentheorie zwar nicht in die klassische Physik zurück, aber über sie hinaus.(31a)
We have seen that Aristotelean movement is characterized by a twofold presence, namely by the presence of the being in its potential (or power or propensity) and the absence of its realization. All physical beings, as beings that can move, have a tendency toward an end in which the potential attains its end. The potential itself, in coming to presence as such is realized and is on its way to attaining its end. In coming to presence as such, the potential is at work, and this situation of being at work is its actual movement toward an end that is absent (where this end may be rest, or a perfected motion, such as circular, or conceivably elliptical or uniform linear, motion). Energy is therefore the Aristotelean term for movement expressing its ontological structure as the being-at-work of a potential under way toward its end, a conception still implicitly underlying modern physics' concepts of work, action and energy, despite modern physicists' being ignorant and arrogantly dismissive of Aristotle. A moving being is therefore not merely present, but also simultaneously absent as under way toward... This being-under-way-toward... may be called its momentum. Momentum itself refers to an absence, to a not yet, so the moving being is both present and absent, i.e. its being as moving is both a presence and an absence together. In motion, a physical being is both here and not here. Therefore, we note first that its position is indeterminate.
A definite position for a physical body in motion cannot be tied down. More generally, a determinate state for a physical body in (one of the four types of) movement cannot be tied down. A definite position is only presence here, i.e. where the physical body in question is now at a certain point in space (amenable to geometrization and mathematization), thus eliminating motion by reducing time to an instant. But motion as motion refers also to the not-now, i.e. to an as yet absent future in which it will be somewhere else, and not here at this point. A physical body in motion is therefore both here-and-now in a position and also there, but not yet. It therefore has no definite, determinate position only now but is, as moving, both here and there, now and then, which is already present in its being withheld. In its motion, it is now here and has also always already left its now-position on its way to somewhere else then. As in motion, it is also futural toward a presence that is still withheld(32) in absence. To reduce the being of that which is to that which can be ascertained to be present at a point (or instant) in time, t, at a certain point positioned in space like a point in a geometrical figure is to deny the very phenomenon of movement altogether - the moving being's being as moving is not only presence but simultaneously its future, which is still absent, and its position is stretched or quivering in a propensity between here and there, now and then.
Similarly, a moving being that is now here is simultaneously just arrived from somewhere else where it has been and is no longer, but which is still present as an absence. Simultaneity here can no longer mean, as it usually does, the coincidence of two now-points or instants, but rather the sameness of time of simultaneity (from L. simul 'at the same time') is here to be understood as the tight togetherness or 'at once' of the three temporal ecstasies of past, present and future within the unified three-dimensional structure of time, where dimension is now not conceived quantitatively as a Cartesian mathematical dimension, but as a space 'measured through' or 'traversed' (from Gk. diametrei=n and metrei=n). This three-dimensional ontological structure cannot be captured by conceiving time as a continuum of successive instants, one after the other, and in truth introduces a conception of time foreign to both classical and quantum physics, whether relativistic or not. Rather than confront itself with the phenomenon of three-dimensional time staring it in the face, today's most advanced quantum gravity theory would rather escape to the esoteric dimensions of 'parallel worlds' in super-string theory, thus also pandering to human curiosity in strange and grotesque sci-fi scenarios. Traditional conceptions of time tacitly presuppose that time itself is determinately present, i.e. the now as the instant of time. This positive conception is complemented by two negations of now-time as time that is no longer and time that is not yet, without the phenomenological sense of these two negations coming into their own as a refusal or withholding of presence. Insofar as the being of time is thought simply as presence and its negation, there is no ambiguity or indeterminacy in time.
But the three-dimensional, ecstatic conception of time introduces an indeterminacy by tying together the three ecstatic dimensions in an inseparable 'at once' of presence-and-absence, so that the absence of later and earlier is present as a specific absence. This is apparently a self-contradictory, logic-defying formulation as long as it is taken for granted that presence and absence exclude each other. The phenomenon of movement itself, however, compels us to learn to see, although this may be difficult, to see that the future, although absent, is present as an absence along with the now in three-dimensional ecstatic time. To put it negatively: it is a misrecognition of the phenomenon of time itself to conceive it as a succession of now-instants, each one relieving the preceding one in coming to presence, just as it is a misrecognition of the phenomenon of movement (in any sort of phase-space) to regard it as movement along a continuous geometrical line composed of now-points, a conception congenial to mathematization.
A being in motion is arrived from where it was, i.e. whence it is come, just as it also has a momentum and is underway toward where it will be, or whither it is going, although both this where and when are still absent, still withheld. Past, present and future are connected and hold on to each other in a togetherness of presence and absence, and this is the continuity of its motion in three-dimensional time. Accordingly, any physical being in motion (and not just sub-atomic, quantum-mechanical entities described by a wave function) does not have a determinate position at a determinate time, but, granting for the moment the questionable three-dimensionality of Euclidean space, is six-dimensionally stretched and quivering potentially into three spatial and three temporal dimensions. This phenomenon of time-space can only be seen ontologically before any mathematization sets in, i.e. before any numbers are lifted off the phenomena related to movement, because the mathematization of time and motion misleads us to conceiving a physical being in motion as being simply in a present state at any instant of time, moving along some sort of continuous geometrical line as time itself moves along its linear time-line of successive now-points. A physical being in motion is under momentum from here-and-now toward there-and-then. It is here-and-gone, underway from now to then when it will be. Similarly, a physical being at rest is here-and-there, quivering in an indeterminacy between now-and-then when it will be what it can be (potential).(33) Whether moving or at rest, a physical being is futural, i.e. it exists also as what it will and can be, ex-sisting, i.e. standing-out three-dimensionally somewhere and three-dimensionally sometime. Temporally, a physical being exists once, now and later, where the verb 'to exist' is conjugated grammatically in its tenses as: it existed once, it exists now and it will exist later, this three-dimensional existence is all at once. This formulation includes the respective negations, such as once it did not exist. To summarize: existence must not be truncated to now-presence. All three moments of a moving physical being's temporal being exist together, at once, in a presence that includes the presence of two kinds of absence, namely, the refusal of what, how, how much and where it was and the withholding of what, how, how much and where it will be.
Time itself is the making-way of movement. Movement requires time as its element, and conversely, time itself is only generated by movement, that is, by the physical (from fu/ein 'to arise'), i.e. emerging, arising, nature of being itself. Hence time is not composed of instants that flow through the now, as if the not-yet and the no-longer were not. Movement requires the as-yet withheld later of the future and the refused earlier of the past in order to be movement traversing 3-D time-space, and the not-yet itself is in the mode of being of being withheld in absence, just as the no-longer or once is in the mode of being of being refused in absence. Both withholding and refusal are also positive modes of temporal being in their own right, and not merely negations of presence now. Therefore, to designate the absent dimensions of time as no-longer and not-yet is inadequate. Any physical being, i.e. any being capable of movement/change, therefore is temporal, i.e. it is not merely at an instant in time, but is or exists only in standing out into a three-dimensional stretch of time. Similarly, it makes no sense, properly speaking, to talk of an instant in time, for this is to deny time's three-dimensionality, and a moving being as moving has no instantaneous position at an instant in time but exists only ever in a three-dimensional temporal ecstasy in which it is both present and absent in a twofold way.
Heisenberg's indeterminacy principle therefore has an Aristotelean interpretation that is closer to the phenomena and reveals to the phenomenologically thinking mind, already at the everyday level without recourse to sub-atomic experiments, a tempero-ontological structure prior to any quantification, measurement and mathematization in equations. The interpretation depends solely on considering the simple, hard-to-see phenomenon of movement itself in which the togetherness of presence and absence in 3-D stretched time is revealed. In other words, the conception of the very being of time has to be revised and recast ontologically to get any further with so-called quantum indeterminacy, and this 'illogical' recasting is not amenable to geometrical or mathematical representation. The corollary of a retrieved Aristotelean phenomenology of movement (which, in our interpretation, stops short of the counting and measuring, and thus the incipient mathematization, of time) with regard to Heisenberg's indeterminacy principle is that a physical body in motion, whether large, very small or middling, does not have a determinate position at a determinate time, t. Furthermore, since rest is only the limiting case of motion, even any physical being at rest is itself an indeterminacy of presence and absence together or 'at once'; it is both here and there, now and then, an undecidable quivering, an 'illogicality', because as potentially moving it is always already stretched both toward its possibility of being elsewhere, a possible presence as absence, as a lack, and also toward its retained history of where it has been.
Determining a physical entity's position more precisely here and now makes its momentum and speed more indeterminate, to the point of complete indeterminacy. That is, the greater the accuracy with which a physical body's present, instantaneous position now is determined, the less its momentum under-way-towards... comes into view, for time itself is thus truncated to an instant. Like Zeno's arrow, it comes to a standstill. In other words, the sharper the focus is on the body's present position now, the more the body's momentum, its stretchedness toward its future position, is lost sight of or obliterated, for a body's velocity, i.e. the rate of change of its position, or momentum shows itself, even within modern physics, only in a span of time, in stretching toward the future, and not frozen stationary at a point-instant in time (as encouraged by the fateful counting of instants in the Aristotelean conception of counting-time according to which only the instantaneous now properly is). This simple phenomenological consideration allows us to see why there is an inverse relationship between the indeterminacy of position and indeterminacy of momentum in the above Heisenbergian mathematical probability equation for indeterminacy. It is a matter of focus, or one-sidedness, of the mind's eye based on a misconceptualization of the phenomenon of time. Such a phenomenological consideration is not merely 'intuitive' or 'heuristic' with the connotation of a lack of rigour that has to be remedied through experimental, quantitative 'verification', but arises prior to mathematization by looking at the simple, simultaneous presence-and-absence, or threefold presencing characteristic of the phenomenon of movement itself.
This threefold presencing is entirely overlooked in modern physics as being beneath serious consideration precisely because it is prior to the dogmatically presumed exactness and rigour of 'scientific' quantification and mathematization, and thus eludes any experimental measurement according to the similarly dogmatic prescripts of modern physics' method. Under these scientific prescripts, time can be only a one-dimensional numerical variable, i.e. a varying number-point itself amenable to mathematical manipulation, i.e. to analytic differentiation in theoretical physics, and obtained experimentally by an apparently precise, finite counting of a regular, periodic physical movement. The three-dimensional conception of time outlined above must be anathema to modern physics, for it defies mathematization and its sham rigour, precision and certitude. Such mathematization as the postulated indispensable mode of access to truth is the totalizing Cartesian prejudice of our age. The phenomenological way of viewing is not 'less exact' than modern science, but sees more, and more simply, namely, the ontological structure of movement itself and its intimate relation with that of multi-dimensional stretched or ecstatic time whence a more adequate sense of being itself can be derived that does not collapse into its tacit traditional sense as standing presence. The phenomenologist has the ontological vision others lack. We will approach the phenomenon of three-dimensional time once again by another route in the next section.
How does this relate to the phenomenon of movement, or rather (loco)motion? Motion grasped mathematically is a continuous function of the three Euclidean dimensions and time f(x, y, z, t) or the position 3-vector r is a continuous function of time t: r = f(t). The continuous function of time traces the movement of a physical body represented as a geometric point through three-dimensional real space. Each point in time maps continuously to a point in 3D real space. The solution of the problem of motion thus becomes the mathematical problem of analyzing the curve traced by the vector equation r = f(t) where the vector function f, in turn, may be derived from physical laws of motion, Newtonian, Einsteinian or quantum mechanical (where, with superposition, the real function f becomes a complex Hermitian matrix). Hence mathematical analysis, i.e. the infinitesimal calculus, a powerful branch of mathematics for grasping motion based on the ontology of time as now-presence.
As we have seen (2.2 Heidegger's review of Aristotle's thinking on modes of connectedness from discreteness to continuity and 2.6 Bridging the gulf between the discrete and the continuous), the infinitesimal calculus makes the geometrical calculable. It does so by calculating derivatives and their inverses, i.e. by differentiating and integrating, both of which require the formation of mathematical limits through an adequate calculus with infinitesimals formalizable as a counting process toward... Without the infinitesimal calculus, there would be no motion along a curve, but only stationary points succeeding one another along a curve. Motion enters the mathematics through differentiation that introduces something like an instantaneous velocity which, strictly speaking, is an illogicality, for there is no motion in an instant, but only in an interval of time. A point on the curve can only indicate a motion mathematically by also being in transition, i.e. here at a single point and also under-way-toward..., this latter aspect being captured by the infinitesimal. The point on the curve is in its co-ordinate position and also infinitesimally removed from itself at an infinitesimally later point of time. Thus is the indeterminacy of a point in motion captured mathematically without, however, the applied mathematician or physicist taking cognisance of the ontology of time he is implicitly presupposing.
Now, if, on the one hand, time is conceived mathematically as a continuous real variable, t, that is continuously increasing, it is always assuming also irrational values. If, on the other, time is also conceived as the counting of a regular, periodic physical movement, no matter how fast, such as the natural wave frequency of a Caesium atom, this counting can never determine a point of time, t, but only ever a counted interval between 'now' and 'now' (whose smallness depends upon the finite frequency of the period taken as counting measure or, equivalently, upon the wavelength of one period) within which t is supposed to lie. For any moving physical entity, within this time interval defined by steady counting, no matter how small, it has moved, and so, assuming Cartesian co-ordinates, its position can be determined only to within a certain segment of co-ordinate space. In particular, since rest is only a limiting case of movement, even a physical body at rest has no determinate space-time co-ordinates, but only ever hovers within a segment of space-time where its here-now and potential there-then are indeterminately 'located'. This indeterminacy is not merely a matter of the accuracy of physical measuring instruments, which may be further refined with the progress of physics (without ever attaining continuity), but is an indeterminacy in principle residing in the postulated continuous nature of movement in relation to the countable, and therefore discrete nature of (clock-)time as conceived by both Aristotelean and modern physics (for time is only ever determined by a finite counting of a regular, periodic movement).
Another way of looking at this is that countable, discrete, rational time only ever defines a segment of space-time containing also irrational numbers and alreadyin principle an irrational number cannot be counted, i.e. it cannot be made present within the counting process, so that continuous motion would have to pass through points in space that are outside assumed countable time! Hence we can conclude that if time is conceived mathematically as a continuous real variable, it cannot be counted, and if it is conceived as countable, i.e. as the counting number lifted off a highly regular, periodic movement, it is not continuous, but is a regular sequence of discrete, temporal 'quantum' leaps (cf. 7.3.3 Excursus 3: On time in (a quantized) special relativity theory (Joy Christian)).
Let us push beyond the problematic duality of continuity and discreteness to consider the complex-continuous superposition of discrete-quantum states (in a Hermitian space of infinite dimensions), which complicates the situation beyond the antinomy between assumed continuous real time and actually measured, discrete, counted clock-time. Modern physics tells us that there is a limit to the divisibility of physical bodies which is reached with sub-atomic entities, which are the smallest of all possible physical entities. These smallest of physical entities, however, even at rest, cannot be pinned down by determinate space-time co-ordinates, but, as quanta, are nonetheless, at any real instant t, a complex superposition of (usually) infinitely many quantum states (expressed as an integral over space). This is the formulation provided by Heisenbergian matrix mechanics. Alternatively, for the physicist, Erwin Schrödinger, on the other hand, who developed wave mechanics, "it is declared that the atom in reality is nothing more than the refraction phenomenon of an electron wave so to speak captured by an atomic nucleus"(34), a wave-mechanical quantum formulation that has been shown to be equivalent to matrix mechanics.I believe there is something we are all missing [...] My guess is that it involves two things: the foundations of quantum mechanics and the nature of time. [...] I have the feeling that quantum theory and general relativity are both deeply wrong about the nature of time. [...] We have to find a way to unfreeze time - to represent time without turning it into space. I have no idea how to do this. I can't conceive of a mathematics that doesn't represent a world as if it were frozen in eternity. It's terribly hard to represent time [...]
The break with the scientific conception of time as a one-dimensional variable, t, that ties time to a real instant must be made to see quantum superposition properly, but at the price of sacrificing the causal-determinist time-evolution of a quantum wave-function provided by the Schrödinger equation. Time itself must be conceived as three-dimensional, and this can be done, for heuristic convenience, in a pseudo-mathematical way, by thinking of it as a complex, rather than as a real variable, as it is in modern physics from Newton to the present day. The 'pseudo' nature of these considerations derives from their having to do, properly speaking, with phenomena of movement, which have to be seen, rather than with mathematical numbers, and functions and matrices thereof. If time is conceived as complex, it has both a real and an imaginary part that are independent of each other, so that time now has two degrees of freedom (on the Argand plane). This can be written down in the pseudo-equation for time, t = b + id, where b and d are real numbers and i is the imaginary number, the square root of -1. If d > 0, it is taken to refer to the future and if d < 0, it is taken to refer to the past, where past and future are absent at the now-instant and in that sense 'imaginary'. The real number b pin-points the present now-instant, but does not exhaust time because now, a real time b is coupled with a continuum of imaginary times d referring to what could be potentially at future time d, if d > 0, or to what could have been at past time d, if d < 0, where future and past are considered from the now-instant, b.
The spatial state of any physical system, whether quantum or classical, is now not a complex superposition of infinitely many spatial basis states at the present instant, as in the multiverse interpretation of quantum mechanics (cf. 7.3.4 Excursus 4: On quantum computing and qubits (David Deutsch)), but an indeterminate complex superposition of infinitely many independent spatio-temporal basis states deriving from complex time, where time past is captured by negative d and time future is captured by positive d in complex time. The spatial state now depends on complex time. At a real instant b, the system's spatial state is given by a linear superposition of a real state and infinitely many imaginary states, both past and future, all of which are situated in an Hermitian space spanned by the basis eigenkets |b> and |d>, where the real component of the instantaneous state at |b> is complemented with the infinity of superposed imaginary states for the temporal infinity b+id, where the free eigenvalue d ranges from minus infinity to plus infinity.
The interpretation for negative d is that, in the past relative to b, the physical system could have been 'historically' in a spatial state that is the complex superposition of the position at the past at the real now-instant b+d and also infinitely many imaginary states corresponding to t = b-d + ie, where now the eigenvalue e ranges over the real numbers. The interpretation for positive d for a given real instant b is likewise a complex temporal superposition of the position at the future real now-instant b+d and the infinitely many imaginary past and future states corresponding to t = b+d + ie, where e ranges over the real numbers, providing quasi-eigenvalues and quasi-eigenkets in a pseudo-Hermitian space in which complex time itself constitutes an infinite-dimensional basis. This is a pseudo-mathematically fancy way of saying that the spatial state of a physical system cannot be separated from its possible past history nor its potential future, both of which are present imaginarily as specific forms of absence in the Hermitian pseudo-ket. In particular, if only an interval of real time terminating with a fixed future point in time, d, is viewed (i.e. b an element of, say, the closed interval [0,d]), this amounts to a twofold focus on now and a fixed, final, future-then. Under such a restricted focus, the system's spatial state for each point of time, b+id, where b is within the real time interval, is a linear superposition of a real present state and relatively independent imaginary states, including the imaginary future final state at d, in a two-dimensional Hermitian space. The superposed imaginary final future state at id is not constant as b varies, nor does it depend tightly on b, so there is no law-like evolution of the instantaneous state at b to the finally realized state at d.
Let's take a simple example outside the mathematics. Suppose I have a tennis ball on the edge of my desk. I see it there now sensuously and realize that it could easily roll off the table and fall onto the floor. I thus see the ball both now and a possible imaginary spatial state for it at a future time (imaginary positive d). Or I see the ball on my desk and tell myself that I must not forget to take it with me tomorrow for my game of tennis with friends. I thus see the ball now and also its imaginary future trajectory tomorrow to the tennis game tomorrow where it will be spatially (imaginary positive d). Or I look for the ball on my desk and now see that it is not there - it is absent. So it has probably fallen onto the floor at a now past time (imaginary negative d). I look on the floor and don't find the ball. I see it absent now both from my desk, where it definitely was at a past time, and also from the floor, so it might have been taken by my dog, who loves chewing on balls. I thus imagine a past time (imaginary negative d) at which the dog might have taken the ball, thus making a movement in space from the area around my desk to an indeterminate place somewhere else. In each of these situations, I have double vision, i.e. I see the situation now of presence or absence (which is not a sensuous seeing), and also an imaginary future or past situation implying a certain movement that is uncertain, and by no means calculable. My temporally twofold vision, however, is intelligent and entirely adequate in the context of everyday life, although non-scientific in the modern sense (cf. 7.4 A mundane example to help see movement in three-dimensional time for another, entirely demathematized example).
The present spatial state now of a dynamical system can only be supposed to be determinate by ignoring the imaginary component of complex time that cannot be brought under calculative control. The complex superposition ranging over the imaginary component of time allows for limitless indeterminacy. A real, (experimentally) observed observable of a dynamic system is no longer approximated by a measurement in rational, countable clock-time, but now must first be conceived as the projective collapse of state in complex-imaginary time onto the one-dimensional determinate state for the real component, b, of complex time from the indeterminacy of the spectrum of imaginary time, both future and past, which then is approximated by clock-time. This approximate clock-time is correlated with a sensuously registered, observed experimental result that is said to be the guarantee of scientific truth according to modern scientific method. There is no sense in which the determinate, real time t=b approximated by an observed, discrete clock measurement, were a definite function, whether probabilistic or not, of time's many superposed associated states in imaginary time given by the d's. Because the components, b and d, of complex-imaginary time are independent of each other, there is no necessary causal-deterministic equation tying the dynamical state at the present instant b to the imaginary dynamical states ranging over an infinity of d's or vice versa. d marks and ranges over an imaginary, immeasurable time, independent of the real time of the present observed and recorded state at this instant, b, of the dynamical system.
This pseudo-mathematical interpretation of complex-imaginary time corresponds to the Aristotelean insight that movement is characterized by a twofold presence, namely, the presence now of a definite state, and the unfinished presence of a future definite state toward which the dynamical system is under way. Likewise, retrospectively, a dynamical system in its present state now is also the absent states in which it has been previously (its 'history') and also the absent states in which it could have been, since we are assuming efficient causal determinism neither prospectively nor retrospectively. The Aristotelean insight can therefore be extended to a threefold presencing of present, future and past as three-dimensional time, where the presencing of the latter two temporal dimensions are forms of absencing.
Modern mathematical physics is characterized by the striving to make a necessary mathematical link, by means of equations (hence the mathematization), between a unique system state at real time b and a real future state at time b+d, with positive d, so that future time loses its imaginary independence and collapses, at b+d, into the real continuum in which b is also situated. Time is then a one-dimensional real continuum of one real instant inexorably and tightly following another rather than a free threefold presencing of the present instant and of other imaginary moments from the future or past. Modern physics is concerned with governing the future physical state of a system from the present moment b, either by bringing it about or by predicting it, so that position can be expressed as a mathematical function of real, present time. Equivalently, the equations can be read backward (for negative d) to determine causally a present state now as effected by states at a past point in time. Because modern physics views movement through mathematics, whose equations can be read either forward or backward, it is confronted with the dilemma of the irreversibility of time that it has manufactured for itself. It seeks a resolution in a particular movement that it claims is one-way, being governed by the second law of thermodynamics, in which entropy is formulated mathematically and therefore in the proper form for scientific truth.
With the Aristotelean insight into movement as a twofold of presence and absence, the complex superposition of quantum entities loses its singular, paradoxical nature because not just sub-atomic entities, but all physical (movable/changeable) beings are characterized by complex-imaginary superposition of present and absent dynamical states, where the imaginary refers to potentiality and the possibilities of what might be and what might have been. The definite real observables observed at a real time b (perhaps, for the sake of scientific 'objectivity', read off measuring instruments and a clock inaccurately as rational numbers) result from their observation by an observer, no matter whether this observer is a physicist-experimenter or somebody else dealing with affairs in everyday life. It is a matter simply of turning one's attention toward the physical state of affairs presently surrounding one and does not depend necessarily on measurement and quantification.
The observer's mind, whether physicist's or not, can and does range temporally over both past and future in imagination, and this is the phenomenological justification for introducing an imaginary positive or negative component into time to denote the focus of attention, employing the pseudo-mathematical notation merely as an heuristic device for those familiar with modern mathematical physics. The mind's attentiveness is itself double: on the one hand, the observer is more or less aware of his present physical surroundings and, on the other, he can be, and usually is, also focused in the imagination on a prospective or retrospective state of affairs that is situated, no matter how vaguely, spatio-temporally in the future or past, over which the mind can range freely. In this sense, and in paradoxical contradiction to a physicist's 'realist' common sense, the observer's physical, sensuously observable surroundings are precisely not present, but absent, and the mind calls to presence a future or past, and therefore absent, state of affairs! The imagination can, and often does, go even further in abstracting also from time-space altogether to turn its attention to wholly abstract thoughts lacking a spatio-temporal place. Such is the power of human imagination (fantasi/a, Vergegenwärtigung, calling-to-mind).
In his eulogy on the greatness of Weyl as a mathematician, scientist, philosopher and highly cultured individual, Wheeler formulates on the first page four questions with affinity to Weyl's concerns that serve as the structure for his talk: "(1) What is the machinery of existence? (2) What is the deeper foundation of the quantum principle? (3) What is the proper position to take about the existence of the 'continuum' of the natural numbers? And (4) what can we do to understand time as an entity, not precise and supplied free of charge from outside physics, but approximate and yet to be derived from within a new and deeper time-free physics? In brief, why time? What about the continuum? Why the quantum? What is existence?" Wheeler's speech ends with the demand, "that we can and must achieve four victories: Understand the quantum as based on an utterly simple and - when we see it - completely obvious idea. Explain existence by the same idea that explains the quantum. Through this larger vision of existence and the quantum, recognize that the continuum of that physical world out there and the bit-by-bit means by which alone we can define that world are not contradictory, but complementary. Reduce time into subjugation to physics." The enumeration of challenging problems here amounts to linking being and time, discreteness and the continuity, and also these two couplets into a quartet in a simple way to achieve "victory". The lead role is given explicitly to the quantum, i.e. to the discrete primal physical entity, which, once understood in a completely simple, hitherto unseen way purportedly will solve also the question concerning "existence", i.e. being.
As discrete, the quantum also provokes the question as to the relation between bits (digitized, in-forming information) and the continuum, which, in turn, leads back to time as a continuum. The "victory" to be achieved amounts to a "subjugation" of time to quantum physics and, as a precondition, a subjugation of the continuum to the discrete. Only in this way will time no longer be a free lunch thankfully received by physics but will itself be "derived from within a new and deeper time-free physics". Hence, time is to be derived from being (as Aristotle did) and ultimately from the quantum and thus a "unity of knowledge" attained. Such is the structure of Wheeler's envisaged research program, and he is presumably speaking in the name of quantum physics with its pretensions to be the foundational science par excellence of the modern age that has already celebrated, at least, its 350th birthday.
The toughest nut to crack in this research program, Wheeler says, is time: "Time, among all concepts in the world of physics, puts up the greatest resistance to being dethroned from ideal continuum to the world of the discrete, of information, of bits."(Likewise in the context of endeavours toward a unified quantum (gravity) theory, Roger Penrose expresses the conviction "that our present picture of physical reality, particularly in relation to the nature of time, is due for a grand shake up - even greater, perhaps, than that which has already been provided by present-day relativity and quantum mechanics" (Penrose 1989, 1999 p. 480).) But why is this reduction to discrete bits necessary? Because, Wheeler goes on, the, "continuum of natural [sic] numbers, Weyl taught us, is an illusion. It is an idealization. It is a dream. With numbers of ever increasing mathematical sophistication we can approach that infinity ever more closely; but we commit a folly if we think we can ever get there." Numbers are thus only ever potentially infinite, on their way to an infinity that can never become actual, i.e. never be 'had' in its end as a perfected, completed presence. Accordingly, time, imagined as a continuum, is as such an illusion, a mirage that continually recedes into the distance the more we approach it, and which has to be reduced to bits, that is, to the hegemony of the lo/goj, which calls beings to presence and itself is present at will and always discrete and hence digitizable, computable, i.e. within the domain of the calculative power of mathematics.
Let us look more closely at what Wheeler means by existence and its link to the quantum. The allusion to "the machinery of existence" indicates some kind of efficient causality, albeit indeterminate, according to quantum-mechanical laws: "Machinery of existence for us means laws of physics under the overarching governance of the quantum principle". Existence itself for Wheeler means, in traditional metaphysical fashion, the thatness of beings as a whole, simply, that they are. But that they are, it is claimed, also has a meaning, and for this totality of existing beings to have a meaning, Wheeler claims, citing Weyl, "it is necessary that the world be governed throughout by simple elementary laws," to wit, by the laws of quantum dynamics, which are assigned the task of accounting for both the sheer existence and also the movements of beings as a whole, now broken down into bits of digitized information. But this elucidation of a purported link between the sheer thatness of existence and its meaning in terms of "simple elementary laws" does not say what existence itself means, but rather presupposes it: that something is means simply that it is. But what does 'is' mean? Does it mean experimental observability?
Once this 'is' of beings is presupposed, their dynamics can be accounted for by laws, hopefully, simple quantum laws. This is the will to power to discover a 'Weltformel', a formula for the world. This all-powerful mathematical formula would make the movement in time of all entities, no matter of what kind, mathematically calculable in a unified way. Hence the title of Wheeler's paper: "Hermann Weyl and the Unity of Knowledge". Furthermore, Wheeler wants quantum laws to account not only for the dynamics of existents, but also for the very thatness of their existence: "Existence? How else is it brought into being except through elementary quantum phenomena?" The coming-into-being is to be explained in terms of quantum phenomena. But even this still does not answer the question concerning the very meaning of this "being" into which beings come.
Putting that aside for the moment, what is meant more precisely by "quantum phenomena"? Wheeler explains with regard to the "objective description" of reality: "Not until the observing sense, or observing device - by its geometry, its layout, and its adjustment - has chosen the question to be asked, and by its registration has made a record long enough lived to produce internal or external action, has an elementary quantum phenomenon taken place that contributes to the formation of what we call reality." An "elementary quantum phenomenon" is therefore an observed measurement in which there is a quantum leap from unobserved indeterminacy to observed determinacy as an ascertained, present measurement, in which the probability wave mathematically describing the quantum-mechanical situation collapses from a superposition of (even uncountably) many states to an eigenvalue. Only on the basis of this observed, present determinacy can there ever be an "objective description" "of what we call reality". Reality (the totality of that which is) is thus conceived as an assemblage of sense impressions gathered and registered by human observation with the aid of experimental apparatus: "There is not a single sight, not a single sound, not a single sense impression which does not derive in the last analysis from one or more elementary quantum phenomena." Accordingly, reality is the perceivable, since science is always referred to the empirically perceivable, even if elaborate experimental apparatuses are required for such perception, and objectivity itself is subjective in the sense that objective reality is that which is perceived by the human subject which, in turn, is subject to the conditions only of experimental scientific method.
The meaning of being tacitly underlying this conception is therefore that of scientifically registered presence for a perceiving subject. It is scientific method that bridges the gap between subject and object, making of merely subjective description an ostensibly objective one. The key feature of scientific method, in turn, is that the experimental experience gone through is amenable to both mathematical quantification and experimental reproducibility. Scientific method makes merely 'subjective' experience exact, rigorous, mathematizable, even though experience, even scientifically methodical, 'objective' experience, can only be experience for a subject. The mathematical quantification prescribes, or precasts, that the observations made must be able to be entered into pre-existing mathematical formulae of a theory modelling reality, and the reproducibility requirement aims at overcoming the opinionated subjectivity of the individual human subject in favour of a collective scientific human subject so that science can be of one opinion. Objectivity is thus such for a human subject experiencing within the bounds of scientific method, and such experiences are to be explicable in terms of quantum laws of physics. A unified physical theory, a Weltformel, would account in a unified way for all the various dynamic forces physics has discovered and thus put precalculable domination of all movements in the universe, of whatever kind, into the hands of scientifically methodical humankind.
The "complementary description of nature as it is seen in quantum theory", which presumably refers to the indeterminate dual nature of sub-atomic entities as both wave and particle, i.e. as both continuous and discrete, is thus claimed to be the "only possible" human experience according to the scientifically valid, mathematically quantifying method, or path, for experiencing "reality". Such quantifiable, 'objective' experience for humanity as subject is a mass of "bits of information", "at most a countable infinity" amenable to calculation and digitization. This, in turn, inevitably throws up the problem of the apparent "existence" of "a continuous infinity of locations for particles, a continuous infinity of field strengths, a continuous infinity of degrees of freedom of dynamic space geometry". Instead of accepting this apparent existence, Wheeler asks: "Do we not do better to recognize that what we call existence consists of countably many iron posts of observation between which we fill in by an elaborate papier-mâché construction of imagination and theory?" In other words, the continuum has to be reduced to the discrete if it is to conform to scientific mathematical method which, ultimately, is digitizable, and it is the encounter with quantum phenomena that induces the scientific conviction that the continuum indeed does collapse to the discrete and finite.
Hence reality is ultimately nothing other than a heap of experimentally accumulated information bits that hangs together by virtue of mathematical equations. "When Bohr tells us that quantum theory gives us the only objective description of nature of which one can possibly conceive, is he not also telling us that no description can make sense which is not founded upon the finite?" Quantum theory therefore is the prescription that reality conform to the digital cast of being. Quantum theory, in turn, is the model based on the experience of observed reality according to mathematico-scientific method. "Encounter with the quantum has taught us, however, that we acquire our knowledge in bits; that the continuum is forever beyond our reach." This "reach" is the reach of "absolute logical rigor" which stands in contradiction with any conception of the continuum beyond such reach. Wheeler, however, does not want to declare a contradiction, but speaks rather of a "complementarity between the continuum and logical rigor" which purportedly has been achieved as a "hard-won power ... to assess correctly the continuum of the natural numbers [growing] out of titanic struggles in the realm of mathematical logic" in which continuity has succumbed to governance by the discrete, i.e. by the exactly calculable mathematical lo/goj.
In these "titanic struggles", which must be regarded as the modern scientific analogon to Plato's gigantomaxi/a peri\ th=j ousi/aj, mathematical logic is said to play the role of the "courageous outpost-cavalry", preparing "the way not only for the main cavalry that is mathematics, but also for the army that is physics". Accordingly, the theatre of war has purportedly shifted historically from ontology, i.e. the question concerning being, to the question in mathematical logic concerning continuity. In its supreme, unquestioning self-confidence, quantum physics, and modern science in general, has entirely lost sight of the question concerning the very meaning of being and its connection with time. All of observable reality, that is, all sense data, seems to be reducible to finite bits according to the scientific program laid down long ago by Democritus which Wheeler paraphrases by quoting Weyl, in turn, citing Democritus: "'the doctrine of the subjectivity of sense qualities has been intimately connected with the progress of science ever since Democritus laid down the principle, >Sweet and bitter, cold and warm, as well as the colors, all these things exist but in opinion and not in reality; what really exist are unchangeable particles, atoms, which move in empty space<' (Philosophy of Mathematics and Natural Science, p. 110). In accordance with this view of Democritus, we understand green today as a characteristic frequency of 5.7 x 1014 vibrations per second," etc. Everything that is, according to this Democritean-Cartesian-Leibnizian cast of being, is reducible to a finite number. In "reality", everything is a bit. The ultimate quanta are the smallest (observable, measurable) bits from which everything else is composed.
Only the continuum of time, which is not simply a sense datum, shows itself to be refractory to this digital cast of being, perversely defying mathematical logic. "But time: how is time to be reduced to more primitive concepts? Reduced from the continuum to something built on bits?" Wheeler thus concludes his survey of the four major questions confronting mathematical quantum physics as a unified theory of all that is and its movement with a conundrum and a deferment: "Of all obstacles to a thoroughly penetrating account of existence, none looms up more dismayingly than 'time.' Explain time? Not without explaining existence. Explain existence? Not without explaining time. To uncover the deep and hidden connection between time and existence, to close on itself our quartet of questions, is a task for the future." The antinomy of the continuum, time, in connection with the question of being (and hence, after all, the logically boggling task of an ontology of time) is said to be a cause for dismay which challenges future quantum physics, fired as it is by a will to power over moving reality, to "achieve four victories", as quoted at the outset of this note. And so we return to the challenge to "[u]nderstand the quantum as based on an utterly simple and - when we see it - completely obvious idea" from which the continuum of time could be derived. Only thus could the will to mathematically calculable power over the dynamics, i.e. the movement in time, of beings as a whole be satisfied.
Someone who has taken up Wheeler's program in his own way by striving to eliminate time from physics is Julian Barbour, an English researcher into the foundations of modern physics. He has done so with his book, The End of Time (Weidenfeld & Nicolson, London, and Oxford University Press, New York 1999), and other papers. For the project of bringing together "Einstein's general theory of relativity and quantum mechanics" into a "single over-arching theory", a "quantum theory of the universe (also called quantum gravity)", Barbour claims, similarly to Wheeler, "the 'problem of time' is perhaps the most severe" (http://www.platonia.com/books.html accessed October 2009). John Wheeler actually voiced a glowing comment on Barbour's book (cf. ibid.). Here, for the sake of simplicity, we shall concentrate on a shorter, prize-winning essay by Barbour, 'The Nature of Time', available at http://www.platonia.com, in which the issue of the elimination of time, at least from Newtonian dynamics, becomes clearly visible. The elimination of time is an important step in Barbour's approach on the way to formulating a unified theory of quantum gravity. As Barbour envisions this theory, "the quantum universe is static. Nothing happens; there is being but no becoming. The flow of time and motion are illusions" (NT op. cit.). This shorter essay of Barbour's will be enough for my purpose here of showing what he skips over in his efforts to eliminate time.
Barbour's essay is ingenious. His attempt to banish time as a fundamental concept from physics, replacing it with spatial difference ("All we need are differences."), in truth deals exclusively with the measurement of time, i.e. with time as quantitative, not with time per se, which he surreptitiously continues simply to assume in his considerations. Barbour proceeds from Newton's conception of absolute time in the Principia of 1687 as duration, and hence does not go back to consider Aristotle's conception of time as developed carefully in the Physics. Such a concept of time as duration which, according to Newton, "flows equably without relation to anything external," immediately leads Barbour to ask the question, "What is a clock?". This question and Newton's positing that "[a]bsolute true and mathematical time [...] by another name is called duration," show that both Newton's and Barbour's focus is on the measurement of time and on time as a mathematical magnitude. Barbour takes as "[t]he best guide to the nature of time [...] the practice of astronomers", proceeding self-evidently from the assumption that astronomers are in the business of predicting the motions of planets (eclipses), etc., and his entire ensuing discussion of Newton and Kepler is therefore in terms of equations with whose aid motion can be predicted, precalculated. But how could a prediction (say, of an eclipse) be at all possible without the temporal dimension of the future being understood a priori and taken for granted by astronomers? By focusing from the outset on scientific attempts to quantitatively measure motion predictively (reduced to difference in position), the phenomenon of time itself is skipped over and taken for granted as self-evident. If, in line with Barbour's research program, there is, in 'scientific truth', no time, then the activity of predicting engaged in by astronomers, which presupposes some such thing as a future dimension, is merely an illusion based on astronomers' self-delusion. But such a temporal dimension is deeper-lying than any conception of duration as measured clock-time, on which Barbour concentrates. Barbour would have to argue explicitly that this deeper-lying temporal dimension, too, is an illusion, which would amount to asserting that all there is is differences in position in a positional state space (with 3N dimensions for a 'universe' with N 'particles'). This assertion, in turn, would give rise to the question as to how change is at all possible in a universe in which there 'is' only static position and no difference in time? Does he deny the phenomenon of change itself and declare it, too, to be merely a human illusion? Does not Barbour end up proposing a static unified theory of all there is, eliminating what is genuinely dynamic? Has not Barbour unwittingly merely reproduced Parmenidean ontology, according to which all ki/nhsij is impossible, i.e. an illusion? Has he not once agin stumbled upon the problem that the preoccupied ancient Greek philosophy from Parmenides through to Aristotle, the problem of how to conceive movement and change as such? We shall see.
Barbour cites approvingly on the first page of his essay Ernst Mach, according to whom, "[i]t is utterly beyond our power to measure the changes of things by time ... time is an abstraction at which we arrive by means of the changes of things;..." Mach thus admits the phenomenon and concept of change, and treats time as an abstraction from such change. This accords also with Aristotle's conception of time, for whom time is the counting number abstracted from movement (ki/nhsij) of which there are four kinds (movement with respect to what, how, how much and where), one of which is change (a)lloi/wsij) and another (loco)motion (ki/nhsij kata\ to/pon). For Aristotle, the counting of time takes place with respect to before and after, which are themselves temporal terms referring to the dimensions of past and future. For Barbour, by contrast, there is no temporal order, no before and after, but only a jumble of atemporal differences in spatial states (to which he likes to refer as 'snapshots') and, apparently, atemporal changes between such states (on which more later).
Barbour's arguments against Newtonian absolute time first concentrate on the difference between solar and sidereal time, demonstrating that, in truth, Newton's absolute time, which is supposed to "flow equably without relation to anything external" turns out to be sidereal time, i.e. time as measured by the motion of the stars relative to the Earth. It is easy therefore to agree with the conclusion of Barbour's argument in this section: "As Newton himself defines it, absolute time is by no means independent of the world; it is a specific motion, the rotation of the earth." This argument, however, does not impinge on a deeper-lying conception of three-dimensional, ecstatic time as enabling all kinds of movement from which, then, counted clock-time is read off.
Barbour's argument then proceeds on the basis of the assumption: "Since time must be deduced from change of position (motion), I shall here take position and differences of position as given,...". This is a first step toward eliminating both time conceived as duration, and also genuine motion, in favour of changes in position. [This first move of Barbour's warrants a recall of Leibniz' critique in 1698 of a contemporary metaphysician of physics, Sturm, who asserted, in a way not dissimilar to Barbour's argument, "Motion... is only the successive existence of the thing in motion at diverse locations" (Motum... esse successivam tantum rei notae in diversis locis existentiam; Gottfried Wilhelm Leibniz 'De Ipsa Natura sive de Vi Insita Actionibusque Creaturarum' (1698) Philosophische Schriften Band IV (ed.) Herbert Herring, Wissenschaftliche Buchgesellschaft, Darmstadt 1992 S. 296), to which Leibniz responds that these different locations is only "what results from motion" (quod ex motu resultat; ibid.) and "the body is not only in a location commensurable to it at the present moment of its motion, but also has the striving or strain to change its location, so that the following state is a consequence of the present state of itself by force of nature" (non tantum corpus praesenti sui motus momento inest in loco sibi commensuarto, sed etiam conatum habet seu nisum mutandi locum, ita ut statu sequens ex praesenti, per se, naturae vi consequatur; ibid.) Leibniz thus shows that he has learned something essential from Aristotle's Physics.]
The second, crucial step for Barbour is actually replacing Newtonian absolute time, t, by "the angle f through which the rotating earth turns relative to a fixed star". The temporal thus becomes spatial, viz. an area swept out by a motion. Barbour then shows that Kepler's discoveries demonstrated that clock-time as measured by the change of f gives the same time as measured by the areas swept out by the planets' motion around the sun, which is simply another angular magnitude. Terrestrial sidereal time (equivalent to an area swept out by the Earth's rotation) and planetary areal-motion around the sun are thus the same measure and the Earth's rotation and planetary areal-motion are equivalent natural clocks. Newton was able to formulate his famous mathematical laws of motion to capture these Keplerian planetary motions, and these laws were generalized axiomatically to all physical motions. "Newton had discovered dynamics," remarks Barbour, and the modern age had a powerful mathematical theory in its hands to predict and control motion of all kinds.
In the next section of his essay, Barbour introduces the conundrum presented to physicists in the 1890s that Newton's laws of motion could not account precisely for the moon's motion, which exhibited "a small but undeniable non-Newtonian acceleration". In an effort to get to the bottom of this anomaly, physicists redefined measurable clock-time so as to fit in with Newton's laws of motion, which thereby become axioms that can be applied to a closed dynamic system. The problematic thus becomes entirely mathematical, a matter of writing and solving equations for a dynamical system. Indeed, to be led by (ever more sophisticated, mind-bending, Magister Ludi) mathematics is the method of modern mathematical physics. The criteria for dealing with dynamical problems then become mathematical, which override any phenomenological considerations (which can then be dismissed as non-mathematical, non-scientific and human self-delusion). Barbour follows this lead of late-nineteenth century physicists by ingeniously proceeding from the Newtonian law of conservation of energy in a closed system. This allows him to write a first equation (1) for the potential energy, V; of a system consisting of a finite number of bodies in terms of the universal gravitational constant G, the masses of the individual bodies and the distances between them. Potential energy V therefore depends only on the masses and relative positions of the bodies. The system's kinetic energy, T, can also be written, classically, as a sum of the individual kinetic energies of the bodies in terms of product of half their masses and the square of their instantaneous velocities. Barbour takes as an approximation to these so-called instantaneous velocities the small distance dx covered divided by the small duration dt taken for such a small change of position. He has thus implicitly (and later explicitly; cf. below) presupposed the notions of an instant of time, of instantaneous velocity, of infinitesimal distances and infinitesimal durations.
By appealing to the axiomatic principle of the conservation of energy, Barbour can now postulate a constant total system energy E which is equal to the sum of V and T. He then proceeds to solve this equation for the infinitesimally small time interval dt, thus obtaining an equation (3) for dt, so-called "ephemeris time", in terms of the individual masses, the individual small distances covered and the difference E-V, where, as we have seen, V depends only on the masses of the individual bodies and the distances between them. Expressed in words, equation (3) says that the time difference dt is equal to the square root of the sum over all the bodies in the system of the product of the mass and the square of the body's displacement all divided by twice the difference between total energy, E, and potential energy, V. For time mathematized in such a way by Barbour's equation (3), there is no before and after, and this is simply because mathematics itself abstracts from phenomena as they show themselves in the physical world, rendering them timeless. The quaking issue for mathematical physics is that mathematical entities are timeless. So it is inadmissible to argue from mathematical equations, which inherently eliminate the temporal, that they are 'time-symmetric' and that therefore time, or the so-called arrow of time, scientifically 'does not exist'.
Employing equation (3), Barbour can then eliminate dt from the equation for the "instantaneous speed of particle i" which is now expressed in terms of instantaneous displacements (i.e. infinitesimal distances), the individual masses, along with E and V. The antinomies inherent in the relationship between discreteness and continuity surface here, without Barbour making any mention of them. In particular, how can Barbour claim that "the ephemeris time defined by (3) runs continuously" whilst at the same time asserting that the time defined by (3) emerges "from observed positions of objects," i.e. from observed finite differences in position which, as observations, can never constitute a continuum? Barbour's self-evidently assuming the continuum, infinitesimals, and the like goes against Wheeler's and Weyl's caveat "that the continuum is forever beyond our reach" (see above in this note).
If Barbour's equation (3) is regarded as one in terms of very small finite differences, rather than in terms of infinitesimals, then it can be used by astronomers to have time "truly emerge from observed positions of objects," namely, of celestial bodies, so that "[t]ime can be read off the heavens" in a finite, discrete, approximating, measuring procedure. This scientific method of determining ephemeris time will only be of use if the motions observed in fact give the same, uniform time. Barbour expresses this condition by referring to "the wonderfully correlated motions that nature exhibits" underlying "how natural clocks can march in step". Without such a marching in step of physical motions, there would be no way of postulating universally applicable, mathematical equations of motion. But are the wonderfully co-ordinated motions of celestial bodies, perhaps even in step with the counting of equally wonderfully construed artificial clocks, not a special case of motions and movements which in general are neither co-ordinated with each other nor uniform and "equable" within themselves? Is not the postulation of such co-ordinated uniformity (of celestial motion) more an axiomatic precondition for formulating mathematical laws of motion to which all kinds of movement then have to be made to somehow fit, or to which they have to be subjugated, to be governable, rather than an empirically verified fact? Why should celestial motion be the yardstick for all kinds of motion and movement and change? And is not the time that "emerges" from such co-ordinated motions only the measurable, mathematical time that suits the scientific will to know and, through this knowledge, to govern motions mathematically?
Barbour's equation (3) depends on the constant E for the total energy of a "perfectly isolated" dynamic system. From such a system, mathematical time dt is said to "emerge". But, "in reality there is no perfectly isolated system except the entire universe." So, strictly speaking, ephemeris time emerges only from the motions of the bodies in the entire universe as expressed in equation (3) which would then comprise a huge, finite number of bodies in motion, and imply a God's-eye view of the universe. Such a God's-eye view of the universe is never to be attained scientifically, quite apart from the question whether the number of celestial bodies is finite, and quite apart from the impossibility of scientifically measuring their masses and displacements between two different instants. Hence equation (3), - which, significantly, is derived from considering astronomers' looking down on the solar system "from a 'crow's nest' very far 'above' the sun" - is an unverifiable and unfalsifiable Gedankenexperiment, just as Newton's first law of motion (the Galilean law of inertia) is. Only approximations to these axioms are to be had scientifically, and physics may well be satisfied that its theory of dynamics delivers very good, experimentally verified approximations. One could then say that Barbour's equation (3) represents the elimination of time from classical Newtonian physics by relying on Newtonian axiomatics. Barbour notes that "[e]ven in Einstein's much more sophisticated general relativity time emerges in much the same way" as in equation (3). The upshot is that time is eliminable from considerations of physical motion, and the calculation of such motion depends only on "snapshots taken [...] in quick succession" of the positional states of a dynamic system at different instants. In such instantaneous snapshots, however, there is also no motion, just as in Zeno's paradox of the arrow instantaneously frozen in flight. From his work overall, Barbour indeed draws the conclusion that "[t]he flow of time and motion are illusions". But one could turn this around and say that mathematical physics, precisely by virtue of its mathematical nature, is unable to truly capture the phenomena of time and motion and must declare them to be illusions. Hence, could not Barbour be accused of saving the mathematics in precedence to saving the phenomena?
After this excursion we return to Wheeler's search for "an utterly simple and - when we see it - completely obvious idea" of the quantum from which the continuum of time could be derived. One option is to go against the mathematical grain and to simply look at the problem the other way round, thus reversing the order of derivation here: time (along with movement of all kinds) itself would then become the originary phenomenon whence the existence of finite quanta would be derived. Time and movement always exceed what can be ascertained in the present as observational data, even and especially by the most elaborate and precise scientific experimental apparatus of this exact science. Time and movement are always beset by the lack that they are also what is not present; time and movement are also a refusal and a withholding never to be made present as observational data. Refusal means that what has been in the past is no longer retrievable as such and within the reach of a will to mathematical cybernetic power. Withholding means that what is yet to come from the future is as yet withheld and also beset by an uncertainty, an indeterminacy evading mathematical precalculability. The dynamic laws of quantum physics, classical mechanics and general relativity are out to calculate motion, a primal phenomenon of the physical that goes hand in hand with time. Calculation is a kind of logic but, as Wheeler himself says, the continuum (and along with it time and movement) defies logic and the dissolution into logical, computable bits. An adequate phenomenology of motion (to which all movement is scientifically reduced in the modern age) shows that anything in motion is both present and absent, so that a mathematical account of motion based on observational data has always already truncated the phenomenon of motion itself to what can be ascertained in presence whence what is yet to come is supposedly governed, or whence what has been can be explained in retrospect as a law-governed motion.
What does this imply for the ambitions of quantum physics, or "quantum gravity", as Barbour puts it, to be the unified foundational science for the truth of all physical beings? The quantum itself seems to be an irreducible phenomenon, a 'hard nut' struck upon when physical entities are divided and divided (almost) endlessly that puts an end to any notion that physical reality is continuous (for continuity implies endless divisibility). Hence Wheeler's insistence on bits as ultimate, and hence also the various attempts at 'digital physics', "a collection of theoretical perspectives that start by assuming that the universe is, at heart, describable by information, and is therefore computable" (http://en.wikipedia.org/wiki/Digital_physics accessed October 2009). In truth, quantum physics claims, physical reality is ultimately, on the Planck level, discrete, and its continuity is merely an illusion arising from our everyday, 'inexact' dealings with the macro-world, which seems to be continuous. Quantum physics tells us that it has experimentally registered the ultimate building blocks of all physical entities, such as the electron and the photon. But at the same time, it has also ascertained that these ultimate physical entities, which are all in motion, cannot be pinned down determinately to a here-now point in space-time, and that this indeterminacy results in a range of possible measurements according to scientific measurement procedure itself, which always has to collapse wavering, superposed indeterminacy into the determinacy of an ascertained measurement. The collapse to observed, registered data through measurement by 'interfering' experimental apparatus is in truth the truncation of time and movement of all kinds to unambiguous, real (not imaginary), 'instantaneous' presence which, incidentally, gives rise to well-known, peculiar paradoxes such as Schrödinger's cat and the quantum Zeno effect. These thought-experiments and other mysteries of quantum mechanics can only be misconceived as long as the tacit understanding of being as unambiguous, logically graspable, standing presence underlying all Western thinking since the ancient Greeks maintains its (strangle)hold on today's scientifically-infected thinking.
(September-December 2009)One might like to object to the critique of the mathematical approach to time presented in this study, which considers both classical and quantum mechanics, that it still has not taken into account the groundbreaking Einsteinian relativity theory in which something as unheard-of as curved space-time has come into view. The quick repudiation of this objection is that relativistic space-time, even when enriched with gravitational forces as in general relativity theory, still operates with a four-dimensional space-time in which t is simply a continuous real linear variable obeying certain equations, and this is preserved in recent advanced physical theories in which relativity is wed with quantum mechanics. Even when a crumbling of time into a grainy discreteness in the region of Planck time is theorized in some recent speculations (see below on Joy Christian's work), the four-dimensional space-time structure remains the mathematical framework. It is nevertheless instructive to take a look at that mysterious relativity of time which continues to exercise as strong a fascination on the physicist's and the layperson's mind as do the paradoxes of quantum mechanics.
In relativity theory, time loses its independence as an 'absolute' phenomenon and becomes 'relative'. Relative to what? Relative to a co-ordinate frame of reference in which the passage of time is measured by a clock for an observer-subject. Time is therefore measurable clock-time, measured by counting the ticks of the clock which is nothing other than a mechanism of some sort exhibiting a strictly regular, periodic movement such as the oscillation of a quartz crystal. Clock-time is the time counted off by a suitable detection mechanism from an underlying, natural or artificial, motion, a countable number of ticks that keeps on increasing endlessly. Each co-ordinate frame of reference has its own clock-motion, such as an oscillating quartz crystal or the circling of the stars, from which it counts off time. Relativity theory is a consequence of the theoretical discovery that the clock motions in reference frames moving at differing velocities differ, even though the clocks used to count the time tell the same time, e.g. are derived from the same underlying crystal oscillation. How can this be? It is the consequence of both how time is ascertained and the discovery made during attempts to experimentally demonstrate an absolute inertial aether-medium for light-travel and then adopted as an axiom of relativity theory, that no entity can move faster than light in a vacuum, so that the movement of light represents an absolute maximum of motion. So relativity theory depends essentially upon postulating an absolute motion whose magnitude is an absolute maximum. Time in physics is the time ascertained within an experimental set-up as measured by a clock. The clock measurements are part of the observations recorded during the course of the experiment to which other events are assigned as having eventuated at such-and-such a time. Each inertial reference frame observes its own time on the ticking clock, and hence, paradoxically, time itself becomes dependent upon the observing subject, in genuinely Protagorean manner, even though the counting of time is performed by an 'objective' mechanism/movement. This Protagorean observer, however, is supposed to be general, i.e. any old observer will do to observe the results of the experimental set-up, thus puportedly guaranteeing 'objectivity'. Nevertheless, despite modern science's claims to 'objective truth' (queerly regarded as unloosed from any subjectivity whatever), relativity theory introduces a subjectivism of time into physics, more on which below.
Because the speed of light or, equivalently, of electromagnetic radiation, is an absolute maximum, the passing of time itself can be measured by the magnitude of the distance covered by light between an earlier and a later point in time. If time is measured this way, time itself can, in a certain way, be regarded as the movement of light (or, more generally, of electromagnetic radiation). All clock-time can be made equivalent to the movement of light by equating the time interval between two ticks to the distance travelled by light in that time interval. One second, for example, becomes the distance from the Earth to the Moon. Time measurements made in two different frames of reference moving uniformly (or, in general relativity theory, non-uniformly) in relation to one another, A and B, depend on the light paths between the two frames to ascertain the time on each other's clocks. A only has access to B's clock by sending an electromagnetic signal to it and receiving back a signal with the information of the time read off B's clock. This signal has to travel a certain distance at the speed of light before A can read B's time, so this light-distance has to be added to the time measurement, and A's clock measures a later time (more ticks) than B's clock at the instant it receives the time signal back from B. Viewed from A, B's clock-time goes more slowly than A's clock-time if time is thus conceived by physics as factually registered clock-times on the basis of the postulate of the speed of light as an absolute maximum.
This is the kernel of relativity theory, which is elaborated mathematically with a focus on magnitudes, i.e. on time-measurements between different frames of reference moving in various ways (toward, away, uniformly, even non-uniformly) in relation to each other. The famous Lorentz transformations, that set up a mathematical relationship between space and time, arise from considering the change of space-time co-ordinates between two reference frames moving uniformly in relation to each other. Because, in relativity theory, time's measurement has become a spatial distance travelled by light, time and space are now interrelated instead of being independent variables in equations of motion for all sorts of physical entities. The more the relative speed between reference frames A and B approaches the speed of light, the longer the light-signal paths between A and B, and hence the greater the clock-times in A and B differ. They differ symmetrically, since the relative velocity between A and B has the same magnitude, differing only in sign: positive or negative.
In general relativity theory, the subjectivism of modern physics gains an added twist, namely, the curvature of space-time. Both special and general relativity theory are based on the postulate or axiom of the absolute nature of the movement of light or, equivalently, any electromagnetic radiation. The only pertinent movement of light that physics can see from its mathematical casting is change of place, i.e. locomotion, or simply motion as measured by change of place in unit clock-time. In special relativity theory, it is only the speed of light moving in a straight line that is of interest. According to both classical Newtonian and Einsteinian relativity theory, the first Newtonian law or axiom is upheld according to which it is proclaimed, without ever any hope of experimental observation, that physical bodies continue to move uniformly forever in a straight line unless acted upon by a net external force. Any change of velocity, i.e. an acceleration in a particular direction, must be accounted for by the action of a net external force (and there is invariably some net external force acting). In general relativity theory, it is precisely accelerating frames of reference that are introduced, and such acceleration is, and must be, accounted for by the action of some external force or other. Any observed acceleration implies a force at work. Relativity physics is based on the (postulated absolute) motion of light relative to the observer subject's frame of reference. Since the speed of light (in a vacuum) is a constant absolute, the only way an acceleration of light can take place is through a change in direction of its vector of motion, i.e. its path is not straight, but curved, and this curvature is accounted for, as it must be, by a force called gravity that is attributed to the massiveness of matter, and mathematized as a force vector proportional to quantitative mass (as well as to the inverse square of distance from the mass). Since, however, light is the absolute motion, gravity can be conceived simply as this motion's equivalent, namely, as a curvature of space itself given by the path of light itself.
Light, or electromagnetic radiation, as the absolute motion, provides the standard reference frame for all time and space. Not only are time and space as mathematical quantities interrelated via the Lorentz transformations of special relativity, but this space-time is also curved, which is equivalent to postulating gravity as the force acting on light, thus changing (accelerating) its motion in a definite, calculable direction. This curvature of space-time is expressed mathematically by a set of differential equations encapsulating the vector force-fields of matter (or, equivalently, energy). Space-time is hence relative to an observer-subject according to how the subject observes the motion of light, or electromagnetic radiation of all kinds. The universe is thus centred on the observing scientific subject receiving electromagnetic signals at a point of observation in space-time from which, with the aid of the appropriate equations of motion expressing causal interrelations among all physical phenomena, it calculates all motions in the universe and hence all events, both past and future. Thus it can be seen that the so-called objectivity of advanced physics as a much admired foundational scientific theory of the mathematico-scientific age goes hand in hand with an extreme subjectivism in a precise sense. As such, relativity theory, both special and general, is an apt name. For relativity theory in its mathematico-Cartesian, ontotheological cast, light as pure motion is the Absolute, and all scientific observers its calculating subjects.
The fundamental postulate of standard Einsteinian special relativity theory, that the speed of light is an absolute maximum that cannot be exceeded by any physical entity, plays a key role in the critical appraisal and further development of quantum mechanics with its postulate that, prior to measurement, the dynamical state of physical entities must be conceived as a superposition of possible or potential states that can actually be measured uniquely by an apparatus in an experimental set-up. An observable difference in measurements comes to light experimentally in factually registered data only at the sub-atomic level, with physical entities inhabiting dimensions in which the Planck constant makes a difference. Any dynamical state of a physical system must be regarded as an imaginary-complex probability distribution of possible states, so that the physical entity or entities in question are not well-defined, not definitely there now with certain determinate properties.
This indeterminacy in the state of a physical entity was repelling to some physicists, including most famously to Einstein who, although having been awarded a Nobel prize precisely on the basis of his work on quantized energy in 1905, together with Podolsky and Rosen, published a thought-experiment in 1935 (Einstein, A., Podolsky, B., and Rosen, N. 'Can Quantum-Mechanical Description of Physical Reality be Considered Complete?' Physical Review 47 1935 pp. 777-780). This paper's paradoxical result was supposed to show that quantum mechanics was incomplete and would have to be supplemented by as yet unknown, hidden variables that would ensure a determinate, rather than an indeterminate dynamical state of a physical entity prior to measurement. EPR argued that "[s]tarting with the assumption [...] that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality". Since non-commuting operators on a system, such as position and momentum, cannot be measured simultaneously, EPR concluded that the quantum-mechanical theory of physical reality must be incomplete. The future task for mathematical physics, therefore, was to attain completeness through the supplement of hidden variables. EPR take a complete theory of physical reality to mean that the parameters (or variables) accounting for a dynamical state must have "simultaneous reality definite values" (p. 778) which, starting from given initial conditions of a dynamical system, can be theoretically predicted. With its non-commutable operators, however, quantum mechanics does not fulfil the condition of "simultaneous reality" of "physical quantities" and instead treats the dynamical state of a system at any time as an indefinite superposition of potential states with complex-imaginary coefficients. Wave states composed by superposition cannot deliver such determinacy, and this was anathema to EPR. It should be noted and underscored that "simultaneous reality" means the determinacy of "physical quantities" at a point in time, t, so the conception of time as consisting of mathematizable, determinate now-points (no matter whether hanging together continuously or discretely separated or both) is fundamental for mathematical physics' conception of reality.
That the conception of time is fundamentally implicated in EPR's charge that the quantum-mechanical theory of physical reality is incomplete has been overlooked in the debate among physicists since 1935. Instead, the focus has been on finding an experimentally testable hypothesis to determine empirically whether quantum-mechanical indeterminacy is tenable. Superposition was approached through the complementarity of the dynamical states of paired physical entities emitted, say, by a change in energy state of an atom, whose states are said to be entangled. Thus, for instance, the spin angular momenta in a given direction of a generated photon pair must sum to zero, i.e. one is the negative of the other. If the one photon is measured as having positive spin, then one can immediately conclude that the other has negative spin. If, however, both photons prior to measurement are an indeterminate superposition of potential dynamical states, assuming a determinate spin only upon actual measurement, then the measured positive spin of photon A, physicists following EPR argued, must 'communicate' its spin to photon B which instantaneously assumes a determinate negative spin. But such an instantaneous communication or teleportation would violate the fundamental principle of relativity that no causative signal can travel faster than light.
Unfortunately for EPR's adherents, the physicist John Stewart Bell proved a theorem that provided a way of experimentally testing whether quantum indeterminacy or realist determinacy pertains to a pair of entangled sub-atomic entities prior to measurement. Bell's theorem shows that the expected value of the probability distribution for superposed dynamical states exceeds the maximum allowable expected value for the dynamical state of a well-defined, determinate entity (cf. Norbert Dragon Geometrie der Relativitätstheorie Chap. 1, Subsection 'Quantenteleportation und Bellsche Ungleichung' accessed August 2009). Such expected values are open to experimental testing by registering statistical frequency. Experiments in a domain dubbed "experimental metaphysics" by Abner Shimony (cf. 'Search for a Worldview which can Accommodate Our Knowledge of Microphysics' in Search for a Naturalistic World View Vol. I, Cambridge U. P., Cambridge 1993) have come down in favour of quantum mechanics and against so-called 'local realist theories' postulating hidden variables that account theoretically for dynamical states at any point in time. So theoretical physicists have set to work in an attempt to reconcile the paradox of apparent instantaneous teleportation of information between entangled quantum entities. Such attempts involve introducing alternative mathematical conceptions, above all related to the role of gravitational force in the so-called collapse, or reduction, of the superposed wave function to a definite measurement caused by an experimental apparatus (Roger Penrose, Joy Christian). The problem of measurement in quantum mechanics consists in understanding theoretically the transition from a superposition of many potential dynamical states of a system to a definite dynamical state as measured determinately by an experimental apparatus. Such a problem, of course, presupposes that there is such a transition from indeterminacy to determinacy. In what sense can it be said that a physical being, which is capable of motion and change, even when 'at rest', is in a definite dynamical state? Repeated measurements on a photon may confirm that it is stably polarized, within a very small range of measuring error, at alpha degrees within the constricted and artificially construed environment of the experimental apparatus. Who is to say, however, that the measuring error is not an indeterminacy of superposed potential states lying beyond the accuracy of the measuring macro-apparatus to measure?
The incompatibility between local realism and quantum-mechanical superposition brings up philosophical issues revolving around what it means for a physical entity to be. For local realism, a physical entity, whether on the quantum scale or not, is a something with definite properties inhering in this something at any given point in time, i.e. at any present instant. This invokes already two elementary Aristotelean categories, ti/ and poio/n, something and quality, or what an entity is and how it is, both of which regarded as present and predicable of an underlying substrate that Aristotle terms the u(pokei/menon or 'subject'. It has been recognized in quantum mechanics, however, in a naive and rather superficial retrieval of Aristotle's Metaphysics, that superposition must be conceived as a wavering bundle of potentialities: "The neo-Aristotelian notion of quantum-mechanical potentiality as a novel metaphysical modality of nature - situated between mere logical possibility and bona fide actuality - was favoured by Heisenberg, and has been exuberantly endorsed by Shimony (1978, 1998)." (Joy Christian 'Potentiality, Entanglement and Passion-at-a-Distance' in Studies in History and Philosophy of Modern Physics 1999, accessed August 2009) What a physical entity is becomes, in modern physics, a mathematical magnitude, and how it is, i.e. its quality, becomes a wave function on a (finitely or infinitely) multidimensional phase space which captures the entity's dynamical state via a vector within that phase space (viz. a unit ket in an Hermitian space).
As we have seen in 2.9 Time and movement in Aristotle's thinking, however, being as potential, or a duna/mei o)/n, has to be conceived as a twofold presence of both presence and lack, and a being can have multiple potentials. Hence Aristotelean du/namij is compatible with quantum mechanical superposition, and indeed, prior to any quantification and mathematization of superposed states as (complex amplitude) probabilities of dynamical states. A twofold of presence and absence, to be sure, is necessarily anathematical to modern physics which tries to cope instead by employing a bundle of wavering complex probabilities at any present point in time to capture indeterminacy, thus salvaging mathematizability. As exposited in the main body of this book, the famous Aristotelean triad of ontological concepts, viz. du/namij, e)ne/rgeia and e)ntele/xeia, which he fashioned to come to grips with the phenomenon of movement (of four kinds), i.e. with the hallmark characteristic of physical beings, needs to become once again an intense focus of attention, even for today's thoroughly mathematized physics.
The basic postulate of Einsteinian relativity theory, that the speed of light is an absolute maximum, is also understood as a "causality condition" (cf. Joy Christian 'Absolute Being vs Relative Becoming' in Relativity and the Dimensionality of the World within the series Fundamental Theories of Physics edited by Vesselin Petkov, Springer, NY 2007, accessed August 2009). Such causality is conceived exclusively as efficient causality, effects being caused via a transmission from one physical entity to another, as a signal or a bit of information, maximally at the speed of light. Hence, modern physics operates overwhelmingly with electromagnetic force-fields to capture the motion of physical bodies causally in force-field equations. Aristotelean material cause is implicitly also acknowledged by modern physics under the head of mass or matter, matter being conceived as the passive stuff on which force-fields act, and mass being cast as quantified matter that appears as a variable in the appropriate equations of motion and above all as the bearer of gravitational force. The other two kinds of Aristotelean cause, the final end of the movement and the mover, are rejected in modern physics as 'subjective' as opposed to 'objective' forces of nature, as if objectivity and subjectivity could be separated.
First of all, note that calling to mind an end is just one way in which the human mind calls beings into the presence of the mind's eye, and such calling to presence (German: Vergegenwärtigung) is not subject to an effective cause that can act only at the speed of light or less (e.g. "the most charming young man in the world is instantly before the imagination of us all." Jane Austen Northanger Abbey end). Such calling to presence in awareness by 'thinking-of' is not merely a fantasizing or an imagining but is the primary way in which beings come to presence for human being. All human action involves calling to presence the matter to be acted upon, for which the sensuous perception of what is present at hand is auxiliary. This holds true both for everyday life and even for the theoretical physicist, for whom physical beings are called to presence in the mind's eye predominantly via the theory that is at the focus of the physicist's practice. Calling to mind can be wordless, or it can be articulated in the lo/goj, i.e. in language, including mathematical language, which addresses beings, thus calling them to presence as such-and-such. In particular, the concepts of a physical theory are the special lo/goi that call beings to presence for the theorist, often prospectively, which thus only shape up for the theorist's understanding in terms of such concepts, i.e., for instance, as masses, forces, force-fields, etc. Since the physicist is so intent on measuring by factual registration and theoretically precalculating effective causes among physical beings, above all in experimental set-ups, he overlooks and takes for granted the calling to presence of beings inherent in calling to mind conceptually in which he is constantly engaged and for which no superluminal restriction applies or even makes sense.
For any experiment to test an hypothesis, the physicist must first prospectively call to mind the experimental set-up in terms of fundamental physical concepts. He has a plan and an end, namely, to determine whether the hypothesis stands up to experimental testing. Hence it can be said that the experiment itself has a teleological cause, namely, to attain an experimental result, and that this teleological cause is not subject to any luminal limit in its action. Calling-to-mind as the hallmark movement of human being is not subject to the upper bound to the motion of physical beings postulated by relativity theory. Otherwise we human beings could not call to mind a star or galaxy millions of light-years away as such. We human beings can reach back in time as such without luminal limit. The as such means here that, say, the light received by a telescope from a distant star is not merely registered, say, on photographic film, as light in the present, but is identified as light that has travelled a certain number of light-years.
We can therefore say that Einsteinian relativity of time is no restriction for the movement of human calling-to-mind (the mind being not the brain, but awareness of the world in its coming to presence prospectively, retrospectively or momentarily). For instance, we can think of the sun in less than the eight minutes that it takes for the sun's light to reach us, and perceiving the sun's light sensuously is not the only way, nor even the usual or predominant or most interesting way in which the sun presents itself to human awareness. Distant galaxies millions of light years away play a role for human being principally in the context of cosmological theories for which quasi-sensuous perception through telescopes of various kinds supply only data. But isn't this the purest subjectivism, grossly at odds with the objectivity aimed for by modern physics for which hard, objective, quantifiable data provide the bedrock of testable physical theories? Don't physical theories have to be much more than a mere 'thinking-of' that is both subjective and anthropocentric? An apt response to such objections consists in asking for whom physical theories are developed, if not for human being, and in pointing out not only that physical theories rely crucially on fundamental theoretical concepts which are ways of thinking of key physical phenomena such as motion, matter, energy, force, etc., but also that all the theoretical and experimental work carried out by a scientist according to the rules of scientific method is carried out by the scientific subject who is motivated by the te/loj of achieving experimental confirmation or falsification of an hypothesis under the impetus of an unbridled will to power over movement that shapes how the physical world shapes up for the mind. It is therefore a self-delusion of modern scientific method as practised today to claim that it has dispensed entirely with the 'superseded' Aristotelean notion of teleological cause and operates exclusively with objective, effective causes. Modern science disseminates obfuscation about the categories of subjectivity and objectivity.
For EPR, as a typical example, there is an "objective reality, which is independent of any theory" (p. 777). But is "objective reality" such an innocent, unprejudiced title for 'out there'? Isn't "reality" already implicated in an understanding of 'out there' as such-and-such, e.g. as matter moving around in space rather than, say, as the gods' playground? If "objective reality" is supposed to be "independent of any theory, and the physical concepts with which the theory operates," but nevertheless, "these concepts are intended to correspond with the objective reality", how is such a correspondence at all possible? And don't these concepts already inevitably involve a preconception, a precasting of reality that opens it to human understanding in the first place? If "the correctness of the theory is judged by the degree of agreement between the conclusions of the theory and human experience" and this "experience, which alone enables us to make inferences about reality, in physics takes the form of experiment and measurement", is not this mode of access to reality not already preconceived and hence massively prejudiced, namely, as the way in which the world shapes up for modern humanity via scientific method?
There can be no physical experiment whatsoever set up without the basic physical concepts in terms of which the experiment is supposed to test what it is set up to test, for otherwise experimenting would be a blind, senseless action. Experimental measurements are always measurements of theoretically preconceived and precast 'quantities', such as mass, energy, momentum, position, etc., and this presupposes, and ensures, that the physical phenomena in question are amenable to a quantitative grasp. Such amenability is a 'correspondence' to reality that can never be experimentally tested, but, on the contrary, is a positing of a theoretical casting of 'out there' through which it becomes visible ('to theorize' means originarily 'to look at') at all to the human mind. Insofar, it is a misconception cherished by modern science to imagine that there is such a thing as "objective reality" independent of human subjectivity. Objectivity is always for a kind of subjectivity that has conceptually precast this objectivity as such-and-such, and human being itself is conceived as subjectivity only within a certain historical epoch, namely, our own Western modern age.
In the present context, the objective world out there is precast on the basis of the experimentally confirmed axiom that no physical motion (and hence no causal effectivity) can exceed the absolute maximum of the speed of light. From this results, first of all, Einstein's theory of special relativity which compels an interlinking of time and space co-ordinates in four-dimensional space-time. This theory stands aloof from quantum theory that posits an ultimate discrete quantization of all physical entities. The holy grail of theoretical physics since the 1920s has been to unify (special and general) relativity theory with quantum mechanics. One such partial attempt on the way to a so-called "Complete Theory of Nature" via a "Quantum Theory of Fields" is presented in Joy Christian's article 'Absolute Being vs Relative Becoming' (op. cit.) which introduces Planck-scale magnitudes as upper and lower bounds in order to demonstrate how time itself is causally generated by the movement of the physical world.
Whereas Einsteinian special relativity deals only with the motion of physical entities relative to different four-dimensional space-time frames of reference moving uniformly with respect to each other, Christian introduces in addition the internal movement or change of physical systems moving within such co-ordinate frames of reference. Hence, in an oblique way, the perspective is widened from modern physics' intense focus on motion (or more precisely: locomotion, change of position) to consider also other kinds of movement covered by the Aristotelean conception of movement of four kinds, namely, change with respect to what (becoming and perishing), how (qualitative change), how much (waxing and waning, growth and shrinkage) and place (locomotion). The difference from the four kinds of Aristotelean movement/change is that, for modern physics, all movement has to be conceived in quantitative, mathematical terms, so that the internal movement of a physical system consisting of N particles is considered as a phase space of 2N+1 dimensions representing the dynamical variables of the particles, position and momentum, plus one-dimensional time, t. Christian therefore calls his proposed theory a "generalized theory of relativity", not to be confused with Einstein's theory of general relativity that takes into account gravitational force and accelerating referential frames. Through this generalizing extension, time itself comes to be conceived as depending mathematically not only upon change of position (expressed by the Lorentz transformation), but also upon the change in phase space of the physical system under consideration. Again, this has an oblique affinity to the Aristotelean conception of time, which is not absolute, as in the Newtonian paradigm, but derivative of movement. Time for Aristotle, namely, is the counting number resulting from counting physical change/movement, e.g. the time counted in months by observing the waxing and waning of the moon.
In Christian's generalized theory of special relativity (quantum special relativity) it is no longer motion taking place in four-dimensional space-time, but, more generally movement/change taking place in a 4+2N dimensional space-time-phasespace. Relativity now means that time is relative not only to a spatial co-ordinate frame moving uniformly with constant velocity v relative to another co-ordinate frame, but in addition to a uniformly changing physical system whose (rate of) change Christian captures with a further constant, w (omega). The transformation factor between reference frames hence becomes more complicated, depending now not only on v, but also on w, providing that one assumes that, just as it is assumed that the upper bound for the rate of change of position is the speed of light, there is also an upper bound for the rate of change of the physical system itself, with this upper bound depending on limiting Planck quanta deriving from quantum mechanics. The speed of light, c, is itself a Planck quantum, namely, c = lP/tP, where lP is the Planck length and tP is the Planck time.
Christian makes a crucial move by noting, "In particular, the Planck time tP is widely thought to be the minimum possible duration. It is then only natural to suspect that the inverse of the Planck time-namely 1/tP, with its approximate value of 10+43 Hertz in ordinary units-must correspond to the absolute upper bound on how fast a physical state can possibly evolve". This postulated absolute maximum rate of change is then incorporated into the (usual Lorentz) transformation factor between inertial reference frames (the square root of the expression 1/[1 - (v/c)2]), yielding a factor with an additional term dependent on both v and w, namely, the square root of the expression 1/[1 - (v/c)2 - (tPw)2], where v is bounded above by c, and w is bounded above by the inverse of tP. With this neatly symmetrical addition there results a mathematically expressible, mutual interdependence among time t, position vector x, and phase-state vector y. In particular, the dependence of t upon y implies that the change in phase state of the physical system, its movement, induces change in t, i.e. it efficiently causes the growth of time.
Christian points out as an argument in favour of his generalized theory that, "unlike in special relativity, in the present theory physical quantities such as lengths, durations, energies, and momenta remain bounded by their respective Planck scale values". A consequence of this boundedness, however, is also a quantization of the physical magnitudes of length and time, in particular, and hence a breaking up of the space-time continuum into discrete time-space elements. How is this to be reconciled with the "instant-states" of the phase space of the physical system, with the "instants of time", with the "infinitesimals" of change of state, change of position and change of time that Christian invokes at various points throughout his argument, not to mention the various acts of integration over infinitesimals of time? If one takes the assertion seriously that "Planck time tP is [widely thought to be] the minimum possible duration", which is crucial to Christian's line of argument, then how can there be any instant of time?
Note first of all that, if there is an absolute minimum time interval, there can be no "clock of unlimited accuracy", as Christian assumes in his reasoning. Secondly, if the time increment, Dt, cannot approach the limit of zero required for infinitesimals, there can be no differentiation or integration with respect to t. This may not be an insurmountable problem if infinitesmals and integrals are replaced by finite differences and sums thereof. Thirdly, and most fundamentally, if tP is indeed an absolute lower bound, there is no way of pin-pointing a point in time, i.e. an instantaneous now, and hence there is an indeterminacy about both the external position and internal state of the physical system under consideration, for neither are instantaneous any longer. That is, there is no way of describing the physical system's dynamical state as a function of the real variable, t. The physical system is in a superposition of infinitely many dynamical states over the infinite continuum of the time interval, tP, which is, although finitely bounded, also composed of a continuous infinity of real numbers. Thus, whereas classical quantum mechanics after Heisenberg posits a superposition of dynamical states at any given instant of time, now there is no longer even the possibility of pin-pointing an instant, and the indeterminancy becomes also temporal. Within the temporal interval tP 'now' and 'then' are indistinguishable, and the physical system quivers or wavers in an indeterminancy with respect to both dynamical state and time. Time itself would have to be conceived as the complex superposition of infinitely many time quanta tP (cf. 7.3 The phenomena of movement and indeterminacy in relation to continuity, discreteness and limit, and complex-imaginary time).
If, on the other hand, one wants to retain temporal instants in a time continuum (as required by differentiation with respect to time), one is faced with another dilemma if tP is to be the absolute lower bound for a temporal interval, for then, an extended space-time-state phase-space either will have an instantaneous state in which it is forever fixed, or it will never be in just one instantaneous state, but both in a state at time t and also in prospective states at times greater than t + tP, i.e. it must straddle the gap between now and then, and in a sense 'be' both in a present state now and future states then. Why is this so? If the physical system has a uniquely determined instantaneous dynamical state now, and its change is to be continuous in time, how can it change continuously if the next instant in time is separated from it by an interval of at least tP? The physical system would be forever frozen in its instantaneous state. Alternatively, the physical system must have always already bridged the temporal gap and 'be' both 'now' and wavering infinitely in all potential future 'thens', each separated by an interval of tP. In other words, being itself would then not be a matter simply of instantaneous being now, but also of prospective being then (and also of retrospectively having been back-then). If a system is simply in an instantaneous state, then it cannot move (continuously), cut off from a state at a quantum leap tP away, and the universe is Parmenidean. For the universe to be physically moving, all physical entities must always be both in a now-state and also all potential prospective then-states, and movement itself would have to be conceived as the wavering of quantum indeterminacy (with one quantum state coming into focus, and then another) rather than as a change from one definite state at one point in time to another at a later point in time. Hence, no matter whether instantaneous time or a minimum time-interval is postulated, the result is the wavering indeterminacy of dynamical states of physical systems over both space and time.
This state of affairs, however, should not be described as Heraclitean as Christian does, for the ontology of 'everything is in movement' is untenable, as Plato already demonstrated (cf. e.g. Sophist 249b and GA19:488). If everything moves, there can be no unmoving 'ideas' which, in the present context, means that there could be no 'unmoving', steady fundamental concepts of physics such as mass, force, energy, position, etc. by means of which the movement of physical entities is theorized. Knowledge must have steady foundations, even if it is a knowledge of movement in its quantum indeterminacy. The lo/goj of scientific knowing, even in advanced quantum physics, must be stable, i.e. well-defined, in its fundamental concepts which, in turn, are called to mind by any theorizing movement/activity of the physicist's mind.
If there is an indeterminacy in the dynamical state of the system together with a temporal indeterminacy as a result of the minimum time-quantum demarcated by Planck time, then there is no way to causally determine a future state of a system starting with an initial state now, for this initial now cannot be singled out and privileged as the governing principle effecting later states. Rather, a dynamic physical system is always already hovering in its present dynamical state together with the infinite multitude of potential future states which, however, do not depend causally in a unique, efficient way on the present state.
The lo/goj of modern mathematical physics comes up against a limit in the Planck scale where the continuity of the physical universe gives way to quantum discreteness and therefore to quantum indeterminacy and indefiniteness. The infinitesimal calculus employed throughout modern mathematical physics since Newton breaks down on the Planck scale, becoming finitely very small where the infinitesimally small should really count. Is this a matter of an empirically validated scientific discovery, or is it an effect of the scientific lo/goj itself in its essentially discrete nature which perennially raises, over and over again and in ever new phenomenal garbs, the ancient antinomy between the continuum and discreteness that points to an unsurpassable limit to knowing the physical, moving world? (Cf. Excursus 1)
7.3.4 Excursus 4: On quantum computing and qubits (David Deutsch)(36)A recent development in quantum physics (Deutsch 1985) opens up the prospect of employing quantum indeterminacy in computing with the aim of increasing the computing power of computers. Computability, or computing power, is the concern of complexity theory which deals not only with what is computable at all (Turing machine theory), but with how much time the computation takes. Quantum computing theory already shows that, if and when a quantum computer can be built, it will significantly reduce the computing time required for computational tasks (e.g. Grover's algorithm), thus, among other things, endangering the security of encrypted code which relies on decryption computations requiring enormous amounts of time, such as years and centuries, to crack a code.
Quantum computing goes hand in hand with a quantum-digital cast of being which postulates that physical reality is dissoluble ultimately into discrete quantum bits so that a "universal quantum computer Q" can be devised "which is capable of perfectly simulating every finite, realizable physical system". (Deutsch 1985 p. 103). The practical-technological objective of quantum computing research is to build a quantum computer that will be even more effective, i.e. faster, than a classical Turing machine in producing its computational result. In conceiving computation as a physical process and physical processes as computations, the theoretical ambition is apparent to conceive human thinking itself as computation and therefore materialistically as a quantum-physical process.
"Like a Turing machine, a model quantum computer Q, consists of two components, a finite processor and an infinite memory, of which only a finite portion is ever used. The computation proceeds in steps of fixed duration T, and during each step only the processor and a finite part of the memory interact, the rest of the memory remaining static." (Deutsch 1985 ibid.) In addition to the finite number of bits in the processor (Turing's machine states, conceived as equivalent to states of mind) and the countable number of bits in the memory, the universal quantum computer Q has specified also an integer "'address' number of the currently scanned tape location" (ibid.) which, of course, can be expressed as a binary number. The input into Q is therefore a ket-vector consisting of an integer memory address, the processor bits and the memory bits. These "computational basis states" (ibid.) form the basis for a Hermitian space H spanned by these "simultaneous eigenvectors" (ibid.). The initial input is transformed, algorithmically one finite step after another, by "a constant unitary operator U" on H. The difference from Turing machines is that Turing machines "are those quantum computers whose dynamics ensure that they remain in a computational basis state at the end of each step, given that they start in one" whereas "Q admits a further class of programs which evolve computational basis states into linear superpositions of each other." (ibid.)
A state of Q thus admits in its processor and memory cells qubits, each of which is a complex superposition of |0> and |1> specifying a unit ket in complex two-dimensional Hilbert space. These superposed kets can also be rotated on the unit sphere in this two-dimensional Hilbert space. The complex superposition amounts to doubling the basic Turing machine. The more qubit cells in Q, the more the doubling into parallel-computing Turing machines. Since the content of each complex superposed qubit cannot be ascertained (an observable must give a real number), Q must be left 'in peace' in its quantum indeterminacy to complete its calculations until finally, after a finite number of unitary transformations of the Hilbert space of Q, the result is output as a computational basis state, which is simply a discrete binary number. As a computer, Q therefore moves from a binary input to a binary output, with many complex-superposed parallel Turing machines in between that are finally collapsed to produce a result. Insofar, quantum computing remains within the digital cast of being.
A qubit is a physical system, each of whose non-trivial observables is Boolean, providing just two observations, that can be coded as, say, 1 and -1. This can be interpreted as meaning that the physical system either has a certain property or not. The qubit's state itself at any time (i.e. finite countable computational step) is a complex superposition of both having the property and not having it. Only if the physical system has assumed, or has been prepared with, one of its eigenstates for a given observable is this ambiguity resolved for this observable, and the observed observable will always give just one of the real eigenvalues 1 or -1. Because of non-commutability, however, other Boolean observables on the same qubit, however, will be a genuine complex superposition of basis states, and the hovering ambiguity will remain: the qubit as physical system both has the property and does not.
This unwittingly retrieves Plato's dialectic in The Sophist, according to which any being that can move is, in a certain way, also what it is not. A movable/changeable being is a mh\ o)/n, and a mh\ o)/n, or non-being, is in a certain way. This perplexing ontological insight into changeable being is also at the heart of Hegel's dialectic: every being is also its negation. The decisive difference from quantum-mechanical computing is that, whereas quantum mechanics must resort to postulating 'many worlds' or, in the case of a qubit, dual worlds at any given real time t - viz. the one in which the property is present and the other in which its negation is present - as we have seen (2.9 Time and movement in Aristotle's thinking), the Aristotelean solution in the Physics, which builds upon Plato's, is to conceive any physical, movable being as a superposition of its present state and the absence of the state(s) toward which it is potentially under way (cf. 7.3.1 From antinomic discrete vs. continuous real time to complex-imaginary time). This solution is only possible because, in contrast to quantum physics, for which only the real time of the present instant is, the Aristotelean insight means that future time, which is not yet present, also is in its own way in being withheld in absence.
To see the phenomenological point more clearly and to complement the above considerations from 'inside' quantum mechanics, let us take an example 'outside', from everyday life, that allows the phenomena to be seen without theoretical constructions obscuring the view. This is necessary because examples from physics are construed from the outset within the mathematical theoretical terms through which physics attempts to calculably grasp the phenomena of movement. Suppose, mundanely enough, I am in the kitchen chopping an onion on the chopping board for the evening meal. Both I and the onion are in movement, not merely in motion. I am chopping the onion to put it in the frying pan that has been heated on the hotplate of the electric stove. I have also got some other vegetables, such as carrots, potatoes and mushrooms, around the chopping board which similarly will be used to make the evening meal. The onion in its movement has come from the onion basket in the pantry, and will continue its movement into the frying pan once it has been chopped. The chopping itself is a kind of movement that changes the onion and does not simply shift its place, as in the case of motion, but its form - into chopped onion. Although the potatoes are presently at rest on the kitchen table, this state of rest is part of a movement from where the potatoes have been in the potato basket in the pantry to their likewise being peeled and chopped and ending up in the frying pan, or in a saucepan to boil and later to be mashed. As the cook, I know where the onion and the potatoes have come from, and I also know where they are going, even though I may not yet have decided whether to sauté or boil the potatoes; they could also be grated and turned into a crisp, fried potato pancake. The future movement of the potatoes is therefore to this extent indeterminate, or open, with a finite spectrum of potential, depending as it does on me, the cook, as mover, and on the ends I set. Therefore, in my cooking activity I have past, present and future together implicitly in view, for otherwise it would be impossible for me to engage in this activity, this everyday movement.It always bothers me that according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space and no matter how tiny a region of time. [...] I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed and the laws will turn out to be simple.
Now suppose that I have finished chopping one half of the onion and have put it in the frying pan when the phone rings, and, after turning the hotplate down to low, I go out of the kitchen to answer it. The other half of the onion is left lying on the chopping board, and I leave the light on in the kitchen. While I am on the phone, my wife comes looking for me in the kitchen, for that is the place where I usually am at this time of day. What she sees is the half onion lying on the chopping board, the other vegetables on the table around the chopping board, the already chopped onion frying on the stove at a low heat. My wife does not merely see a present state at an instant of time, but sees the movement of cooking, even though it is presently at rest. Without knowing the details, she sees where the onion has come from, namely, from its usual place in the pantry, and she sees where it is headed, namely, either into the frying pan or into the fridge to be used tomorrow. She also sees the movement of the potatoes, carrots and mushrooms, whence they came and whither they are going, although it is indeterminate what future awaits them from a finite spectrum of possibilities. She doesn't know whether the potatoes will be peeled, chopped and added to the onion in the frying pan, whether they will be peeled and then boiled in a saucepan and finally mashed, or whether they will be peeled, grated and fried separately. A description of future possibilites in terms of real space-time co-ordinates, which are of their nature uncountably infinite, would be an instance of exact-scientific overkill.
She also sees my absence. My absence is present to her. But not only that. She sees where I have been a short time ago and she sees where I will be coming back to in a short time, namely, the kitchen. So she sees, in the present, but as an absence, both my past and future movement. She also sees where I am at present, albeit indeterminately, namely, somewhere else in the flat, probably in my office or in the bathroom. All this she sees 3D-temporally by viewing the situation of movement at rest in the kitchen, with the onions simmering at low heat on the stove, the other vegetables ready for being prepared for cooking, and so on. She takes this situation in at a glance and understands it without having to make explicit her perceptions and draw syllogistic conclusions from them. The situation she understands is one of movement involving me, onions, potatoes, carrots, mushrooms, the stove, etc. The presence, past and future of the movements in the situation are understood, albeit indeterminately, but within the three temporal ecstacies that are taken for granted by understanding. She sees at a glance that the chopped onion is on its way to its future in which it will be part of an evening meal, and she sees that I will soon be coming back to the kitchen, i.e. that that is the future destination of my movement in the short term, and that soon the evening meal will be on the table. A stranger to our flat, such as a burglar, would also understand the situation of the empty kitchen as one of various movements associated with cooking and also that the cook will soon be coming back to the kitchen, but the burglar's understanding of the past and future movements of cooking ingredients and the cook would be more indeterminate than my wife's because he is not familiar with the particularities of our household world.
My wife needs no laws of physics to predict and precalculate either my movements or the onion's. Such laws of physical motion, especially laws of quantum mechanics, are not only superfluous, but also useless for understanding the movements comprised by the situation. If applied, such laws of physical motion would only obfuscate through a theoretical construction laid over the phenomena. The situation and its movements are understood already, and with a certain indeterminacy, before any scientific physical view of it could ever be formulated by reducing its everyday context and meaning to get an entirely artificial situation involving entities conceived of as extended in some fashion and subject to various force fields amenable to mathematical formulation. In other words, the scientific physical description of the situation can only make us dumber than we are as beings at home in a world which we always already understand and to which we are attuned. To treat the situation in the vacant kitchen as a physical system in a certain classical or quantum state to which certain dynamical variables apply would indeed generate a mathematical problem in physics, which may or may not be soluble, determinately or otherwise, but the imposition of the mathematical physical problem would obliterate the situation itself and make it not only altogether incomprehensible but also entirely invisible. The situation itself, involving movements in the timespace of our shared household world, would have dropped out of sight. That is indeed the danger of the modern scientific mode of access to the world, that it bamboozles us with ideas bearing the hallmark of scientific seriousness and backed up by powerful institutions of learning premised upon scientific method as self-evident.
A physicist might object by asking whether we are to be content with the mere description of a banal situation as given above as all we can hope for, as opposed to digging deeper into the phenomenal situation to uncover its fundamental, underlying laws of motion. Surely, he would say, we can do better than merely reiterate a trivial description of a banal situation. The response to this objection is that the above is not merely description, but lays bare (an aspect of) the inconspicuous temporal structure of the world we human beings inhabit and which we simply take for granted. Expressly noticing how and that we perceive movement itself as stretched out into three temporal dimensions in a logic-defying unity must be a cause for wonderment and the starting-point for an explicit phenomenological ontology. How is it possible that in a situation of rest we can see movement? And yet we do without thinking twice about it. Any movement taking place takes place in both space and three-dimensional time. The determinacy or indeterminacy of movement is a phenomenon that can only occur within this time-space in the transition from the present to the future, from here-and-now to there-and-then, or from the past to the present. Moveover, the future there-and-then is present as an absence in the present situation, a perplexing circumstance. What is as yet withheld from the here-and-now is nevertheless present, albeit to a greater or lesser degree of indeterminacy.
Once this is seen, it would be folly to assume as a matter of principle that there are laws of movement, whether known or as yet unknown, governing this transition in every case and for every kind of movement. And yet, modern physics believes that, in principle, it has within reach the ultimate truth about movement and change for all that is, encapsulated in fundamental, mathematical physical laws of motion. It has been digging itself into this hole since the seventeenth century. By contrast, on the basis of a phenomenological insight, we should not be surprised that in the attempt to formulate mathematical laws of motion, modern physics strikes upon an indeterminacy that challenges the universal validity of a rigorous principle of efficient causality.
There is a reluctance among today's scientists and analytic philosophers to seriously and radically pose the question concerning time, even though it is clearly on the agenda. Nevertheless, philosophically it is time for dissidents to raise their voices against the regime of modern science that has been in power for well over three hundred years.
10a. Cf. also Met. 1009a32ff: "Namely, being [presencing] is said in a twofold way, so that in one way a being [a present] admits becoming something out of a non-being [a non-present], and a way it does not; and the same can be [presence] and not be [not presence] at the same time, but not according to the same mode of being [presencing]; potentially,  namely, the same can admit being [presencing] at the same time as its opposite, but not in actual, finished presence." (to\ ga\r o)\n le/getai dixw=j, w(/st' e)stin o(\n tro/pon e)nde/xetai gi/gnesqai/ ti e)k tou= mh\ o)/ntoj, e)sti d' o(\n ou)/, kai\ a(/ma to\ au)to\ ei)=nai kai\ o)\n kai\ mh\ o)/n, a)ll' ou) kata\ tau/to\ [o)/n]: duna/mei  me\n ga\r e)nde/xetai a(/ma tau/to\ ei)=nai ta\ e)nanti/a, e)ntelexei/# d' ou)/.) If (the meaning of) being is confined to presence in the present, then the principle of non-contradiction (Met. G 3;1005b30) holds water, but if (the meaning of) being encompasses also the modes of absencing, as in the case of potentiality, then a contradiction can 'be', namely, as a twofold presence-and-absence. Potentiality is the presence of a future presence that is as yet absent which, however, is also present in the present being [present] as an absence. Anything capable of change/movement must have this twofold presence. Aristotle goes on immediately to admit also another kind of (ever-)presencing that is compatible with Parmenides by "assuming also another being of beings [another mode of presencing of presents] as springing up thoroughly without movement or decay or becoming" (u(polamba/nein kai\ a)/llhn tina\ ou)si/an ei)=nai tw=n o)/ntwn $(= ou)/te ki/nhsij u(pa/rxein ou)/te fqora\ ou)/te ge/nesij to\ para/pan. 1009a36f). Back to 10a
23a. In his exchange with Makoto Nakada in
'A Dialogue on Intercultural Angeletics' (2011), Rafael Capurro says, "Although
we mostly live immersed in the given openness of everyday existence, exchanging
messages and maintaining communication through the phatic function, we
have the potentiality to grasp a given (historical) disclosure of Being
as a possible one, that is to say, to change its truth. [...] The radical
questioning of a given world-openness by a messenger of Being that makes
explicit this ontological or structural relation between Being and messenger,
can lead to strong opposition from the defenders of the status quo and
– to condemnation of the messenger, as in the case of Socrates." Back
31a. "Finally, the same mistake is made when,
as is done in quantum theory, time is described as a real parameter. Time
is measurable by clocks. A point-in-time is a fiction. It could, in turn,
only be determined by an irreversible process, and that only with finite
imprecision. Through these unsolved problems, quantum theory does not point
back to classical physics, but beyond it." C-F. von Weizäcker
der Physik 1985 Kap. 13: Jenseits der Quantentheorie, pp. 588ff, as
communicated by Rafael Capurro on 11 August 2010. Back