Twice the story about

ACHILLES AND THE TORTOISE

AND THE PROBLEM OF TIME[1]

Zenonian mathematics

The Aristotelian idea of causality was based on the ancient techné, the art of the handicrafts, poetry etc. earning different purposes or goals. Therefore Aristotle of course also operated with final causes.  The ancients used levels, rollers, wedges, and slopes for their work. Newton surely learned from such technique but also from the more rational science of statics. The increasing empiricism in the course of history confirmed the value of the mathematical way of thinking and, eventually, the endeavours of gaining rational knowledge turned itself into a special ‘art’ on mathematical basis for constructing appropriate models of thought. In this way, the subject intellectualised itself, thereby neglecting the ‘arts’ of the hands aiming at common purposes. An exception was here of course the interest in concrete experimental laboratory work to test the mathematical models.

The new rational way of life and especially mathematical physics was based on ideas of (advanced) time and causality. Not to acknowledge the importance of these concepts would seriously have threatened the whole ‘Newtonian’ way of thinking, which had proven so useful in science as well as in production and economy ever since. But scientific theory and general praxis in society and laboratory differed in essential points. Essential decisions by measurement of movements had to be done at certain time points experienced as NOWs of action and indicated by clocks, calendars etc. On one hand, in mathematical terms to calculate such NOWs and the state of affairs at such time points (on the basis of their development functions) demanded differentiation and integration. To develop such methods and the relevant concepts was certainly a historical achievement that in no way was a matter of course, and for which preconditions we therefore seriously have to ask.

On the other hand, the shortcoming of ancient intellectualism was already felt at the time of, for example, Zeno of Elea. He therefore proposed his famous paradox about the race of Achilles and the tortoise. Of course the high-speed Achilles would never have lost a race against the tortoise if this low-speed animal was not allowed a certain lead - and especially had Achilles himself been a little more practical-minded! Hence we can see this paradox of Zeno as an ironic commentary at that time to the growing intellectualism of the Greek mind itself. According to Zeno the first business for Achilles - in terms of the story - was to reach the place of the tortoise, this surely being the very first precondition of overtaking the other. And that was just the problem! Why? The story was really told in a quite cunning way. When arriving at the place where Achilles had lastly seen the tortoise, this point of the racing track was actually empty; the tortoise had just left it for crawling a little bit further. This tactical mistake of Achilles’ in fact happened again and again. The geometrical problem was just these line segments in Zeno’s story (in more modern terms) being defined open; consequently, this line segment series could never be definitely ended – Achilles no-longer moving, not-yet ending. The ancients of course missed the limes concept of the modern calculus. Therefore, in Achilles’ own eyes, his partial goals were always empty points, and so the possibly final point of this race itself would necessarily be empty, undecidable – or, as the Greeks said themselves, ‘unsay-able’ – that is, unimaginable in the eyes of silly Achilles the ‘philosopher’. The race was made up by a recursively infinite number of stages defined as relations no one of which really defining the final goal (for instance like the never ending calculating of the diagonal of the square). So, without being definitely conscious about his own goal and the way to achieve it, Achilles - of course! - could not end his race as the winner. What was missing by fast-running Achilles? Simply, he just couldn’t imagine nor define his own unsay-able goal, and neither could he imagine the real NOW of being on a par with the tortoise. He neither conceptualised any external (objective) time valid to both agents thereby reducing the subjective NOW into an abstract time differential, a NOW defining the theoretical endpoint of their race as a collective action. To determine such a NOW would just be a conscious confirmation of a fact in common agreement, philosophically a ‘setting’. And to this Achilles was just absolutely unable.

- o – 0 – o -

Zenonian metaphysics

The anticipational basis of temporal coherence is the complex second-personal co-operative experience of the I-Thou assuming the possibility of duplicity of co-ordinated mutually overlapping event series describable by parallel progressing, re-entering etc. The epistemological and metaphysical problem of time is on this starting point to create the abstract continuous and quasi-linear (Euclidean) time model – and eventually to give up all reflection of its complexity.

We have already once discussed Zeno’s famous paradox of Achilles and the Tortoise. This race could not have been modelled in any temporally quasi-linear form, even if the distance to be run certainly could; in fact, the racetrack itself was just such a simple Euclidean line segment. The race had a complex structure based on I-Thou co-operation (but as mentioned excluding any final agreement). Seen from a third-person stance the race would of course make no problem at all. With a modern concept of velocity you could certainly calculate the spatial-temporal point where the competitors were on a par, and even without such a concept you simply as the third person would have seen who the winner was. Not so in Zeno’s case! Here we lack just such a third-person vantage point at all. No third person was even mentioned in the story (e.g. as a judge). At issue was therefore not to ascertain that someone ran faster (all knew that!), but rather, exactly from the second-person stance, to ascertain the impossibility of deciding the crucial NOW of the two runners being on a par. Lacking an ‘objective’ concept of velocity defined from the third-person stance - just the lack of the temporal concept of movement - was the real problem. Of course all knew what fast running meant. Certainly Achilles ran faster, but exactly in this Zenonian model of the race he could never be the declared winner! That was the matter.

With a real, but abstract concept of velocity there would have been no problem at all. Further, the Zenonian race would easily have been won by Achilles, if they – undisturbed by arithmetical troubles – had run a little bit further to give time for the runners’ preserved excess energy to be transformed into heat, and even in the light of the equally preserved self-referential consciousness to decide the result of the race. Achilles then had simply overcome the tortoise. In such cases, the post-race experience would clearly have been sufficient to confirm the previous anticipation of the situation once both being on a par; this confirmation being just the aim of the running.

But exactly this being on a par would in this narrative definition of the race never be the case. Its definition had the consequence that the contradiction characterised by the ‘open’ point of the undecided ‘no longer & not yet’, that is the missing limes concept could never be definitely cancelled. This opened up for the behavioural indefiniteness that absolutely prevented any final equalisation from the vantage point of the third-person stance, for instance a ‘normal’ self-referential consciousness of both actors. Such an ‘objective’ valuating or judging (self-judging) stance could under the rudimentary first/second-person conditions of participation – without any possibility of mutual agreement – never be ‘construed’.

You could ask why this Zenonian narrative in the course of time attracted so much attention. What is the really ‘paradoxical’ in this odd story? Even the later Euclid avoided all kinds of ‘flowing’ points, otherwise not absolutely unknown to ancient thinking to be involved in eleatic quarrels. So geometry in the classical antiquity was constructed as an absolutely spatial but also absolutely a-temporal science, which for the next two millennia was assumed rather axiomatically valid.

Let us resume. The Zenonian narrative about Achilles and the tortoise was based on the assumption of an iterative never ending spatial divisibility but, on the other hand, excluding any moment of anticipation naturally including the temporal aspect, so that the contradiction contained in the formula of anticipating the coming NOW was in principle unsolvable. This offered the Euclidean geometers the possibility of quasi-temporal calculating a chain of dividing points on any line-segment; in this way certainly recognised as coherent, but, at the same time contradicting the generally accepted Euclidean (a-temporal) concept of the continuous space. That is, there was an unsolvable contradiction between the proposed intentionality of the narrative and the generally realised but not yet conceptualised intentionality based on fully developed conscious anticipation from the third-person stance. Missing was the third person in the narrative, and missing was the concept of temporal movement and therefore of time as such, too.

However, you might ask: why had this understanding not yet been possible at Zeno’s time – and which historical conditions later made it possible? The last question was attempted answered above (that is in the manus; see also my book), and the answer to the first is given at the same time. I presume that the political, economical, and theological quarrels in late Antiquity was necessary to force the social attention to focus on this general contradiction especially actualised, first by the social transition from using free artisans instead of slave artisans (especially in the great towns of northern Africa, for instance Antiochia), these now demanding payments and demanding the right to agree in matters of payments - that is the “right price” for their work, this further giving rise also to the first formulations of the impetus theory (Philoponos a. o.) Later in the Middle Ages, also the co-operative work in the manufactures – especially analysing movements of work as such along the time scale under the condition of decisions by the workshop master/owner – further developed this theory as a forerunner of the theory of inertia.

All of this led to a coherence view of the time parameter. Cancelling the above mentioned contradiction of the ‘no longer & not yet’, when coming to the end of an open line segment, was no longer a real problem, because all humans of course now have ‘normal’ (modern) intentional experiences of their life and work including the self-referential observation (through self-consciousness as essential third-person phenomenon, but not at all mentioned in the Zenonian narrative) of their work that have to be finished and paid for in a certain time. So the missing instances at the time of Zeno were all in place in the overall coherent social ‘universe’ of modern times. The real recognition of material and energetic flows in mills and the industrial mode of production simply necessitated the conceptualisation of the temporality of movements and so of time as such. The final break-through of this task was, as we know, eventually done by Newton.

Thus the ‘objectified’ third-person view of work and velocity – the view of an observer standing principally outside the world! – had far-reaching conceptual consequences. The transformation of the quasi-linear temporal coherence into a differentiable time continuum (with the present NOW ‘flowing’ with the velocity of…?) demanded, respectively conditioned the new model of all three continuities into one, that of the line, that of the time, and the third one – just that of the most difficult moment to ‘objectify’ – that of the self-consciousness as such. Physical velocity - now easily defined as v = dq/dt at any selected NOW – determines, by means of the fully conceptualised limes transition, the temporal development of the actual movement – exactly in the form of a mathematical decision-making as limes transgression absolutely impossible to silly old Achilles.

So, seemingly, the matter is quite simple. But today to ‘solve’ Zeno’s paradox by means of differential analysis is of course pure anachronism.