It was the pre-Socratic thinker Parmenides who first mooted the idea (as far as we know) in a document, only fragments of which survive in the writings of later philosophers,^{1} that all movement and development is illusory. His disciple Zeno developed this insight through a series of subtle paradoxes, over which philosophers and logicians have been arguing ever since, but there is no doubt that he made a unique contribution to the form of argument known as *reductio ad absurdam*, the pursuit of an argument until it confounds all common sense by resulting in a contradiction.

One of famous paradoxes is known as Achilles and the tortoise. Achilles is in a race with a tortoise and gives the tortoise a head start. Zeno maintained that logically Achilles would never be able to catch up, showing that movement is illusory. He reasoned thus: in order for Achilles to catch the tortoise he would first have to pass the point the tortoise had reached when he (Achilles) began running; but by the time he had reached that point, the tortoise would have advanced further to a new point; and by the time he had reached *that* point…and so on *ad infinitum*. A variation of this paradox is that in order to walk a given length of path, one must first reach the halfway mark; but in order to reach this point, one must first reach the halfway mark of the halfway mark, and so on in an infinite regress. Thus, Zeno argued, movement is an illusion as it defies reason.

From a purely logical point of view, it seems that the paradoxes are insoluble.^{2} That is because they are a product of pure reasoning. To illustrate this, consider a variation on this theme, a Zeno-like paradox: the impossibility of getting anything whatsoever done.^{3} Take a simple, everyday action, such as making a cup of tea. In order to make a cup of tea, I must first fill the kettle; in order to fill the kettle, I must first turn on the tap, before which I must reach out to the tap, preceded by the decision to reach out to the tap; between these two events there is a number – undetermined and possibly infinitely extendible – of describable stages involving neurons, nerves, muscles, sinews and various bodily appendages. And this is to get only as far as filling the kettle; actually, only as far as turning on the tap.

The ‘solution’ of the paradox in this example should be fairly obvious; the labelling of each stage of the process (a stage being, moreover, a somewhat arbitrary choice) requires a conscious amassing, ordering and expressing of verbal information to describe actions most of which take place unconsciously. To carry out a simple action in the real world takes a finite amount of time. To undertake a detailed description of every possible stage, both conscious and unconscious, of that action in the real world (and descriptions can only be undertaken in the real world) takes potentially an infinite amount of time, and certainly much longer than actually carrying it out.

A similar objection can be levelled at Zeno’s paradoxes. Zeno takes an everyday action – I think we can stretch our credulity a little to accept a race with a tortoise an everyday action – and divides it infinitesimally, not into descriptive utterances in this case, but into fractional expansions ordered in mathematical series, which are infinite. Zeno and his protagonists do not even have to enumerate them beyond the first, as denoting the function is sufficient, such is the invariable rigour of mathematics. Pure reasoning can be preserved without, however, any relationship to the real world.

No one doubts the extreme usefulness of logic and mathematics in underpinning the natural sciences. However, the foundations of logic remain unchanged after 2500 years and are rooted in the ontological suppositions of antiquity regarding the nature of reality. This has determined the course of Western philosophy, allowed an extremely sophisticated dialogue to take place within its parameters, but also limited its applicability to describing the real world and solving real world issues. It is extraordinary, for example, that within the Western philosophical tradition categories such as movement, change and – particularly – relationship are difficult to discuss, despite the fact that they constitute essential elements of all reality.^{4}

To some extent, the same problem exists with mathematics, perhaps even in a more extreme form. Mathematics is pure Platonism: forms or ideals bearing no relation to objects in the real world. As pure abstraction mathematics has enabled explanations of the nature of reality in an unsurpassed level of sophistication, which has also unlocked unprecedented levels of mastery of the world through technological innovation. Pure reasoning, though, gives rise to paradoxes, as both Kant and Gödel in their own ways have demonstrated.^{5} Moreover, technological mastery is limited by the theoretical conception of reality, and this is limited in a more profound and perhaps insoluble limitation: that even the most exquisite mathematical models can only ever be an approximation of reality.^{6}

NOTES

- Details of the historical transmission of the Parmenides’ and Zeno’s views can be found at: Palmer, John, “Parmenides”, The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2016/entries/parmenides/>.
- Mathematicians would hold that the paradox has been solved by integral calculus, which sums an infinite series and thus models the real world.
- Many in the modern workplace would probably say that this is a fact of life rather than a philosophical conundrum, given our ever-expanding capacity for generating and consuming information.
- For example, the law of contradiction, which states that p and not-p cannot be true of the same thing, is rooted in a very concrete and reified conception of reality: a thing cannot be the same a something else which is not the thing. Except of course, when it is an element of a system, something of which the ancient Greek world had no understanding. Taoism, by contrast, has concepts of relativity and relationality at its core.
- Kant’s antinomies, discussed in
*The Critique of Pure Reason*; Gödel’s*Incompleteness Theorem*. - Discussion of the status of scientific theories in the mid-to-late twentieth century was dominated by the ideas of Karl Popper and Thomas Kuhn. Popper, reacting against the logical atomism of the Vienna Circle, declared that theories could only ever be tentative as they were permanently awaiting falsification. Kuhn, a historian of science, noted the fact that every theory was in time superseded. They differed, though, in their interpretation of this transformation. For Popper, old theories existed as limiting cases of better theories, while for Kuhn a fundamental reconceptualization of fundamentals was required, which meant the change was stochastic .

Zeno’s paradox is arguably addressed by the advent of differential calculus in the late 17th century and the concomitant notion of the limit of an infinite sequence or series. However a century and a half elapsed before this concept was made mathematically rigorous by Bernard Bolzano and the way paved for a consistent new framework of analysis to be established, testimony to the non-trivial nature of the Zeno paradox.