Chapter 42

It’s a repeat occasion that I come across the view being expressed that with the invention of calculus, some of Zeno’s famous paradoxes, specifically having to do with the traversal of an infinite series of points, were solved. This claim actually serves to underscore a critical disconnect between mathematicians and metaphysicians. Firstly, to be fair, I will confess that when I was young, Zeno was something of an intellectual hero to me. He remained unchallenged until I actually took the time to learn about his master, Parmenides. However, all sentimental fondness aside, the issue is a basic one: calculus is designed to perform specific empirical tasks, not to investigate the problem of void.

When taking on the measurement of velocity, for example, calculus can make use of a “convergent series” which produces a way, mathematically, to account for the infinite series, obstacle-free, and produce useful values. In calculus equations, the tasks are admittedly practical ones, but in order to complete them, the question of the nature of infinity has been avoided. In other words, thanks to calculus, the problem of infinity in a metaphysical sense simply gets deferred, allowing mathematics to get on with its various applications.

Zeno was a champion of “reductio ad absurdum” which, like “via negativa”, “apagogical arguments”, “neti neti”, and the various other things this technique was called in other cultures, aims to adopt a position in order to determine that was proceeds from it is a falsity. In this regard, Zeno never made any positive claims about anything. The context of his paradoxes was the problem of void, which was posited at the time as a requirement for plurality. If there is an object A and an object B, with space in between, it is this “void” which is the nature of their separation. This void was considered to be infinite. One version of Zeno’s paradoxes (simplified) involves an archer shooting an arrow from point A to point B. If the arrow is to reach point B, it must first traverse 1/2 of the distance between point A and point B. To get 1/2 of the distance to point B, it must first traverse 1/2 of that, or 1/4. This is repeated, becoming an infinite regress. How then can void be infinite, when experience shows that the arrow indeed reaches its destination?

The atomists counter with the theory that if void is not infinite, then it is necessarily finite. Following this logic, in which there is a smallest unit of measure, an “atom” (having nothing to do with the particle), also results in self-contradiction. If there is a smallest unit of measure, then why can we not take 1/2 of that measurement? Well, the atomists would simply say that such a thing isn’t possible, as it is already the smallest it can be. It goes without saying that no one has discovered such a measurement, but the atomists would say that our instruments cannot detect such a small measurement. However, imagine you had a one inch circle. If you reduce that a little bit, its diameter becomes a fraction of an inch, and so on. At what point do you reach a point where you can no longer reduce the diameter? Theoretically, a point. However, you must necessarily make the claim that by definition a point has no diameter to be reduced. If such is the case, then a point (0-diameter) next to another point (0-diameter) equals the same measurement (0-diameter), and this then makes the point useless as an additive unit of measure which is responsible for the distance between point A and point B.

All such roads lead to nowhere for Zeno. However, behind this man was Parmenides whose poem, which has since been called “On Nature”, paints a completely different picture. It is a work in three sections, the proem, The Way of Truth (Aletheia), and The Way of Opinion (doxa). While this is no place for details on his work, the significance to Zeno’s paradoxes is the concept of being as one (i.e. a denial of plurality). While Zeno’s “reductio ad absurdum” intentionally did not make positive claims about this (for reasons which On Nature illustrates), he cleverly refuted each and every claim which supported plurality (along with time, motion, etc.), demonstrating that such claims are self-contradictory and therefore must be dismissed. There is a chronic misunderstanding for mathematicians, which have gone so far as to believe Zeno a lunatic for questioning things that are apparently irrefutable and obvious, making misinformed comments like “… one must ask whether Zeno actually believed it [...] One wonders whether he got around much.” Furthermore, this illustrates a distinction between science, which qualifies and quantifies our observations, and metaphysics which does not hold that the appearance and experience of phenomena is indicative of reality. There is quite a bit more to this story, but that will suffice for the sake of the point at hand…

Now back to calculus. It should be clear that the task to which calculus is applied is a far different matter, mathematical in nature, as opposed to metaphysical. Before calculus, there was a fire between metaphysicians and epistemologists, but a buffer appeared safely in between thereafter. Ironically, the bridge re-formed in 1958 when a soviet physicist termed a phenomenon in quantum physics the “quantum Zeno effect” after one of Zeno’s paradoxes. As science in recent years has boldly crossed a threshold which was previously seen as the edge of epistemology, a new appreciation is developing for metaphysics within science. Mysterious phenomena suddenly seem to have more in common with philosophical traditions stretching back for thousands of years. It’s a peculiar alliance, though perhaps a long time in coming.

Tags: calculus, metaphysics, zeno

You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.