Zeno’s Zeros

Zero is a scientific advancement.  It may seem basic, even fundamental, to any knowledge of numbers and mathematics, but consider where you begin counting – I suspect that a scant handful of people begin counting from zero on a regular basis (and that most of them are computer programmers).  Inquiry into the nature of nullity, absence, nothing is the province of civilizations with the capacity for thought in areas far astray from the business of living (or a strong imperative (religious, perhaps) to embrace a concept of zero).

It speaks, therefore, to the psyche of ancient Greece circa 500 CE that philosophers actively engaged with both the concept of zero and the concept of infinity, each mathematically complex topics which were not rigorously described until the nineteenth century.  While more recent work with infinity and zero is largely driven by application of those mathematical entities to problems in physics or other fields, the Greek philosophers of Aristotle’s ilk engaged with them to explore metaphysical questions that on the surface appear preposterous and indulgent.  Specifically, I am referring to Zeno’s Paradoxes.

I came across a reference to Zeno’s Paradoxes, though I now don’t remember where, and so of course I had to do some research.  For once, I did not dive into the original texts that reference Zeno’s work (Plato’s Parmenides, Aristotle’s Physics (among other works), and Simplicius’ On Aristotle’s Physics), and instead contented myself with an entry from Stanford’s encyclopedia of philosophy.  I quickly found that, while Zeno’s Paradoxes appear ridiculous, even silly, on the surface, considering them with greater seriousness reveals that they begin to plumb concepts of physics and mathematics that remain unresolved today.  Like quantum physics, Zeno makes claims that initially seem contradictory to lived experience and common sense, but are nonetheless legitimate queries of the nature of the universe.

The Stanford piece identifies nine paradoxes: the argument of denseness, the argument of finite size, the argument of complete divisibility, the dichotomy, Achilles and the Tortoise, the arrow, the stadium, the paradox of place, and the grain of millet.  I do not intend to explore each of them in detail in this post, since you can just go read the Stanford piece if you are interested in such a treatment, but there are similarities amongst many of them that I want to discuss, related to the concepts of zero and infinity.  Plato’s Parmenides is an argumentative piece in which Plato argues against the views of Parmenides, who “rejected pluralism and the reality of any kind of change.”  Zeno, in turn, is thought to have conceived his paradoxes in response to the anti-Parmenides arguments, perhaps especially those lodged by the Pythagoreans.

Of the nine paradoxes, several can be grouped together as pertaining to similar ideas and arguments, beginning with the arguments of denseness, finite size, and complete divisibility, which together are often referred to as the paradoxes of plurality, since they study the notion of what we might call, in the modern parlance, discrete versus continuous entity.  These arguments reminded me strongly of concepts from calculus, despite that calculus would not be developed until almost two thousand years after Zeno conceived his paradoxes.  Specifically, they contemplate the notion of infinite sums and infinite divisions.

Remember, Zeno is arguing against the pluralists, who hold that there exists a multitude of distinct entities – that is, that the universe is composed of a number of discrete components.  This would be a definite number, meaning that someone with the resources and patience could conceivably count each and every piece of the universe.  The argument of denseness points out that in order for each of these entities to possess a discrete existence, there must be something else which separates them, and that separating entity would also need to be counted, and additional separating entities would then be required, and so and so forth, ad infinitum.  Thus, an argument for plurality is an argument for infinity, which, until you gain an understanding of orders of infinity, is nonsensical.

Although the first of the paradoxes of plurality can be addressed with modern mathematics, since we now have the tools to understand and analyze orders of infinity, the argument of finite size speaks more directly to the fundamental concepts of calculus and the discrete versus continuous debate.  To me, it might be the most compelling of all of Zeno’s Paradoxes.  Consider how the integral operation in calculus purports to obtain the area under a curve.  Most introductory calculus courses begin with a Reimann sum, which approximates the area under a curve by dividing that area into approximate, geometric sections (rectangles).  The smaller you make the base width of each rectangle in the Reimann sum, the closer will be the result to the actual value of the area under the curve.  An integral, it is then explained, does the same thing, except that it makes the rectangles infinitely small.  This results in the actual area under the curve.

This explanation, if you stop and think about it, is nonsensical, because one of two things must be true.  Either a) the base width of the rectangles involves is zero, and thus the result of the integral should always be zero, or b) the base width of the rectangles is non-zero, in which case an infinite sum of them will always produce an infinite result.  This is Zeno’s argument framed in terms of calculus and Reimann sums.  If the horizontal span of the curve is one, consider dividing this line in half, and then each half in half, and so on and so forth unto infinity.  Intuition says that the sum of all of these fractions ought to be one, but mathematically an infinity of equal extended parts – meaning parts, like these rectangles, that possess some width – is infinitely large.

No easy resolution exists to this paradox.  While we can prove that the infinite sum ½+¼+… is one, dividing each half in half again with each iteration results in a different sum, with an infinite result, which implies that the natural extension of the Reimann sum scheme used as the basis of calculus instruction is that the area under every curve is infinite, a result that is intuitively nonsensical.  The Stanford piece glosses over this issue by redefining the problem and moving on – its authors clearly do not believe Zeno’s Paradoxes are valid – but I consider this a question worth deeper consideration.  It leads directly into the third paradox of plurality, which asserts that a body subdivided to an infinite and dimensionless extent is either nothing, or an illusion.

As the inverse of the previous discussion, this argument, instead of assuming that the infinite divisions of a whole must have some dimension, assumes that the infinite divisions are dimensionless, and since the sum of an infinite number of zeros is still zero, anything that can be infinitely divided in this fashion does not actually exist; instead of there being an infinite area under a finite curve, there is no area under a finite curve, a conclusion equally at odds with our lived experience and an intuitive understanding of geometry.  I am not mathematician enough to plumb the intricacies of ordered infinities or derive the proofs of Cauchy, Grunbaum, and Cantor, so I do not claim to provide answers to the questions Zeno’s work raises; my interest in Zeno’s Paradoxes is more rooted in their logical interest and the mental gymnastics required to probe them.  If it is resolutions you seek, Stanford’s piece on the Paradoxes goes out of its way to provide “solutions” to the Paradoxes.

If the paradoxes of plurality deal with the mathematics of zero and infinity, the paradoxes of motion enter the realms of physics.  Those of you readers familiar with the conflicts between relativity and quantum physics will find significant resonance between those debates and the issues raised by Zeno’s paradoxes of motion, the first of which asserts that motion is also an illusion because the motion can be divided infinitely, just like we divided objects in the paradoxes of plurality, and thus we have another infinite sum to address.  Similar logic can therefore be applied, so where the paradoxes of motion become really interesting is with the third paradox, the paradox of the arrow.

In the arrow paradox, Zeno explains that a flying arrow is actually at rest, since time is composed of moments, and the arrow cannot be moving within a moment since no time is passing, and therefore the flying arrow cannot be moving.  In other words, if time is discrete, then nothing can ever move, because nothing can move within each discrete instant.  The counterargument provided by Stanford is, predictably at this point, to assert that it is “fallacious to conclude from the fact that the arrow doesn’t travel any distance in an instant that it is at rest; whether it is in motion at an instant or not depends on whether it travels any distance in a finite interval.”  This argument, however, only tangentially addressed Zeno’s issue with motion.  If time is discrete, and composed of infinitely small instants, then the arrow would not exhibit continuous motion, but rather discontinuous motion.  My response, then, is that the arrow could move, but it would be moving in a series of discrete applications of instantaneous, infinite velocity that, when averaged over a finite distance, produce a perceived velocity.

If, when you started reading a post about the concept of zero, you anticipated that the discussion would come to involve teleporting arrows, you’re probably not a new visitor to the site.  So yes, my direct answer to Zeno’s arrow paradox is that discrete time does not preclude motion via teleportation.  More usefully, my indirect answer is that time is not discrete.  Since time can be thought of simply as a measuring of the natural progression of entropy, the idea of infinitely short instants is inapplicable, which is why motion can occur.  Or maybe I’m wrong, and arrows actually conduct stepwise teleports to their destinations.

Zeno’s stadium paradox, the final in the paradoxes of motion, is a useful thought experiment for understanding relative motion and reference frames, and the paradox of place provides an interesting framework for inquiring into the nature of space and locality, but I think teleporting arrows is a suitable point at which to conclude this discussion.  ‘Conclusion’ might be too strong a word, though; these paradoxes, regardless of whether you consider their assertions legitimate or nonsensical, are an infinite gold mine for mind-bending considerations of reality.  These were fascinating to consider, and I would be thrilled to engage with you in the comments on any of these paradoxes.  This article barely scratches the surface of everything there is to ponder based on Zeno’s considerations.  It’s been some twenty-five hundred years, and his challenges remain insightful for those with an open mind.