Zeno and Infinity

No person can create a true infinity. There is no number associated with it, only an idea of never-ending numbers. Therefore, logic says a bounded line would contain a finite number of points rather than an infinite number. Zeno decided that this was not accurate, in that there is always a half-way point on any bounded line, no matter how many times you chop it up.

Zeno of Elea. Image: Jean Housen, via Wikimedia Commons.

Zeno of Elea. Image: Jean Housen, via Wikimedia Commons, from the Achilleon in Corfu.

Zeno (or Xeno) of Elea was a Greek philosopher of the Eleatic School, easily best known for his paradoxical arguments having to do with time and space and the impossibility of motion in any logical sense. Mentioned in writings of both Plato and Aristotle, he showed that viewing space as a multitude of points and time as a multitude of “moments” concludes that motion is an illusion. But thanks to common sense we can dispute that through our own sensory perception, motion is indeed a thing (though some philosophers would love to argue that our senses are merely deceiving us on a number of levels). Nevertheless, Zeno proposed four arguments against motion, among them The Dichotomy.

The Dichotomy is a very similar argument to the bisecting line mentioned before. Zeno’s example describes a horse attempting to cover the distance between points A and B. Now, in order for the horse to reach its end point, it must first reach a midpoint, right? Of course. But in order to reach that midpoint, it must reach a quarter point, and so on. Therefore, if space truly contains an infinite set of points, the horse must cross an infinite number of these points in a finite amount of time. This statement is paradoxical, so by this reasoning the horse will never reach point B, because it cannot even move from point A.

As mentioned before, infinity is merely a concept. But it is a concept that mathematicians have, if begrudgingly, come to accept and even apply. So thanks to modern calculus, we do have a mathematical solution to Zeno’s The Dichotomy: the distance the horse is to cross can be explained as a series such as ½ + ¼ + 1/8 + … When it comes to a series, an easy way to solve this one in particular is by using partial sums. With partial sums, one could recognize that the sequence, otherwise expressed as 1/2 + 1/(2^2) + 1/(2^3) + …, follows a specific pattern that results in the sum after n terms being 1-(1/2^n). Bringing in limits and the concept of infinity, and assuming that the horse runs at a fairly constant speed, we can conclude that these sums become increasingly close to 1. In fact, the series converges to 1, meaning the horse does not require an infinite amount of time to complete its task and movement is indeed possible.

Because of its useful application in problems such as The Dichotomy, infinity has become a well-used and helpful tool, so long as you don’t look too closely at the actual implication behind the concept. To say an infinite set of numbers will “eventually” reach a finite number seems like such a preposterous conclusion to draw, but alas we can. Physicists will also chime in on such a paradox as this with the fact that if you look close enough, you will not find infinite space, but a finite, albeit incomprehensibly large, set of atoms and other such building blocks of nature.  These microscopic materials do indeed seem to be Mother Nature’s way of saying infinity is a silly concept in real world application, though the term is still used in regular calculations throughout physics and engineering. As my calculus teacher is fond of mentioning, Engineers and Physicists will look at an application of infinity such as the one used to solve The Dichotomy, shrug, and say “it’s close enough”. As mathematicians, we may nod in agreement, but we know better.