Zeno & Infinity

Pivotal moments in one’s intellectual development come unexpectedly. For me the key moment arrived in 9th grade English class when Miss Blumenstock gave a brief run-down of Zeno’s “theory of motion” [see footnote] and asked us to write a paper supporting or refuting him. Never could I have guessed it would lead to atheism.

That is exactly where it led, though it would take 5 1/2 years to get there.

Zeno’s “theory”, as she presented it, was that motion was not continuous but rather consisted of discrete segments. The path of an arrow shot across the horizon would actually, according to Zeno, not be smooth (although it might appear so to our eyes) but would in fact jump from segment to segment.

Why didn’t Zeno think motion was smooth and continuous? The answer is mathematics. Zeno realized there could not be an actual infinity of numbers between point a and point b on a numberline: numbers by their nature were inherently finite and countable, and therefore the path of an arrow across the sky had to consist of finite, countable steps.

If we think about it, we realize Zeno’s arrow was an early call for the Cosmological argument, which hinges on the assertion that there cannot be an actual infinity. There can’t be, per the Cosmological argument, an infinite regress of physical causes and there can’t be, per Zeno, an infinite number of steps in the motion of any object.

Just as there are two types of infinity — the macro infinity of going on and on to higher numbers and the micro infinity of more numbers between any two numbers on a number line — so there are two types of physical infinities which one can deny in the world. Zeno denied one, the Cosmological argument denies the other.

In my paper for Miss Blumenstock, I argued that there had to be an infinite number of arrow segments across the sky, otherwise scientists (not to mention camera enthusiasts) would have discovered a shutter speed which either hit between segments of the arrow’s path (resulting in no arrow in the photo) or else caused a blurred image as the arrow suddenly jumped from one segment to the next during the exposure. (I imagined that an example of my proposed effect could be seen by taking high-speed exposures of a movie screen at a theatre.) Yet neither scientists nor photographers had discovered such an anomaiy, which I took as a good enough implication that Zeno was wrong.

Miss Blumenstock liked my paper, but argued that if there were infinite segments then I had actually proved Zeno’s contention about segments correct! But my intuition was the opposite: infinite segments meant no segments at all. It meant that movement was continuous. It meant that there was a mismatch between all attempts to segment & number the path of the arrow, and the actual physical movement of the arrow itself (which was in its physical reality infinite and segmentless).

I came back to Zeno’s theory of motion again and again during my high school career, convinced that Zeno had mistakenly conflated mathematics and the physical world. And not just Zeno. I found it to be a popular error, even in the 20th century.

In short, by the time I graduated from high school I rejected anything that conflated the nature of mathematics with the nature of the physical world. I recognized such conflation as a repeat of Zeno’s error. To be sure, I agreed with Zeno’s rejection of an actual infinity in respect to numbers, but I maintained that the same rejection could not be applied to the physical world, for the simple reason that the physical world had a different sort of essence. We mapped numbers and mathematical formulas onto the world, because it was useful, but the world itself was nevertheless not mathematical.

When I found myself introduced to Aquinas and the Cosmological argument as a freshman in college, I was reminded of Zeno all over again. As I mulled over the argument I drew the conclusion that God could not be a “mind” who created the physical world by “thinking” it into existence — for the simple reason that the world wasn’t that sort of thing. But then if so, what was God’s nature?

And then it happened. I came across a bit of humor in Reader’s Digest about a Sunday School kid who, told that God had created the world, asked “Who created God?” To Readers Digest it was a funny story to entertain their readers, but to me it was a challenge — a threat — that had to be parried.

God is a special case, I explained to myself. You can’t ask who made God because God is infinite and eternal. But the world is finite and temporal, so who made the world is a fair question.

That didn’t work, and unconsciously I knew it. It didn’t work because it relied on the error of conflating mathematics with the world and declaring actual infinity is impossible. But if my analysis of Zeno was correct — and I knew it was — then the physical world stood in the same existential position God did. The physical world could be infinite and could be eternal — in fact if my intuition about Zeno was correct it had to be.

Therefore “Who made God?” was as valid a question as “Who made the world?” If both were valid, God was no final answer; if neither were valid, God was redundant.

Instantly, I was an atheist. I realized immediately that it was the physical world, not God, that was the proper object of worship. Eden, not heaven. Life, not afterlife. Bodies, not bodiless souls.


Footnote (added 10/11/2006):

Actually, Zeno did not present a “theory of motion” but rather a number of paradoxes designed to demonstrate that motion is impossible and an illusion. My 9th grade understanding of Zeno’s arrow was, to put it politely, garbled. Kevin Brown’s book Reflections on Relativity has a fascinating chapter on Zeno’s paradoxes. The arrow paradox, Brown writes

. . .focuses on the instantaneous physical properties of a moving arrow. He notes that if physical objects exist discretely at a sequence of discrete instants of time, and if no motion occurs in an instant, then we must conclude that there is no motion in any given instant. (As Bertrand Russell commented, this is simply “a plain statement of an elementary fact”.) But if there is literally no physical difference between a moving and a non-moving arrow in any given discrete instant, then how does the arrow know from one instant to the next if it is moving? In other words, how is causality transmitted forward in time through a sequence of instants, in each of which motion does not exist?

Instead of arguing that motion consists of discrete segments, Zeno is actually demonstrating that it cannot. He goes further, implying that if time is composed of “instants” then motion is impossible. Brown concludes,

If . . . we insist on adhering to the view of the entire physical world as a purely spatial expanse, existing in and progressing through a sequence of instants, then we again run into the problem of how a quality that exists only over a range of instants can be causally conveyed through any given instant in which it has no form of existence. Before blithely dismissing this concern as non-sensical, it’s worth noting that modern physics has concluded (along with Zeno) that the classical image of space and time was fundamentally wrong, and in fact motion would not be possible in a universe constructed according to the classical model. We now recognize that position and momentum are incompatible variables, in the sense that an exact determination of either one of them leaves the other completely undetermined. According to quantum mechanics, the eigenvalues of spatial position are incompatible with the eigenvalues of momentum so, just as Zeno’s arguments suggest, it really is inconceivable for an object to have a definite position and momentum (motion) simultaneously.

I would recommend the entire chapter on Zeno in Brown’s book to anyone interested in Zeno’s paradox of motion.

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