What is Half of Half?

Finally, what’s half of half? 25 percent. What’s half of that? 12.5 percent. What happens when you get to infinitesimal numbers? Do you ever reach zero? Nope. But is that reflective of reality? Nope. That’s what Zeno’s paradoxes sought to explain: what seems to be apparent and obvious on paper often has no bearing to reality. Once again, the map is not the territory. Zeno’s paradoxes play out at the interface between math and life, between concept and reality. Importantly, they may show us how uncomfortable the fit really is: what can seem “intuitively” correct can be shown to be completely wrong mathematically, statistically or theoretically. Could it be that the conception of “infinity” itself is flawed, and so every story based on it will eventually contain strange paradoxes like these?

What’s Half of Half?

The Greek philosopher Zeno’s so-called paradoxes have a koan-like quality about them, and do this work of highlighting the very tools we use to engage with the world—specifically how they are tools in the first place, and are not perfect.

Zeno’s claim: an arrow shot from a bow will never reach its destination. Why? Because first it has to travel halfway. Once it has, it still has to travel half of the remaining distance, and once it has, it still has to travel half of that remaining distance, and so on to infinity.

The arrow will get closer and closer, but never actually arrive. It will pass through an infinite number of halfway points, which (you can see where this is going) will take an infinite amount of time to pass.

There are variations on this theme: Imagine Achilles races with a tortoise, but agrees to give him a hundred-meter head start. By the time Achilles catches up with the tortoise, he finds in the meantime the tortoise has actually moved a further ten meters. He runs to close this gap but finds the tortoise still has a lead, since in the time it took him to run the ten meters the tortoise moved a further one meter. Achilles is the faster runner but will seemingly never catch up to the slower tortoise.

A final variant concerns arrows or moving objects. At any point along an arrow’s trajectory, for an infinitely small window of time (i.e. no time) the arrow can be said to be motionless. If the arrow’s path is composed of an infinite number of motionless moments, how can it ever move?

When does the arrow ever have time to move? You might have noticed that Zeno’s paradoxes predict and predate Heisenberg’s uncertainty principle—we cannot measure an object’s position and its velocity simultaneously. Perhaps Zeno was so frustrated with this line of thought that he reincarnated as Heinsenberg to finish his work…

So, what shall we make of all this? This is called a paradox because although you could agree with every statement Zeno makes to arrive at this bizarre conclusion, you can plainly see that the conclusion itself is false. Why?

Let’s take a closer look at the world that Zeno lived in. As a proponent of the Eleatic school of philosophy, Zeno supposed that physical phenomenon in the world were illusions, and that reality was essentially a single Being that moved perpetually.

When we see something that looks like movement, we are actually only witnessing different perspectives of this one great entity. Zeno’s paradox was therefore a way to support this worldview—i.e., “movement” as we understand it is impossible. The paradoxes prove this because we want to say that both the premise and the conclusion are correct, whereas they cannot both be.

Something is fishy, but what? It’s easy to disprove the claim “movement is not possible”—simply move. But then what about the argument Zeno makes—what’s wrong with it? The problem here may not have anything to do with whether movement is possible or not, or the exercise in halving progressively smaller increments. The problem may be (again) in the translation of abstract concepts into “real world” experience. Perhaps Zeno merely reminded himself and others that the map (i.e., the language and ideas we use to talk about space) is not the territory (reality itself, in its full, unsymbolized nature).

Zeno’s paradoxes play out at the interface between maths and life, between concept and reality. Importantly, they may show us how uncomfortable the fit really is: what can seem “intuitively” correct can be shown to be completely wrong mathematically, statistically or theoretically. Could it be that the conception of “infinity” itself is flawed, and so every story based on it will eventually contain strange paradoxes like these?

The ancient Greeks wrestled with similar concepts when they wondered what would happen if you continually cut a thing in half. Physics now tells us that you do not get infinitely smaller and smaller pieces, but at some point, the “rules” change and we can no longer talk of physical particles at all, but probabilistic, phenomenological ones. Perhaps in the same way, Zeno merely invited us to take a closer look at the threshold where one expression of reality borders another.

Physicists have actually tried to solve this problem (like the researchers who literally gave monkeys typewriters) and drew on knowledge of the smallest length possible given the size of electrons, the speed of light and the gravitational constant. They say that there in fact aren’t infinite lengths to travel along a path—we can calculate multiples of what’s called the Planck length. Others have countered: well, if the Planck length is the smallest length possible, you can imagine a triangle with its longest side equal to the Planck length, so then how long are the other sides?

Perhaps you are already getting a sense of what classical philosophers and mathematicians have had to force themselves to accept: that math is not reality. That an expression such as 1/3 is a mathematical, not a physical phenomenon.

What does Zeno teach us about our own thinking? That hypothetical, abstract, theoretically sound conceptions of life are not the same as life. In fact, this is a very good argument for not conducting thought experiments at all! We could jokingly say that all of physics is the strange compulsion to break the universe into imaginary chunks, and then the effort of wondering what the rules are that govern the interaction between the chunks, and then assuming you’ve discovered something mind-blowing when you realize that you are unable to. The problem is not that the chunks (time, space, numbers) are behaving erratically, but in our original error: the claim that there are chunks in the first place.

If you are someone who cares about literally and actually improving your real lived experience, at some point you may tire of thought experiments and doubt their applicability. After all, there is and will never be a Trolley Problem set you before you, nobody has yet found an “edge” to existence, and movement clearly occurs. But this is precisely the value of thought experiments: they illuminate, in crystal clarity, the raw value of our mental models, our ideology, our language. They show us the outlines of our thinking—even in places where we didn’t think we were thinking, but assumed we were engaging with reality itself in earnest.

What is the proper place of the word (the literal pixels or ink on the page, and the shapes of the letters that make up this word in modern-day English) “infinity”? Let’s not engage with the concept of infinity, but instead look behind the scenes, at the code or architecture of the word itself—to what phenomena have we assigned this word?

What assumptions have we made in doing so, and are we correct? More than correct—is this label useful? What does it tell us about ourselves, and our universe? If we find an unhappy mix between symbol and referent, does that mean we throw away the symbol, or decide that the referent is wrong somehow? Both? Neither?

There are no answers to these questions, which is universally the case with modern philosophy. But strange verbal and intellectual tangles like these are not just games—they show us important things about how our brains work. Significantly, they give us the option and opportunity to decide if we’d like to continue on in this realm of thought, or think something else.