Finding the sum of Infinite series using Zeno’s paradox (and not the other way around).

I’ll provide a very succinct overview of what Zeno’s paradox actually is (you can read about it in depth at Zeno’s paradoxes – Wikipedia). One of its multiple versions is as follows: Imagine that Achilles (the famous Greek Hero) and a Tortoise are, for whatever reasons, playing tag with each other with the tortoise having a headstart of 100 meters. Also, imagine that the tortoise can cover 50 meters in a second while Achilles is twice as fast and can cover 100 meters in 1 second.

After the start of the run, Achilles covers the distance that’s between him and the tortoise (100 meters initially). However, the tortoise, too, has moved in the meanwhile (50 meters to be precise). When Achilles runs to cover that remaining 50m, the tortoise has further moved by 25 m and ad infinitum. The paradox here is that no matter how many times Achilles tries to catch up with the tortoise, he can never quite seem to surpass the tortoise because in the duration that it takes for him to cover the distance between the two, the tortoise would’ve always moved some more.

Achilles and the Tortoise

Just to reframe this situation in a more concrete fashion, here’s a table that outlines how things develop:

StepAchilles coversTime Achilles takes (T)Tortoise covers in T
Progress of the race

The question that interests us, then, is does the race ever end? If yes, at what time? Many great minds were positively troubled by this question as they couldn’t really find any flaw in Zeno’s argument. But then, if you do proceed according to it, you never reach the exact time when Achilles meets up with the Tortoise because you’re stuck with forever adding the times of the rows of the table above to arrive to your answer. More concretely, the numbers that you are trying to sum, are never-ending!

Having established what the Zeno’s paradox is, now let’s switch gears and get into the theme of this essay. Imagine that you are an ancient mathematician who is trying to find the sum of an infinite series using the knowledge of your time. So, if the series you are trying to work with is (say) S = 1 + 0.5 + 0.25 + … then how do you obtain the sum? Since you don’t know anything about calculus and geometric series, you’d probably be stuck and call the problem preposterous (unless you’re particularly gifted, you may even be able to figure it out by yourself). However, a very intriguing (and easy) approach would be to just use a calculator. But wait, how do you get a calculator in an ancient world? And for that matter, what kind of calculator would even be capable of handling such an infinite sum? The answer is really simple: the only calculator that can reliably calculate such a sum, in whichever era, is the world itself!

Here’s a way you could think about it. Imagine that I give you all the information about a person (let’s call him Greg) including his age, BMI, muscular and skeletal structure, caloric intake, metabolism, and a thousand different related variables about his body. Now, if I ask you this question: How far can Greg throw a pebble of 100 grams, will you be able to answer it? For sure, it would include thousands of complicated differential equations and millions of variables that would have to accurately capture how the excitation of all of the neurons of Greg’s body causes contraction of each individual muscle fibers which then combines with his bone structures to contribute to the final thrust on the pebble imparted by his hand and for sure, given a sufficiently powerful computer, you could use numeric methods to solve the differential equations, finally obtaining an answer to a satisfactory level of accuracy. But ask yourself this, wouldn’t just giving Greg $100 and asking him to throw the pebble with as much force as he can possibly muster produce a better (and admittedly faster) result?

Even now, we spend millions of dollars in creating and simulating computational models of how atoms interact, how interstellar collisions happen, how weather plays out, and how the stock market moves, but the simple fact of the matter is that all that we are ever trying to do is to just imitate a computation that’s already happening! Yes, the world itself is a computer that, at each instance, is calculating the next position, velocity, momentum, and a thousand different parameters of individual atoms, galaxies and black holes. And like I said, all that we ever do by building our fancy models and expanding our modelling techniques (mathematics) is to understand how the world does it.

Circling back to our original question then, with you as an ancient philosopher trying to find out the sum of the infinite series S = 1 + 0.5 + 0.25 + …, can you reliably use the massive computational power that’s omnipresent to get your answer? Yes! Setup, a Zeno experiment with two racers of desired speeds, give them a similar headstart, and then … just measure if and when they meet and there’s your answer right there! (Just for the sake of completeness, it’s 2 seconds). If you’re asking, how does this even make sense, congratulations on asking the right question. I, unfortunately, am too unqualified to answer why the world works the way it does or even why Mathematics so succinctly and accurately correlates with it. But I can tell you this: If you say that the time it takes for Achilles and Turtle to meet is given by T = 1 + 0.5 + 0.25 + … and then you observe that time to be T = 2 seconds. Then it must logically follow that in this universe:

$$1 + 0.5 + 0.25~+ ~… ~= ~2$$

This is how the world-computer works, at least. But before we wrap-up, there’s another question that’s worth asking. Is this what we always want? Which is to say, do we want our mathematics to always be tied up to this universe? Or do we want it to be independent of the universe and the laws of physics? I understand that this is a confusing question, so let me illustrate what I’m saying with a different example.

In this example, you, once again, are the ancient mathematician who is trying to obtain a different sum i.e. S = 1 + 1 + 2 + 4 + 8 + … to 60 steps. Calculating this long sequence is a very difficult and tedious (albeit not impossible) task. So once again, you come up with a clever way to exploit the world to do your calculation for you. You invoke Achilles who is a superhuman runner and ask him to run for an hour starting with the speed of 1 m/s, and to double his speed every second. So, Achilles starts with the speed of 1m/s, at the next second, he adds 1 to his speed and doubles his speed to 2, and so on. Here’s a table that shows what happens:

Time (seconds)Current SpeedAdded SpeedNew Speed
Achilles’ speed chart

At the end of an hour, you can expect the final speed of Achilles to be V = 1 (initial) + 1 (added at T=1) + 2 (added at T=2) + 4 + 8 + …

So you have basically recreated your sum using Achilles and to obtain the answer, you’d let him run and you’d just ask him (because he’d know) what his final speed was at the end of the hour. But the funny thing is, if you added the sum by hand, you’d expect Achilles’ speed to be somewhere around 1152921504606846975 m/s but that is not what Achilles would report his speed to be. He would actually say that his speed never quite exceeded 299792458 m/s even till the end of the hour! This is because, in the calculations done by the world-computer, there doesn’t exist a larger speed than c the velocity of the light in vacuum! So, no matter what Achilles does, he will never be able to exceed that threshold and thus, this is one instance where your calculation would not match up with the world computer. So, what do you do? Do you decouple your mathematics from how the world works? And can you even do that? Is it even mathematics if it cannot be used to reason about this world? Or perhaps you don’t agree with it and think that Mathematics is something that’s even more abstract and higher than the observable universe itself and exists independently of the laws of causality? What do you think?

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