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January 5th, 2019
I say ‘Marco’, you say ‘Polo’. And based on what I hear, which direction it sounds like ‘Polo’ comes from and how faint or loud it is, I move in that direction a certain distance. I repeat it and call out ‘Marco’ again, and hopefully the response sounds a little closer.
A basic search algorithm for a set of numbers that seeks to find just one number will query the half way point to see if that half way point is smaller or greater than the number it seeks to find. Having done this, the search algorithm has eliminated half of the dataset from it’s search. It then performs the same function on just the remaining half, and it repeats this until it finds the number.
Each time it queries whether the halfway point is larger or smaller than the sought after number, it’s somewhat like calling out ‘Marco’ in the game of Marco-Polo.
Then there is Zeno’s Paradox, which states: if you are travelling from point A to point B, then you have to pass the halfway point between A and B, and while traversing the second half, you have to pass the halfway point of this segment, and in order to traverse the second half of this segment, you again have to cross the halfway point. And since we can always cut something in two in theory, we never arrive at B because we are busy slicing the remaining distance in half. Zeno’s paradox holds if there isn’t some kind of smallest possible distance that cannot be divided into two, but this is beyond our scope, and question of large debate among physicists. Clearly we can get from point A to B but this does not some how negate the usefulness of Zeno’s paradox.
Mathematically, Zeno’s paradox is describing an asymptote. We can picture a curve on a graph that is always getting closer to a particular number, but never actually arriving. While Zeno’s paradox is always framed as a curious thought experiment about distance and space, it’s more accurately applied to finding, learning and strategizing.
We might think of a given skill that we can learn, like playing pool. Our first hour playing might be awful, but as we figure things out, our brain is constantly pivoting back and forth in order to find a method of operating that works in closer concert to our goal. Quickly the accuracy of our shot goes from not even hitting our indented target on the pool table to trying to get it to ricochet at just the right angle. We are making smaller and smaller adjustments as we move forward.
Or we might think of a sculptor, whether adding clay or removing stone, large amounts of material are being moved to rough out a basic shape, and as the sculptor continues less material is moved with each action until the finishing touches are barely perceptible changes in the form and feature of the sculpture.
Or we might think of switchbacks that lead up a mountain. At the base each switchback is long, but as we get closer to the top there is literally less room for each switchback so we pivot faster, more often but on a much smaller scale.
In the case of pool, each shot being taken is slowly zeroing in on a better understanding of the geometry involved. For the sculpture, the artist slowly zeroes in on just the right size and shape in order to convey the object they portray, and for the hiker, it’s as simple as zeroing in on the top.
All learning incorporates a kind of search algorithm like this that helps us zero in our skills and fine-tune our efforts. We start with large leaps, cutting half the dataset out of our perspective, and then slowly work our way towards tiny steps, until we are fine-tuning on such a small scale that there is nearly nothing left to cut in half, or certainly nothing we can easily perceive that can be improved upon.
This is why, when a friend shows an impressive project, we might be astonished but the creator is likely to disagree somewhat and say that it could be better. As a mere observer our ability to recognize the interaction of all details pales in comparison to the creator who is zoomed in to the project and sees it with a fine-tooth comb.
It may seem like Zeno made a mistake by applying a principle to the wrong domain, but it’s clear his creative command of concepts zeroed in on just the kind of brain teaser that would last for millennia.
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