What is Mathematics?

When I was in math grad school, every time I hung out with colleagues to a bar or restaurant, we had a hard time splitting the check, because, as somebody would eventually mention, mathematicians are horrible at doing arithmetics, and basic math in general.

Also surprising is the fact that everyone who studies advanced math ends up knowing the whole Greek alphabet. In fact, Greek letters are so used in math and sciences requiring mathematical modeling, that it led cartoonist Zach Weiner to joke: “Isn’t it weird how the Greek language is entirely made up of physics symbols?”

Now, of course, getting to know the Greek alphabet as a byproduct, and not being able to do basic math, are not a very good description for what a mathematician is. Perhaps here Paul Erdos gets a bit closer: “a mathematician is a device for turning coffee into theorems.”

And what about mathematics itself? While it is true that it’s something concerned with arithmetics, involving Greek letters, and centered on theorems, those are merely some descriptive features. Let’s get a little more conceptual.

To begin with, math is often regarded as a science. Carl Gauss, one of the greatest mathematicians of all time, went even further: “mathematics is the queen of the sciences,” he’s reported to have said. Well, Gauss, I’m sorry to inform you that this is not the case: math is not a science. The reason is that mathematical theories, once stablished, are not falsifiable. And “falsifiability,” a concept introduced by philosopher of science Karl Popper, is widely regarded as one of the main characteristics of a scientific theory.

Another common notion associated with mathematics is that it is a language. Cut to Galileo Galilei: “The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.”

No, Mr. Galilei. And if the Vatican decides to disagree with you on that, I have to support them this time around. A language is simply a tool for communication, for information transfer. Although it is true that mathematical notation allows talking about things in an idiosyncratic fashion, this is a very narrow picture of what mathematics is about.

Mathematics is a representation of reality, particularly suitable for computations. More then describing an object, math is concerned with what you can (and often, what you can’t) do with it. Take the circle of radius 1, centered at (0,0), for example. There are many different ways of representing it. Here are three. (1) the “parametric” way: how the function that turns x into the pair (cos(x),sin(x)) acts on the line of real numbers. (2) the “implicit” way: the set of points (x,y) in the plane whose distance to the point (0,0) is exactly 1. (3) the “limit” way: a circle is the limit of an infinite sequence of regular polygons subscribed in it, where every polygon on the sequence has one more edge with respect to its predecessor.

Some representations are more useful then others in certain situations. For instance, a representation like (3) is preferred in computer graphics for drawing a circle on a screen. On the other hand, (1), extended to an arbitrary circle, is better to compute the centripetal force in a curve.

An important fact to notice is that our brains also “implement” representations of objects, although the way in which we mentally represent concepts, and use them in computations, remain a mystery.

In a sense, mathematical constructs are an extension of that representation. When our ancestors were using rocks to help counting cheep (what is one of the earliest manifestations of mathematics), they were, so to speak, extending the representation and computing capabilities of their brains to outside their heads.

Next time you feel like having an extracorporeal experience, go solve a mathematical puzzle.

(An updated version of this article appeared in Washington Square News.)

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