Love it or hate it, no one can deny the usefulness of mathematics. It has been used to do everything from keeping track of sheep, to building pyramids, to landing a robot on Mars, millions of kilometres away. 

While none of those are exercises of pure math — counting objects, engineering, and spaceflight are all activities in the physical world — they suggest that mathematical concepts and methodologies have a place in physical reality and are effective in describing the universe around us.

The difference between mathematics and physical sciences lies in math’s lack of an empirical component. For instance, a mathematician cannot observe the tangent function’s derivative like the biologist can observe and study cell division or the physicist can observe planetary orbits.

The dichotomy between the abstractness of math and its use in describing reality raises the question of whether math is something humans discover or create. Trying to find the answer to this centuries-old debate might tell us more about ourselves than the origin of math.

Math might not actually exist

To the delight of many students around the world wishing on stars during particularly gruelling homework assignments, mathematical truth may not be as true as it seems, and math might not actually be real. 

This does not mean, however, that one is equal to two or that we can divide by zero and everything instantly collapses. Instead, those who hold this view treat mathematics as a kind of fiction — or, as James Brown, a professor emeritus of philosophy at U of T, put it during a conversation we had about the topic, “Mathematics is a game. It’s something that we have created.”

According to this ‘fictionalist’ view, all mathematical truths and ideas that we know and love — parabolas, the square root of negative one, pi — are not really true in an objective sense, but are parts of the story or the game of math. Consequently, Brown said that, in this view, the rules of math “have the same status as a rule of chess, such as ‘Bishops move diagonally,’ and that’s not an objective truth; it’s simply a tradition of playing chess.”

Therefore, ‘four times two is eight’ is as true a statement as ‘Bishop takes g5, checkmate.’ Within the framework of the game, the rules make perfect sense, but outside of the game, four, two, eight, the bishop, g5, and the checkmate aren’t actually real.

An interesting consequence of this view is that the rules of math can be changed, regardless of whether the game becomes better or worse for it. Math operates on sets of basic rules, called axioms, upon which all other mathematical theorems are built. If these rules change, math changes with them. 

This leads to the conclusion that all math is based on an arbitrary set of rules, invented to fit our needs. But what does it mean to invent mathematical rules?

Inventing math

Mathematicians invent math all the time, formalizing observations they make about the mathematical world into rules. To understand what that might be like, take for example the famous paradox of infinite sums first proposed by the Greek philosopher Zeno. 

Say you’re sitting at your desk and contemplating the nature of distances between objects, and you notice that no matter how close you bring two objects together, there always seems to be more distance to go. You set these objects at two points on a number line, labelled zero and one, and you start thinking about what it means to cover the distance between them. 

You can imagine that a person walking from zero to one must, at one point, pass through the point halfway between them. Then, you notice that the same applies when going between that halfway point and one, and so on.

If you start keeping track of the points visited on the way to one — a half, then a half plus a quarter, then a half plus a quarter plus an eighth, and so on — then you notice that the sums are approaching one. This makes sense because your imaginary person is walking toward the number one. 

The sums are, in fact, getting closer to two or three as well, but you know their final result is going to be one. Since you can endlessly divide the distance in half, your calculations tell you that adding infinitely many such increments results in one.

This idea seems ridiculous at first — you can’t possibly add infinitely many numbers — but you don’t let that stop you because you’re an eager mathematician who has just stumbled upon something promising. You set out to formalize what you just found, and you define the notion of approaching a number. Then, you define an infinite sum so that it equals the value its partial sums are approaching.

Through this process, you invented some new math to make sense of what the universe was showing you. It never felt like you were making things up out of nowhere; it felt more like a discovery.

Existence, truth, and Harry Potter

The view that many philosophers and working mathematicians hold is that mathematical concepts and ideas exist independently of humans. The idea is inspired by Plato’s abstract ideals and is called ‘Platonism.’

Platonists argue that mathematical objects exist, just as you and I, a table, or the Robarts Library exist; however, mathematical objects are abstract, existing in some other corner of reality we can’t physically interact with, similar to Plato’s idea of abstract forms.

Brown thinks of it as an analogy. He said that trying to describe an observed structure or a pattern in the world is akin to “[pointing] to something in Plato’s heaven and [saying], ‘That mathematical structure [is the same] as this bit of physical structure, so it works.’ ” He also noted that this process is not infallible and requires lots of iteration to get it right.

While this idea is all well and good and lets mathematicians get on with their lives, it moves the debate one step along to the question of what it means for something to exist.

If we’re taking the claim that abstract objects exist at face value, then things need not be observable for them to be true. For instance, Harry Potter is a fictional character and just as abstract as the idea of a number. Just like with math, the story of Harry Potter exists in our collective consciousness and has a set of properties and characteristics that everyone agrees upon.

The creation of new math, then, becomes a kind of fan fiction — working with established rules and patterns to create or discover new stories. Just as characters in a story have established patterns of behaviour, functions or formulas behave in certain ways that, when put into new situations, lead to new discoveries.

Infinity and the library of everything

A consequence of Platonism is that there are more mathematical concepts and ideas waiting to be discovered beyond the math we currently know, some of which — for whatever reason — may never be discovered. 

A similar idea is encapsulated in the short story by the Argentinian author, Jorge Luis Borges, called “The Library of Babel,” in which the narrator lives in a library of hexagonal rooms that contain every combination of every possible thing that can be written using 25 basic characters.

The narrator describes the joy of the initial discovery that they and everyone else in the library are in the possession of any and all knowledge; all they need to do is to find the book in which that knowledge is written. That joy is quickly diminished, however, since for every piece of legible text, the library contains millions of pages of utter nonsense. 

The sheer improbability of stumbling upon a 300-page book of random letters arranged, by chance, into something legible, let alone coherent, drives the narrator into despair. Even with a vast world of undiscovered mathematical truths out there, the work of finding and formalizing these ideas can be a long and difficult process filled with many dead ends.

Humans might try endlessly to discover all that is possible to discover, but there may always be an elusive thought or an idea just out of reach. And until we find it, there’s no sure way to know whether it exists or not. After all, if a tree falls in a forest but there’s no one there to see it, did it really happen?