It’s long been debated whether mathematical truths are “discovered” or “invented”. The answer becomes clearer when you dig into what math actually is.
In my opinion, math is a language for describing patterns, relationships and rules. It’s just like any language except (1) it’s far more precise than normal language, and (2) unlike human languages, its domain is limited to only describing patterns and relationships and rules. In English, you can talk about birds AND equations related to bird flight, but in math you can only talk about equations and rules that may or may not model bird flight.
Mathematical statements that make claims are essentially always (implicitly) of the form: “if we adopt these axioms and follow the rules of logic, these things follow”.
Take a mathematical statement such as the Pythagorean theorem:
“For a right triangle with sides a, b, and c, a² + b² = c².” This is a powerful, non-obvious (unless you’ve thought about it for a while), and useful idea, but it’s true BY DEFINITION (once you adopt a set of axioms and the rules of logic). Literally, it could not be otherwise unless we change the definition of what a “right triangle” is.
This isn’t unique to math. English also has statements of this form, such as: “if we adopt the rules of chess, then a checkmate in one move from the start of the game is impossible.” This couldn’t be otherwise because once we define the rules of chess, they imply it.
I don’t think math statements that make claims are any different than that essentially.
Another way to think about it: we invent the set of rules we want to use (such as chess, or ZFC axioms, or postulates of geometry), then we “discover” (or figure out) the implications that we didn’t initially realize (such as that a one-move checkmate is impossible, that 1+1=2, and a^2 + b^2 = c^2 for a right triangle).
“Discovering” is when we realize something we didn’t know by investigating something. In math, it happens when we discover that a definition implies something we hadn’t realized.
Pure mathematics consists of exploring formal structures we’ve defined, how they relate to each other, and what follows from those definitions. We invent the structures we want to consider and decide what constitutes valid inference and truth related to them. After that, we discover consequences we did not foresee. They are discovered (rather than invented) only because we didn’t mean to choose them – we bump up against them as consequences of the parts we meant to invent.
Unlike discovery, “invention” is when we create something on purpose. We invent the systems in which math operates (and the meta systems allowing us to interpret and define truth within them). There are many such systems and subsystems (ZFC being a famous one because a lot of math can be built on its axioms).
The mathematical consequences of a definition only appear objective because they follow from the structure we’ve defined and rules we’ve decided on for inferring truth using those structures.
If someone were to observe that the rules of chess imply the king can only move one square, then they’d be right, because that’s part of the rules. Those rules also imply you can’t checkmate your opponent on the first move of the game, even though the rules don’t state that; they just imply it.
Can you have chess where kings move two squares? No, that would be a different game, even if it’s very chess-like. Similarly, if checkmate is possible on the first turn, you’re not playing chess. Why in chess can’t you win on the first turn? Because if you could, you wouldn’t be playing chess. Of course you can choose to play that other game instead. Or you can redefine the word “chess” to mean something else – but if by “chess” you mean what I refer to as “chess,” checkmating on the first turn of the game is impossible, and this follows from the definition.
But doesn’t mathematics already exist “out there” somewhere before anyone invents it?
Well, did chess already exist “out there” before someone invented it? What about flaming hot Cheetos? Were they out there in the Platonic realm before a human had ever considered them? I’d say no, they didn’t exist, even as a concept, until humans invented them (though they obviously were a potential concept – a concept that a mind could have).
What about the fact that the axioms of math imply that 1+1=2? Did that predate the human invention of math?
Well, did the fact that “if you’re playing chess you can’t do a checkmate on the first move” predate the human invention of chess? In my view, no, and this is no different than the math question. 1+1=2 is true in arithmetic simply because of the definition of arithmetic and the rules of inference we adopt.
But if math is a set of definitions we choose, why does it work to do things in the world? Because we choose the structures we’re working with specifically so that they capture elements of the way the world is. And we choose which mathematical ideas to deploy based on what we’re trying to model about the world.
Adding one ball to a bag and then another leads to two balls in the bag. 1+1=2 models aspects of this situation well. But it’s not the same as 1+1=2; it models aspects of the situation, but it fails to model other aspects, like the movement or the balls as they go into the bag. If they were mud balls that stuck together when added to the bag, 1+1=2 wouldn’t be a good model anymore for that situation.
So is math discovered or invented? In my view, we invent the axioms, definitions, and rules of inference, and discover the implications of those choices.
This piece was first written on June 26, 2020, and first appeared on my website on September 14, 2026.
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