What Does It Mean for Math to Be a Human Activity?

Here’s a strange thing about mathematics: most mathematicians act as if they’re discovering truths that already exist somewhere, like explorers finding new islands. But what if mathematics is actually something we create—like music or stories? What if there are mathematical statements that aren’t true or false until someone actually thinks them through?

A Dutch mathematician named L.E.J. Brouwer (say it like “Brow-er”) took this idea seriously enough to rebuild large parts of mathematics from scratch. And in doing so, he threw one of the most famous rules of logic out the window—a rule so basic that most people never even notice it’s there.

The Rule Brouwer Questioned

The rule is called the Law of the Excluded Middle. It says: for any statement, either that statement is true, or its opposite is true. There’s no middle ground. “Either it’s raining outside, or it’s not raining.” That seems obvious, right? Every statement is either true or false.

But Brouwer asked: does this apply to every statement, even ones nobody has ever checked?

Imagine someone says: “The number of grains of sand on this beach is even.” Nobody has ever counted them. Is that statement either true or false? Most people would say yes—it has to be one or the other, even if we don’t know which. The truth is just out there, waiting to be discovered.

Brouwer disagreed. He said: the statement isn’t true until someone has proved it in their own mind. And it isn’t false until someone has proved it false. Without a proof, the statement simply doesn’t have a truth value yet. It’s like asking whether the third movement of a symphony that hasn’t been written yet has a beautiful melody. The question doesn’t make sense—the symphony doesn’t exist.

This might sound like a small quibble. But Brouwer pushed this idea to places that made many mathematicians deeply uncomfortable.

Where Mathematics Comes From

Brouwer believed that mathematics is fundamentally something humans do—it’s an activity, not a collection of facts floating around in some invisible realm. When you think about numbers, you’re not reaching into a Platonic heaven to grab them. You’re building them in your mind, step by step.

Think about the number 1. You can imagine a single object—one apple, one idea. Now think about 2. You get 2 by starting from 1 and adding another 1. Then 3 is 2 plus another 1. Every natural number is built this way: start at 1, then keep going. Mathematics, for Brouwer, is this kind of mental construction. It’s something you do, not something you find.

This means proofs aren’t just ways to convince other people. They’re the very thing that makes mathematical truths exist. A mathematical statement without a proof is like a house without a builder: it’s nothing at all.

This gets complicated when Brouwer explains where our sense of mathematics comes from. He said it’s based on what he called the “pure intuition of time”—the basic experience that moments flow one after another. You have the present moment, then another moment, then another. This sequence of “now, then, then” is the foundation of counting. For Brouwer, time isn’t just something we measure. It’s the structure that makes mathematics possible in the first place.

The Battle Over the Continuum

This part gets technical, but here’s what it accomplishes: it shows how Brouwer’s philosophy leads to genuinely strange mathematics.

In normal (“classical”) mathematics, the real numbers—numbers like √2, π, and 0.333…—form a smooth, continuous line. Every point on that line is already there, like a solid piece of string. But for Brouwer, you can’t just say there’s a real number at every point. You have to be able to construct it.

So Brouwer invented something called choice sequences. Imagine someone slowly reading out digits of a number, one at a time. But here’s the twist: the person making the choices can decide what digit comes next based on things that haven’t happened yet. For example, they might say: “I’ll keep reading out 3’s until someone proves the Riemann Hypothesis. If they prove it, I’ll switch to 4. If they disprove it, I’ll switch to 5.”

This is a weird number. Does it have a decimal expansion? Well, sort of—you know the rules for building it. But you can’t actually write the whole thing down, because you don’t know if or when someone will prove the Riemann Hypothesis.

For Brouwer, this counts as a perfectly legitimate number. It’s a number that gradually comes into existence as choices get made. And here’s the shocking result: some of these numbers behave in ways that classical mathematicians thought were impossible.

For instance, Brouwer proved that—according to his system—every function defined on the interval from 0 to 1 is continuous. That means no sudden jumps, no breaks, no weird gaps. In classical mathematics, there are plenty of functions with jumps. But in Brouwer’s system, you can’t construct one. This isn’t a limitation Brouwer regretted. It was a discovery: when you build mathematics from the ground up as a human activity, certain things turn out to be impossible.

What Happened to the Law of the Excluded Middle?

So where does that law—“every statement is either true or false”—end up in this system?

For simple statements about finite things, like “this apple is in my hand or it isn’t,” the law works fine. You can check. But for statements about infinite things—like “all even numbers greater than 2 can be written as the sum of two primes” (Goldbach’s Conjecture)—the law becomes problematic. Nobody knows if that statement is true or false. For Brouwer, that means it isn’t either one until someone proves it. You can’t just assume the universe has already decided.

Brouwer’s students and followers developed what’s now called intuitionistic logic, which drops the Law of the Excluded Middle. In this logic, proving that a statement isn’t false isn’t the same as proving it’s true. And “either P or not-P” is no longer automatically assumed to be true for every P.

This isn’t just a philosophical curiosity. Computer scientists have discovered that intuitionistic logic corresponds beautifully to computer programs. When you prove something in intuitionistic logic, you haven’t just argued that it’s true—you’ve actually shown how to construct whatever it is you’re talking about. That’s incredibly useful for programming.

The Fight

Brouwer’s ideas put him in direct conflict with David Hilbert, one of the most powerful mathematicians of the early 1900s. Hilbert believed mathematics could be placed on a secure foundation by turning it into a formal game: write down rules, manipulate symbols according to those rules, and don’t worry about what the symbols “mean.”

Brouwer thought this was nonsense. If mathematics is just symbol-pushing, what makes it about anything? Hilbert’s response was essentially: “Who cares? It works.”

The conflict between them became personal and ugly. Hilbert had Brouwer removed from the editorial board of a major mathematics journal, using underhanded tactics. Brouwer was devastated; he never fully recovered creatively. For years afterward, he published very little new mathematics.

But Brouwer’s ideas didn’t die. Other mathematicians took up his cause and developed intuitionistic mathematics further. The questions he raised—about what it means for a statement to be true, about the role of human thinking in mathematics, about whether math is discovered or created—are still debated today.

So What Does This Mean?

Nobody has settled the debate Brouwer started. Most mathematicians today work in the classical tradition, using the Law of the Excluded Middle without a second thought. But a significant minority take intuitionism seriously, and even mathematicians who reject Brouwer’s conclusions often respect his questions.

The deepest issue might be this: if mathematics is a human activity, then it’s limited by what humans can do. We can’t check infinitely many cases. We can’t decide every question. But if mathematics is discovered, like a landscape that exists whether or not anyone explores it, then every statement has a truth value—we just might not know it.

Which view is right? That’s not a question mathematics itself can answer. It’s a philosophical question. And Brouwer’s life work was showing that the answer you choose makes a real difference—to what you can prove, to what you believe, and to what kinds of mathematical worlds you can build.

Appendices

Key Terms

Term	What it does in the debate
Law of the Excluded Middle	The logical principle that every statement must be either true or false; Brouwer rejected its general use
Choice sequence	A number built step by step, where later digits can depend on events that haven’t happened yet
Intuitionism	Brouwer’s view that mathematics is a human mental activity, not a collection of discovered facts
Continuum	The smooth line of real numbers; Brouwer argued it could only be understood through constructive processes
Proof	For Brouwer, the mental construction that actually makes a mathematical statement true
Pure intuition of time	Brouwer’s proposed foundation for mathematics: the basic experience of one moment following another

Key People

L.E.J. Brouwer – Dutch mathematician and philosopher who founded intuitionism and had a famously combative personality; argued that mathematics must be built from mental constructions, not abstract truths.
David Hilbert – Powerful German mathematician who believed mathematics could be reduced to formal symbol manipulation; became Brouwer’s bitter enemy and had him removed from a journal’s editorial board.

Things to Think About

If nobody has ever counted the number of hairs on your head, is “the number of hairs on your head is even” actually true or false? Or is the question not yet settled? What difference does it make which answer you give?
Brouwer thought mathematics was based on the experience of time passing. Could there be a mathematics based on a different experience—like space, or sound, or touch? What would that look like?
In a video game, do mathematical truths exist in the game’s code before the game is run, or do they only exist when the game actually runs? How is this like Brouwer’s view of mathematics?
If you accept Brouwer’s view, you lose the ability to prove things by contradiction (assuming the opposite leads to a problem, so the original must be true). Are there things you’d be willing to give up to avoid that loss? Are there things you’d be unwilling to give up?

Where This Shows Up

Computer programming: Intuitionistic logic is the basis for “type theory,” which underlies some programming languages and is used to formally verify that software works correctly.
Artificial intelligence: The question of whether a machine can “do mathematics” depends in part on whether you think mathematics is about mental activity or about abstract truths.
Game design: Some games like Baba Is You let players create and break rules in ways that mirror Brouwer’s idea that mathematical objects are constructed, not discovered.
Everyday reasoning: When people argue about things that aren’t yet decided—like whether a new technology will be good or bad—Brouwer’s attitude says “wait until we have evidence” rather than assuming it must already be one or the other.