Philosophical Mathematics
I’m kind of a math nerd in the sense that I love math, but not in the sense that I’m brilliant at it. I took calculus in my junior year of high school, a full year before my classmates, . However, like so many of my contemporaries, I have forgotten much of my math. This led to one of my most hilarious job interviews, ever.
My Hilarious Job Interview
In 2007, I was living in London and job hunting, and someone from a local company known ominously as “the place Perl programmers go to die” was interviewing me. After a short interview, I was giving a programming test. “Write a function that, given a number, finds the prime factors.” I had to do this with pencil and paper.
My first question: “Can I assume I have a prime number generator or would you like me to write one?”
The look of shock on the interviewers face was priceless, but he told me I could assume I have one.
After a couple of extra questions, I started writing the function. At one point, I used an array an array to capture the prime factors and then erased it, replacing it with a hash (a “dictionary”, in many other languages). The interviewer corrected me, saying I wanted an array, and I replied that with an array, I would have to count the number of each of the prime factors, but with a hash, I would merely need to increment the values, with the keys being the primes. The interviewer was again surprised.
After I was done, he desk-checked my solution to verify it. I had quickly, with almost no thought, written the solution. I looked brilliant.
What the interviewer didn’t know what was that about a month ago, I was disappointed that I had forgotten most of my math and bought a book to teach it to myself again. About a week prior to the interview, I had written a program, complete with a prime number generator, to solve this very problem.
I didn’t get a job offer because they wanted to pay me a junior developer salary. This has happened to me more than once.
So I still haven’t relearned the math I once knew, but I found myself getting fascinated by mathematical philosophies and how the approach the idea of whether math is “real” or merely a description of real.
In The Beginning
When we do mathematics, it’s easy to forget that we’re standing on millennia of philosophical debate. Most of us were taught that math is “just true,” but behind every proof lies a question: what is mathematics?
Over the last 150 years, mathematicians and philosophers have offered competing answers. Some see math as eternal truth, others as symbol manipulation, others as mental construction, and still others as a pragmatic tool that survives only because it works. Understanding these foundations matters because they shape how we build proofs, interpret results, and even design new mathematics. But first, we need to understand Plato .
Plato lived roughly 400 BCE and to him, math was something to be discovered, not invented. He viewed mathematics as fundamental to understanding reality and knowledge and believed mathematical objects like numbers and geometric forms existed as perfect, eternal forms in a realm beyond the physical world. These mathematical truths were absolute and unchanging and the concept of a perfect circle or the number 2 existed independently of any physical circles or pairs of objects.
In short, math to Plato was a weird blend of mysticism and science and for the next 2000 years, that’s mostly where math remained. Yes, we made tremendous advances and there were philosophical differences from Aristotle’s realism to Kant’s synthetic a priori , but these were largely philosophical debates, not mathematical. That leads us to one of the great leaps of modern math philosophies, Logicism.
The Great Philosophies.
Here’s a timeline of these philosophies to get you started.
Logicism
Logicism was a school of thought pioneered by Gottlob Frege in the late 19th century and argues that mathematics is fundamentally reducible to logic. Frege’s groundbreaking work, especially in his Begriffsschrift and The Foundations of Arithmetic, tried to show that numbers and arithmetic truths are not dependent on human intuition or experience but can be derived entirely from logical principles. Although Frege’s system was famously undermined by Bertrand Russell’s paradox , his project reshaped the philosophy of mathematic.
Both Frege and Russel had hoped to reduce math to pure logic, Gödel’s incompleteness theorems showed that there are always “true” statements about natural numbers which cannot be proven. Worse, Gödel’s second theorem proved no consistent set of axioms can prove its own consistency.
Formalism
Hilbert’s formalism, developed by German mathematician David Hilbert in the early twentieth century, was an ambitious attempt to establish mathematics on completely secure foundations by treating it as a purely formal system of symbol manipulation. Hilbert wanted to show the consistency of all mathematical systems by formalizing them using a specific language that included variables, quantifiers, equality, logical connectives, and undefined parameters, with the goal of proving that every theorem could be derived using nothing more than axioms and formal rules without any reference to meaning or interpretation.
While logicists believed that numbers are genuine logical objects with inherent meaning and that mathematical statements express logical truths, formalism treats mathematics as purely syntactic manipulation of meaningless symbols according to explicit rules, like an elaborate game of chess. Where logicists sought to ground mathematics in logical truth and abstract reality, formalists explicitly abandon any semantic content, viewing mathematical validity purely in terms of rule-following within formal systems without any insistence on the existence of mathematical objects or their connection to logic.
As you might guess, Gödel’s incompleteness theorems wrecked formalism, too.
Intuitionism
Perhaps my favorite mathematical philosophy is intuitionism , created by L. E. J. Brouwer , intuitionism argues that math is only a construct of the mind. Brouwer’s was a rebel philosophy that thumbed its nose at the big mathematical worldviews of his time.
Platonists like Russell believe mathematicians is discovering eternal truths that exist independently in some abstract realm. Formalists like Hilbert see math as an elaborate game of symbol manipulation governed by rules (with no need to worry about what the symbols actually “mean”). Logicists try to reduce mathematics to pure logic.
Brouwer said “nope” to all of it. His radical claim was that mathematical objects don’t pre-exist waiting to be discovered (contra Platonism), aren’t just meaningless symbols pushed around according to rules (contra formalism), and can’t be reduced to logical principles (contra logicism). Brouwer argued that mathematical objects come into existence through mental construction. When you prove something or define a number, you’re not finding it, playing with symbols, or deriving it from logic, you’re actually creating it in your mind through intuitive mental activity.
He rejected fundamental principles that the other schools took for granted, like the law of excluded middle, because for Brouwer, you can’t assume something exists mathematically unless you can constructively build it step by step. It was a philosophy that made math deeply personal and creative, while simultaneously making it much more restrictive than classical mathematics allowed.
Wait? What the heck is the law of the excluded middle ?
In classical logic, every proposition is either true or false. Brouwer rejects that until you can actually construct a proof that shows which one it is. Think about it this way: imagine I claim “there exists a prime number with exactly 1,000 digits that has some weird property X.” In classical math, most people would say this statement is either true or false; we just don’t know which yet. Brouwer would say “hold up, that statement doesn’t have a truth value until someone actually constructs such a number or proves none exists.” It’s the mathematical equivalent of “pics or it didn’t happen.”
Or to put it another way, I’m currently sitting in Châteauneuf-Grasse, in the south of France. I could easily say, “it is raining here a year from now.” But is that true? It is false? Brouwer would argue that it’s neither until we observe (construct) whether it’s raining.
This is a key insight of Brouwer’s: math is essentially about time. Mathematical truth isn’t some eternal, timeless thing floating in Plato’s heaven. Math unfolds through our mental activity over time. When we construct a proof, we’re not discovering a pre-existing truth, we’re creating truth through a temporal process of mental construction.
This sounds reasonable enough until you realize how many classical proofs rely on saying “well, either P is true or not-P is true, so let’s consider both cases...” Brouwer basically banned this move unless you could constructively demonstrate which case actually holds. The result was a much more restrictive but philosophically purer mathematics where every theorem came with a recipe for actually building or computing the objects it talked about. In fact, while most of basic math remains the same, Brouwer’s insistence upon time meant that he rejected the concept of actual infinity.
I’ve chatted with mathematicians about this and their general opinion, if they’ve ever heard about Brouwer, is that the ideas are useless but since so much of math depends on the law of the excluded middle (and proof by contradition, something else intuitionism rejects), that intuitionism isn’t practical.
But what about irrational numbers like π (pi)? They have infinitely many digits after the decimal point, right? Well, we’ve calculated trillions of them, but even NASA only uses the first 15 digits after the decimal point .
This fits well with my thoughts on the philosophy of math. It’s not so much that we arrive at the correct answer as we arrive at the useful answer. For example, $\sqrt{25}$ is equal to $5$ or $-5$. Which is correct? If you’re using this to measure distances, $-5$ doesn’t make any sense, even thought it’s an “answer.”
More Philophical Approaches.
Set-theoretic Platonism (ZFC)
This is today’s “working foundation” for most mathematicians. Everything is built from sets. It’s powerful but incomplete. Questions like the Continuum Hypothesis can’t be resolved within it.
Structuralism
Instead of objects, math is about relations. The number 2, for example, is just a position in the natural number structure. This view resonates with category theory and modern abstract algebra.
Finitism and Constructivism
More cautious views that reject infinite or circular definitions. They anticipate today’s emphasis on computability. (Note: Constructivism was invented after Intuitionism, but the latter is considered a subset of the former).
Mathematical Naturalism
W. V. Quine and Hilary Putnam argued that math doesn’t need metaphysical justification. It survives because science can’t do without it.
Why It Matters
You might wonder: isn’t this just philosophy? Does it affect actual math? The answer is yes. These foundations influence:
- What counts as a proof. An intuitionist won’t accept a non-constructive proof, while a formalist doesn’t care as long as the rules are followed.
- How we build new fields. Set theory, category theory, and constructive type theory all stem from these debates.
- How math connects to computers and science. Constructive philosophies align with algorithms, while empiricism connects math to physics and engineering.
The result is not just an academic quarrel, but a living set of options that shape mathematics itself.
Do Mathematicians Care?
Not really. Most mathematicians don’t spend their days debating whether they’re Platonists or Formalists. In practice, they adopt a kind of pragmatic Platonism, treating numbers, sets, and functions as if they exist “out there” and using them without worrying too much about metaphysics.
But the foundations aren’t just idle philosophy. They matter in three key moments:
- When crises arise: At the turn of the 20th century, paradoxes in set theory forced Hilbert, Brouwer, and Russell to rethink the entire basis of mathematics.
- When new frameworks are born: Category theory, constructive type theory, and other modern tools grew out of foundational debates and now drive progress in algebra, logic, and computer science.
- When limits are exposed: Independence results (like the Continuum Hypothesis) raise the question of whether some problems are even meaningful, depending on your stance.
So while most mathematicians can happily ignore these questions day-to-day, the edges of mathematics, where paradoxes and new ideas appear, are grown from the foundational soil beneath.
Closing Thought
So is mathematics discovered or invented? Despite the Platonist intuition that math exists “out there,” to me the evidence points toward invention: sophisticated mental constructions we create to describe what we observe.
Consider how we’ve repeatedly adjusted our mathematical frameworks when reality didn’t cooperate. Euclidean geometry seemed eternal and absolute until we discovered that parallel lines actually do meet in curved spacetime, leading to non-Euclidean geometries that better describe our universe. Newtonian mechanics worked perfectly until quantum phenomena and relativistic effects demanded entirely new mathematical languages.
Brouwer was onto something profound: mathematics unfolds through human mental activity over time. We don’t find pre-existing mathematical truths; we construct tools that help us make sense of patterns in the world. When those tools fail to capture what we observe, we invent better ones.
More importantly, ideas such as Brouwer’s intuitionism might have real-world applications. Here’s quantum physicist Nicolas Gisin using intuitionism to show that time might have a direction after all.
Or if you prefer reading, here’s a short article explaining Gisin’s point of view . Or here’s his short and easy-to-read paper on the topic (pdf).
Unfortunately, the Brouwer-Hilbert debate was won by Hilbert and time was essentially ejected from physics (read the Gisin paper).