Can Philosophers Tell Mathematicians They’re Wrong About Math?
The Man Who Wanted to Erase Most of Math
In 1908, the Dutch mathematician L.E.J. Brouwer (1881–1966) made a shocking claim: the math you learn in school is wrong. Not just a few equations — most of it. He believed that standard mathematics relied on a principle that has no solid proof: that every statement is either true or false. (This is called the law of excluded middle.) He also objected to definitions that talk about a thing by referring to a whole collection that includes the thing itself (impredicative definitions). Brouwer wanted to rebuild math from scratch, throwing out anything that didn’t meet his strict, intuition-based rules. His approach became known as intuitionism.
To many scientists and mathematicians, this sounded like a disaster. Classical math was working beautifully — physics, engineering, and everyday life all depended on it. Why should a philosopher’s doubts tear down centuries of success? This reaction gave birth to a powerful idea in the philosophy of math: methodological naturalism. It says, roughly, that when we ask what’s true in mathematics, we shouldn’t listen to philosophical armchair reasoning. We should only trust the standards that have proven themselves in the real work of science and mathematics.
Three Ways to Be a Naturalist About Math
If you’re a methodological naturalist, you believe that some set of authoritative standards should settle questions about math. But which standards? Philosophers have three main answers.
Scientific naturalists say that the only standards that matter are those of the natural sciences — physics, biology, chemistry, and so on. When a mathematical claim is needed for our best scientific theories, we accept it. The American philosopher W.V.O. Quine (1908–2000) championed this view. For him, even the existence of numbers and sets was to be judged by whether science required them.
Mathematical naturalists flip this around. They argue that mathematics should answer only to its own internal rules — proofs, consistency, and the kinds of reasons mathematicians actually use. On this view, if practicing mathematicians accept a certain set theory or a controversial axiom, philosophers have no business telling them they’re wrong.
Mathematical-cum-scientific naturalists take both sets of standards seriously. When math and science speak with one voice, that’s the final word. John Burgess (20th–21st century) is a well-known defender of this blended view.
You can also slide these views on a strength scale. A very strict naturalist might say: “Accept a claim if and only if the standards approve it.” A milder one says: “Believe it if the standards approve it, but other ways of knowing might sometimes count too.” The mildest version, accepted by most philosophers today, is just: “Don’t believe something that clearly clashes with well-established math or science.”
The Blurry Line Where Math Meets Everything Else
Naturalists face a tricky puzzle: where exactly does mathematics end and philosophy begin? After a seminar, when mathematicians gather for coffee and argue about which unsolved problem is the most important in the field, are they still doing math — or is that a personal judgment outside mathematics proper? There’s no sharp boundary. Yet the naturalist needs one, because she’s claiming that only certain kinds of standards (scientific, mathematical, or both) should be the judge.
Many naturalists respond that a blurred line isn’t fatal. A forest has no exact edge, but we still know when we’re deep inside it. Mathematicians have an implicit sense of what counts as a mathematical reason, even if they can’t write down a neat definition. The philosopher Penelope Maddy (born 1950) argues that, in practice, philosophical considerations rarely succeed in overturning established math — and when they seem to, a closer look usually shows that purely mathematical reasons did the real work.
A tougher problem emerges when mathematical and scientific standards pull in opposite directions. Suppose a scientific rule says “don’t believe in more things than you need” (a kind of ontological economy). But a deep principle inside set theory says “the universe of sets should be as rich and large as possible” (something Maddy calls MAXIMIZE). If a theory satisfies the mathematicians but offends the simplicity-loving scientists, a naturalist who trusts both camps has to decide which rule to follow. No one has yet written a clear recipe for solving such clashes.
Why Should We Crown Science and Math?
You might wonder: why should we hand the crown to science and math in the first place? Naturalists often appeal to success. Physics lands rovers on Mars; number theory gave us the encryption that keeps your messages private. Philosophy, by contrast, seems to spin in circles. If you want real answers, trust the methods that actually deliver.
This argument has bite, but it’s weaker than it looks. Imagine a made-up discipline called “Guru-ology.” Its rule is: “Ask the Guru a question, and whatever the Guru says counts as true.” If the Guru never contradicts herself, Guru-ology can be wildly “successful” by its own lights — it answers every question it asks. But that doesn’t make Guru-ology believable. Success measured by your own rules doesn’t automatically impress outsiders. So a naturalist needs a non-question-begging way of measuring success — and that’s hard to find.
The track-record argument says: scientific and mathematical standards have a proven history of uncovering truths, while philosophical standards have led to endless fights. But as opponents point out, philosophers belong to different schools. A medieval theologian might think his tradition’s track record is spotless because his standard includes harmony with scripture. A modern philosopher’s standard might include only logic and evidence. So it’s not clear that “philosophy as a whole” has a uniformly bad record — and even if it did, that might say nothing about your own particular philosophical standards.
Finally, even if science is fantastic at answering scientific questions, why should we trust it to answer philosophical questions about math? A world-class sprinter isn’t automatically the best person to referee a chess tournament. The naturalist needs a bridge from one domain to the other, and building that bridge turns out to be surprisingly difficult.
Maddy’s Two-Hat Naturalism: Leave Math Alone, Then Think Scientifically
Penelope Maddy offers an interesting compromise. She is a heterogeneous naturalist: she puts on one hat for mathematics itself and a different hat for the philosophy of mathematics.
When mathematicians prove theorems, choose axioms, or decide what counts as a good definition, Maddy says, we should respect their own internal, mathematical standards. Physics or philosophy can’t overrule them. If set theorists find that large cardinal axioms make set theory deeper and more unified, that’s all the justification they need — even if those axioms never get used in a physics lab. This is the autonomy thesis: mathematics runs itself.
But when we step back and ask what math is really about — are numbers abstract objects? is set theory a literally true story? — Maddy says we must use the standards of natural science. Here, Quine’s rule applies: we look at our best scientific picture of the world and include whatever that picture needs. If science works fine without treating numbers as real objects, then a philosopher might adopt an “arealist” view, seeing math as a powerful game of “if-then” rather than a report of eternal truths.
Critics push back: if math is allowed to rule itself, why should we then turn around and let science tell us whether math is true? That seems a bit like letting a club make up its own rules, then asking a completely different club to say whether the first club’s games mean anything. Maddy’s more recent work explores the idea that pure math isn’t in the business of being “true” at all — it’s in the business of being mathematically interesting and useful. But whether that fully escapes the tension is still hotly debated.
Who Gets the Last Word on 2+2=4?
This isn’t just a fight among professors. It touches something you deal with every day. You learn in school that two plus two equals four. Then you might come across a philosopher who asks, “But does the number four really exist, or is it just a useful thought?” Most of us feel that the math works, so the philosopher’s question shouldn’t make us panic. But why do we feel that? Is it because math’s proven success gives it authority over abstract puzzles? Or because we just haven’t thought deeply enough?
Methodological naturalism tries to answer that: it claims that the very standards of math and science are the ultimate authority. Yet, as we’ve seen, that answer itself must be defended without sneaking in the very standards it’s trying to promote. The debate remains wide open, and that is good news for anyone who wants to keep thinking for themselves.
Think about it
- Your math teacher says a statement is proven true. A philosopher says the objects the statement is about don’t exist, so it can’t be “true” in the usual sense. Which voice would you trust more, and why?
- Suppose a new form of “number therapy” claims to cure sadness, but its only evidence is that its creator says it works. Would you accept it, or would you demand the kind of evidence science uses? What does that tell you about which standards you already trust?
- If a discipline can answer all its own questions perfectly but never connects to anything outside itself, should we treat its answers as real knowledge? (Think of a board game with perfectly consistent rules but no link to the physical world.)
