Pebble Piles and Shrinking Cores

The Student's Certainty

Eli: Around 1979 or 1980, at the University of Texas at Dallas, a mathematician named Klaus Truemper is talking with one of his PhD students. He asks him a simple thing. Was mathematics created, or discovered? The student doesn't hesitate. Discovered, he says. Not created. Truemper tells this story himself, and his words are: 'I was stunned that he was so sure about his answer.' And then: 'So right then I decided it was worthwhile to pursue an answer for this question in earnest.'

Mara: Which is the question a listener sent in, almost word for word: is maths invented or discovered?

Eli: That's a lot riding on one confident answer. The conversation led Truemper to investigate the question in earnest. And the student's side didn't come out of nowhere.

Mara: And Truemper points out how old his student's question is. People have argued it since Plato, about 2,400 years. Today's angle is a forty-year intellectual shift from mathematical realism to seeing all math as invented.

Eli: The student's side has a name: mathematical Platonism. A clear definition comes from Hartry Field. A mathematical realist, or platonist, believes that numbers, functions and sets exist, and believes them 'mind-independent and language-independent.' Realism, put plainly, says a true equation is true the way a mountain's height is true, whether anyone measures it or not.

Mara: So numbers aren't objects in a drawer, and they aren't opinions either. Real, just nowhere physical.

Eli: Nowhere physical, and nowhere in time. G. H. Hardy took that view in A Mathematician's Apology, in 1940. He wrote that 'mathematical reality lies outside us,' and that theorems are 'simply our notes of our observations.' Numbers like 2 or 317, he said, have nothing to do with sensation.

Mara: Which leaves the hard part. If that reality has no place and no clock, it's not obvious how anyone gets there to take the notes.

Eli: Here's the case realists reach for. In 1854, a German mathematician named Bernhard Riemann lays out a geometry for space that isn't flat. He even separates space being unbounded from space being infinite in extent. It's pure mathematics, worked out on paper.

Mara: Then, later, Riemann's geometry is credited with paving the way for Einstein's general theory of relativity. A piece of pure maths, finished and waiting, ends up as the frame for a theory of gravity.

Eli: So either Riemann uncovered something real, Mara, or he built a tool the universe happened to fit.

The Unreasonable Fit

Mara: And Riemann isn't a one-off. In 1960, Eugene Wigner gave the whole pattern a name: the unreasonable effectiveness of mathematics. In plain terms, maths that people work out for its own sake keeps turning up as the exact language of physics, and no one can say why. Einstein had asked the same thing. Wigner called it 'a wonderful gift which we neither understand nor deserve.'

Eli: That's the fit Truemper's student could point to, around 1980.

Mara: Truemper lays out how the other side answers. The creation argument says humans built maths gradually, 'starting with pebble piles that came to represent numbers.' The discovery argument says it 'exists already, possibly in some metaphysical space.' And he describes a camp in between. Humans create the axioms, the starting rules. Once that's done, every result that could be proved from them comes into existence and gets found, bit by bit. Invent the rules, discover what follows. It's a contested position, not a settlement.

Eli: Formalism pushes further toward invention. It treats maths as symbols moved by rules, with nothing behind them. David Hilbert argued that higher mathematics is 'no more than a formal game': uninterpreted strings of symbols moved around by fixed rules. No realm required. Then comes the awkward case. Roger Penrose points to i, the square root of minus one, and says 'that looks like a pure invention.' Yet quantum mechanics, in his words, 'turns out to be fundamentally based on these complex numbers.' Without them, he says, it couldn't be done.

Mara: So the most invented-looking thing in maths is the one physics can't drop. Truemper had decades with that puzzle still ahead of him. Nobody yet knows why the fit works. And whether it means maths is out there, or only that the rules people chose run very deep, is the part still standing.

The Shrinking Core

Eli: Here's where those decades went. Truemper spent about forty years on his student's question, studying the philosopher Ludwig Wittgenstein and a long run of books on the history of mathematics. At the start, he was already convinced the advanced material was created. What he couldn't settle was the elementary part, like the natural numbers. One, two, three. In his words, 'as time went on, the part that might have been discovered, kept on shrinking.'

Mara: So he began in the middle.

Eli: Then, about six years before he told this story, he's invited to a workshop in Aussois, France, to present what he'd gathered. The talk is one hour. Preparing it takes him a full year. Afterward, someone urges him to write it up as an extended paper for the German Mathematical Society. He can't see the arguments fitting into even thirty or forty pages. So he writes a book instead, and the writing takes two years. By then the shrinking part is gone. 'Today, I am convinced that all of mathematics has been, and continues to be, created.'

Mara: Even the counting numbers, then. His student's certainty, turned exactly around.

Eli: His shift suggests that digging into maths' origins wears away belief in maths apart from minds.

Mara: There's a line of research that digs there directly. It's called embodied cognition: the idea that abstract thought grows out of having a body and a brain like ours. In a 2000 book called Where Mathematics Comes From, two authors argue that even real numbers and sets have their origins in 'our specific embodiment.' Maths, they say, runs on metaphors, like a point 'moving' toward infinity. A set gets pictured as a container. And work from 1996 suggests container schemas appear to be realized in the brain: in maps of the visual field, center-surround receptive fields and gating circuitry.

Eli: Which leaves a strange gap, Mara. If it's all built, the people who do maths still describe finding it.

Invented Rules, Discovered Landscapes

Mara: Truemper speaks to that gap directly.

Eli: He says mathematicians of the nineteenth and early twentieth century 'had a mix of opinions.' And now, 'Today, most mathematicians claim discovery.' His explanation is training: 'the professional training matters a great deal!' That's the sociology of mathematical practice, how the daily habits of a job shape what people believe the job is about. Think back to his student, so sure, and to Hardy calling theorems notes on observations. Immersion in day-to-day problem-solving seems to build a strong sense of discovery, one that drifts apart from the theoretical case for invention.

Mara: On Truemper's account, the feeling of discovery says something about training.

Eli: Roger Penrose holds the other reading, Mara. 'Well I certainly belong to the school of thought that it's discovery,' he says, while adding that one has to be a little careful about it. On that side with him are Hardy, Andrew Wiles and George Ellis.

Mara: And on the invention side, Truemper has company. The philosopher Ludwig Wittgenstein put it flatly: 'the mathematician is not a discoverer: he is an inventor.' On his account, the path from axioms to a theorem doesn't exist until someone builds it. Marcelo Gleiser goes further. 'I think that mathematics is a human invention,' he says, a product of evolving on this particular planet to make sense of things and survive. A grand mathematical plan in nature, to him, sounds 'very crypto-religious.'

Eli: So Truemper's forty years ended with that same conclusion.

Mara: So, plainly, for the listener who asked: it's still a live divide. Realists like Penrose say the structures are there and get found. Wittgenstein, Gleiser and Truemper say it's all made. And a prominent middle reading, a philosophical synthesis that keeps a half of each, says humans invent the axioms and then discover what has to follow from them.

Eli: What would change the picture is a test, and nobody has one for whether numbers exist outside the physical universe.

Mara: And the other signal to watch: why maths built with no physics in mind keeps predicting physics, from Riemann's geometry to the square root of minus one. That still has no explanation.

Eli: For listeners diving in, there is mathematical Platonism.

Mara: Follow science with us as researchers question whether reality is found or made.

Eli: Thanks for spending this time with us. I'm Eli.

Mara: And I'm Mara. Until next time, this has been Angle.