For most of us, our mathematical education involved a lot of names. There’s Pythagoras’s Theorem; the Riemann Hypothesis; Hilbert’s Hotel, and so many more concepts and theorems intrinsically linked to the scholars who first discovered them. Or should we say invented them?
It’s a serious question. We’re so used to math being this indispensable tool, letting us unravel the fundamental laws of the Universe, get rich quick, or simply plan the most efficient party guest list, that we don’t often stop to consider where it comes from.
So which is it? Is math invented, with all those theorems and concepts so rigorously proved throughout millennia just a byproduct of human perception; or is it discovered, prompting the notion of some factual, real “six” out there in the cosmos somewhere?
It’s a trickier question than you might think.
Real numbers, or imaginary?
If you’ve ever found yourself wondering whether a2 + b2 = c2 is some cosmic and immutable truth, or just something we’ve decided is handy for bridge building, you’re in good company. The question of whether math is discovered or invented goes back a very long time: “since at least Plato’s time (4th century [BCE]),” Alexander Paseau, Professor of Mathematical Philosophy at the University of Oxford, tells IFLScience.
“And probably before that,” he adds. “The Pythagoreans had a philosophy of number.”
Even today, one of the major schools of thought on the question is known as Platonism, after the wrestler-turned-philosopher whose ideas it’s based on. It’s fair to say followers came down pretty firmly on one side of the argument: Platonists hold that “numbers, mathematical objects, in fact everything, is real,” explains Joe Morrison, a Teaching Associate in the University of Sheffield’s Department of Philosophy.
“They’re mind-independent,” he tells IFLScience. “They don’t depend on anyone thinking about them, or naming them, or coming up with them, or having any concepts; they’re really out there.”
On the spectrum from “discovered” to “invented”, then, Platonism plumps decisively for “discovered.” A statement like, for example, “the Riemann hypothesis is true” is just as dependent on factual reality as one like “all apples are red”; if all human life were wiped out from existence today, it wouldn’t change the fact that 2 + 2 = 4 any more than it would alter the number of atoms in a molecule of water.
It’s just one branch of what’s known as mathematical realism. There are other ways to believe in the real-life existence of mathematical entities: “Aristotelians think maths is about the physical world,” Paseau notes, while, say, object realism removes the Platonic requirement for independence. But all are agreed: mathematical objects, like numbers and sets, are real (if abstract) entities which we as intelligent agents discover, rather than invent.
To grasp the transcendental
Mathematical realism has attracted some fairly famous names over the centuries – even Kurt Gödel, possibly the person you’d least expect to take a hard “math is real” stance, ascribed to the view. But it’s far from universally accepted: “there’s a huge load of people who say Platonism is preposterous,” Morrison says; “it’s overblown, it’s bloated, and it tells us that we have to accept into our ontology – into our inventory of what exists – all of these really strange entities, these mathematical entities which are just not like anything else.”
And the problems with Platonism run even deeper than that. At first, the discoverability of mathematical realities may seem like a strength of the stance – who can truly say they aren’t a bit swayed by the argument that “two” will equal two regardless of whether or not we say it does?
But “if mathematical truths are somehow really out there,” Morrison explains, “and they’re these eternal non-spatiotemporal, universal structures; they don’t exist in time, they don’t exist in space, they just somehow are real… then how is it that humans are supposed to ever have any knowledge of them?”
If not real, then what?
So, maybe math isn’t made up of real objects and truths just waiting for us to somehow discover them. But what’s the alternative? Well, fittingly enough for a view that opposes the idea that math is fact, it’s called fictionalism – and at first glance, it probably seems a little weird.
“Fictionalism is the idea that maths is a fiction: ‘1 + 2 = 3’ may be true in the story of maths, in the same way that it’s true in Dickens’s fiction that Oliver Twist lives in London,” Paseau explains. “But it’s not literally true, in the same way that it’s not literally true that Oliver Twist lives in London – because he doesn’t even exist.”
As you might imagine, this view is kind of controversial. “When one first hears the fictionalist hypothesis, it can seem a bit crazy,” writes Mark Balaguer, a researcher in mathematical philosophy at California State University, Los Angeles, in The Stanford Encyclopedia of Philosophy entry on the topic.
“Are we really supposed to believe that sentences like ‘3 is prime’ and ‘2 + 2 = 4’ are false?” he asks. “But the appeal of fictionalism starts to emerge when we realize what the alternatives are. By thinking carefully about the issues surrounding the interpretation of mathematical discourse, it can start to seem that fictionalism is actually very plausible, and indeed, that it might just be the least crazy view out there.”
It can seem like kind of a cheat – as Bertrand Russell once famously protested, it’s akin to reaping “the advantages of theft over honest toil,” Morrison says. But, fictionalists would argue, we really do this kind of thing all the time: “we talk about Lord of the Rings, and we can say things like ‘do hobbits live in The Shire?’” he points out. “And the answer is yes, it’s true that hobbits live in The Shire […] hobbits live in The Shire, and hobbits don’t exist.”
The same is true, fictionalists believe, about mathematical statements. This is where the weird idea that “3 is prime” is a false statement comes from – not because a fictionalist thinks 3 has factors other than itself and one, but because things like “3” and “being prime” and “factors” simply aren’t things which exist.
“Mathematical fictionalists are keen to say, look, we’re not denying that 2 + 2 = 4, we’re not saying it’s false,” Morrison explains. “There’s a sense in which it’s true, but it’s true in the same way that, like, Bilbo Baggins lived in The Shire. It’s true according to the fiction.”
A question of surprise
Just like realism, though, fictionalism has its opponents. After all, if math is all simply invention, then, well – how come it works so well?
It’s not a trivial question. How is it that, say, a simple equality linking the radius of a circle to its circumference should also be integral to modeling the dynamics of financial derivatives over time? Why on Earth would the physical movement of a pendulum even be related at all to the presentation of numbers as sums of four squares, let alone be dependent on the exact same equation? How do we even begin to explain analytic geometry, or Monstrous Moonshine?
“That’s taken to be a real challenge to fictionalists. Like, a substantial challenge,” Morrison says. Philosophers call it the unreasonable efficacy of mathematics: the ever-present, seemingly inexplicable, and often entirely unpredictable ability of math to explain apparently any scientific phenomenon.
“So, we can put people on the moon – if you watch the film Hidden Figures, they’re literally doing, you know, long form mathematics […], very complicated equations which were effective, which actually allowed two moving objects to coincide in a very effective way,” Morrison explains.
“And the question would be, if all of this maths talk is in some sense false, or in some sense not real, if it’s not picking up real properties of reality – how can it be so effective?”
Or consider, for example, the periodical cicada species of North America which lurk underground for years before emerging to engulf whole towns in their rush to eat, breed, and drop. There are seven different species of the insects, and all of them keep to a strictly mathematical regimen, only popping up after seven, 13, or 17 years under the earth.
“How come these are prime number life cycles? And the answer is just a bit of number theory,” Morrison says. “It’s just to do with lowest common multiples. It’s a biological explanation, but it’s a biological explanation [that] indispensably relies on properties of numbers. The primeness of the life cycles is clearly making a physical difference.”
It is, on the face of it, a persuasive argument in favor of realism – or at least, against fictionalism, which is not the same thing. Because while a fictionalist may not be able to answer the question perfectly, there’s no guarantee the realist can either.
“That’s really where philosophy gets interesting,” Morrison says. Fictionalists will “probably try and either weasel out of the problem […] or there’s a pushback – ‘well, how does the other view deal with this? Why should it be so straightforward for realism to make sense of surprising coincidences?’”
Quod erat demonstrandum
So, what does this mean for math? Neither invented, nor discovered, but some secret third thing, indecipherable to even those devoting their lives to the question.
“Somewhere in between the two – that has to be the right answer,” Morrison suggests. “My instinct is that some mathematical structures will have the appearance of being objective, but will turn out to be less than fully objective.”
In fact, we’ve barely scratched the surface of possible solutions. “Some people think the objects of maths are mental,” Paseau explains; there are the Formalists, who “see mathematics as meaningless symbol-manipulation. Not a very popular view these days,” he says, and the Deductivists, and “a host of other, relatively unpopular views.”
“Most contemporary philosophers of mathematics, including me, are structuralists,” he adds – although “the question is what sort of structuralism to go for.”
Will we ever know the answer for sure? It’s possible. For all the navel-gazing philosophy is famous for, there’s plenty of practical and experimental work going on as well – and as technology advances, we may start seeing the kinds of breakthroughs you might not expect from a subject sometimes jokingly described as “math without the trashcan.”
Neuroscience might yet yield insights into mathematical cognition, Morrison suggests, separating what in math is just a feature of human thought and what is somehow external to us; the increasing use of computers to invent and prove theorems, Paseau posits, is equally likely to shift how we think about math.
But ultimately, this may be one of those questions we just have to live with not knowing the answer to – no matter how long and hard we think about it.
“My gut instinct is that maths is invented,” Paseau says. “That’s what I believed before going into philosophy.”
“It’s not what I think now,” he adds. “Over time, I’ve come to think that it’s discovered. That’s where the arguments seem to lead.”
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