For most of us, our mathematical education involved a lot of names. There’s

It’s a serious question. We’re so used to math being this indispensable tool, letting us unravel the

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 *a*2 + *b*2 = *c*2 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

“And probably before that,” he adds. “The

Even today, one of the major schools of thought on the question is known as *Platonism,* after

“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,

**To grasp the transcendental**

Mathematical realism has attracted some fairly famous names over the centuries – even Kurt Gödel, possibly the person you’d

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

“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 *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 *all* to the presentation of numbers as sums of four squares, let alone be

“That’s taken to be a real challenge to fictionalists. Like, a substantial challenge,” Morrison says. Philosophers *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*, *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?”

“How come these are prime number life cycles? And the answer is

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

“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

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.”

*All “explainer” articles are confirmed by **fact checkers** to be correct at time of publishing. Text, images, and links may be edited, removed, or added to at a later date to keep information current. *