What does intuition really mean? For some it’s a mysterious mode of penetrating truths that we can’t fully explain or understand, while for others it’s simply a word that gets invoked when we stumble on a provocative coincidence. I might have an inkling that it’s going to rain even though there’s not a cloud in the sky and congratulate myself on my intuitive powers a few hours later when I reach for my umbrella, but the sceptical observer will pointedly ask me how many times I have forgotten similar premonitions when it didn’t rain.
So intuition is a notoriously tricky business, and most people of a scientific persuasion unhesitatingly ignore it altogether. Except, of course, when it comes to mathematics, where dismissing intuition is still a common practice, but inevitably comes with a certain amount of queasiness associated with it.
To non-mathematicians, this might seem very odd indeed. Intuition? In mathematics? Surely if there was a domain that was protected from fuzzy speculations and mysterious forces it would be the high-precision realm of mathematics. Well, not really.
The confusion stems, I think, from the fact that many of us don’t fully appreciate what mathematics is and what mathematicians do. They naturally assume that it is some sort of extension of their own experiences, which tend to be heavily skewed towards their personal struggles with arithmetic. There is, of course, nothing terribly deep or interesting about long division or factoring, per se. But dig just a little deeper, and things suddenly get very profound in a hurry, with metaphysical conundrums lurking behind every corner.
James Robert Brown, a philosopher with keen interest in the metaphysics of mathematics, shows us the way:
“Think of all the mathematical things that you already believe – two plus three equals five, that there is an infinitude of numbers, that there is an infinitude of prime numbers – and ask yourself, ‘Are all those facts, those mathematical facts, are they like physics?’
“When I say, “protons are heavier than electrons”, I’m saying – I hope – something true about the world: a statement of something that is independent of me, something that’s out there that physicists have discovered. Is math like that? Or is it more like this: bishops move diagonally in the game of chess. That’s also true, but it’s clearly something that we’ve simply created.
“So the first question is, ‘Which side are you on?’”
James is an unabashed mathematical Platonist, believing wholeheartedly in the independent reality of mathematical truths. The problem, of course, is that, in striking contrast to protons and electrons, it’s hard to see where, exactly, mathematical truths can be found; and, relatedly, by what concrete mechanism we gain access to them.
“This is the embarrassing question,” James admits. “Clearly they are not in the physical world. The number two isn’t to be found anywhere in the world the way the metre stick is to be found in Paris under glass where we can go and actually examine it. If they let us, we could hold it; we could check other metre sticks to make sure they’re the right length and so on. Numbers aren’t like that. So, the big question is, How do we get to know anything about them? How do we grasp them?
“I don’t know how it’s done. It is a mystery. It’s an embarrassment, frankly; and it’s the number one issue to hammer a Platonist with. You just say, ‘Well, how the hell can you do this?’”
But just because we don’t have an iron-clad answer, doesn’t mean that we can’t say anything meaningful at all. Indeed, probing our conception of mathematical truth inevitably leads us to grapple with what we mean by a proof in the first place. Which, in turn, directly leads us straight to the notion of mathematical intuition.
“A great English mathematician named G.H. Hardy used some provocative imagery to describe how he did mathematics. ‘Proofs don’t mean anything,’ he said. ‘Proofs are just rhetorical flourishes, showing off.’ I think he used the word “gas”. The real facts, he said are like this: it’s as if I can see a distant mountain range and I see a peak and then a couple more peaks.
“And then I pull you over and say, ‘I want you to just look over here. You see this peak that you already know of? Now look a little bit to the right and just behind it. You see that thing there? I can point out something new to you that you’ve never seen before, and then you’ll say, ‘Oh yeah, now I get it.’”
Our notion of a valid mathematical proof, in other words, our conviction of the objective truth of a mathematical statement, can actually be regarded as a form of perception that we somehow intuit.
This might seem, I well recognize, a bit abstract. But James gives a simple and compelling example of such mathematical intuition. Consider the following mathematical statement: 1 + 2 + 3 + … + N = (½)N² + (½)N.
And now look closely at the following diagram. A few moments of studied reflection demonstrate that this diagram is not simply a helpful aid or calculation device.
It is, in a very concrete way, nothing less than an iron-clad demonstration of the truth of the the algebraic statement: 1 + 2 + 3 + … + N = (½)N² + (½)N.
The diagram, in other words, is a tangible example of the process of mathematical intuition.
“If you look at that diagram, you see how powerful it is, because when you see how the diagram works, you understand how the equation works generally. You see that it works not just for N=5, but N=6 as well. It works for N=2. It works for every single number N, all infinitely many of them.
“And that’s truly remarkable. I don’t know exactly what’s going on that allows us to see the truth of that theorem just by looking at that diagram, but when you’ve studied it long enough you become convinced that it’s a real proof. The normal proof of this statement is by a process called ‘mathematical induction’, which all professional mathematicians are happy to accept. The normal ideology of mathematics is that you should stay away from pictures. They can be psychologically helpful, but they’re not actually evidence.
“But when you look at this example, you begin to think, No, that’s a real proof. That does it, That’s just as convincing. The evidence is every bit as strong in favour of that theorem through that picture of a “normal proof” by induction.
“And if you say, ‘Oh, I see that it works for 5, but I’m not sure about 13. Show me 13.’ Then we’d say, ‘No, no, you haven’t got it yet. Let’s study the N=5 case a bit longer and then the light will go on and you’ll see it works for every single number N. Somehow or other you will gain that little bit of perception, that little grasp, that little seeing with the mind’s eye.”
Mathematical intuition is thus a deeply slippery and subtle concept, and likely will always remain so. But that hardly means it doesn’t exist.
Howard Burton, firstname.lastname@example.org
To best explore the topics raised in this post for teaching and learning, please see related video and print resources that are part of Ideas Roadshow’s IBDP Portal: TOK Sampler: Extending Experience (Mathematical Intuition), Clips: Proof by Picture, All in the Same Boat, Appeals to Authority, Compilation: Invented or Discovered? Mathematical Platonism, Full-length video and eBook: Plato’s Heaven: A User’s Guide, and more.