We’re always rationally explaining and articulating things. But we’re at our most intelligent in the moment just before we start to explain or articulate. Great art occurs—or doesn’t—in that instant. What we turn to art for is precisely this moment, when we “know” something (we feel it) but can’t articulate it because it’s too complex and multiple. But the “knowing” at such moments, though happening without language, is real. I’d say this is what art is for: to remind us that this other sort of knowing is not only real, it’s superior to our usual (conceptual, reductive) way.

Using concreteness as a foundation for abstraction is not just good for mathematical instruction; it is a basic principle of understanding. Novices crave concreteness. Have you ever read an academic paper or a technical article or even a memo and found yourself so flummoxed by the fancy abstract language that you were crying out for an example?

Or maybe you’ve experienced the frustration of cooking from a recipe that was too abstract: “Cook until the mixture reaches a hearty consistency.” Huh? Just tell me how many minutes to stir! Show me a picture of what it looks like! After we’ve cooked the dish a few times, then the phrase “hearty consistency” might start to make sense. We build a sensory image of what that phrase represents. But the first time it’s as meaningless as 3 + 2 + 1 would be to a three-year-old.

This is how concreteness helps us understand—it helps us construct higher, more abstract insights on the building blocks of our existing knowledge and perceptions. Abstraction demands some concrete foundation. Trying to teach an abstract principle without concrete foundations is like trying to start a house by building a roof in the air.

Knowledge, and especially disagreeable knowledge, cannot by any art be totally excluded even from those who do not seek it. Wisdom, said Aeschylus long ago, comes to men whether they will or no. The house of delusions is cheap to build, but draughty to live in, and ready at any instant to fall; and it is surely truer prudence to move our furniture betimes into the open air than to stay indoors until our tenement tumbles about our ears. It is and it must in the long run be better for a man to see things as they are than to be ignorant of them.

• A. E. Housman.

Mathematics, which doesn’t include the tawdry efforts of statistics or probability, pure mathematics, the product of the human mind turning to face itself, turning into itself, and finding in the realm of necessary consequences, where no contingent fact is to be seen or heard or smelled or tasted or touched—it discloses a beauty that exhausts human comprehension and a certainty the senses can never touch.

Friedrich Hayek described the issues well in his 1974 Nobel Prize lecture, which he delivered under the title ‘The Pretence of Knowledge’. Hayek observed:

I regard it in fact as the great advantage of the mathematical technique that it allows us to describe, by means of algebraic equations, the general character of a pattern even where we are ignorant of the numerical values which will determine its particular manifestation. ... It has led to the illusion, however, that we can use this technique for the determination and prediction of the numerical values of those magnitudes; and this has led to a vain search for quantitative or numerical constants.

He continued:

compared with the precise predictions we have learnt to expect in the physical sciences, this sort of mere pattern predictions is a second best with which one does not like to have to be content. Yet the danger of which I want to warn is precisely the belief that in order to have a claim to be accepted as scientific it is necessary to achieve more. This way lies charlatanism and worse. To act on the belief that we possess the knowledge and the power which enable us to shape the processes of society entirely to our liking, knowledge which in fact we do not possess, is likely to make us do much harm.