by Jochen Szangolies

In the last column, I have argued against the idea that understanding in mathematics and physics is transmitted via genius leaps of insight into obscure texts rife with definitions and abstract symbols. Rather, it is more like learning to cook: even if you have memorized the cookbook, your first soufflé might well fall in on itself. You need to experiment a little, get a feel for ingredients, temperatures, resting times and the way they interact before you get things right. Or take learning to ride a bike: the best textbook instructions won’t keep you from skinning your knees on your first try.

Practical skills are acquired through practice, and doing maths is just such a skill. However, mathematics (and physics by extension) may be unique in that it likes to pretend otherwise: that understanding is gleaned from definitions; that the manipulation of symbols on a page according to fixed rules is all there is to it. But no: just as you need an internal, intuitive model of yourself on a bike, its reactions to shifts in weight and ways to counteract developing instabilities, the skilled mathematician has an intuition of the mathematical objects under their study, and only later is that intuition cast into definitions and theorems.

At the risk of digressing too far, this is a general feature of human thought: we always start with an intuitive conception, only to later dress it up in formal garb to parade it before the judgment of others. We are not logical, but ‘analogical’ beings, our thoughts progressing as a series of dimly-grasped associations rather than crisp step-by-step derivations. If we do find ourselves engaged in the latter, then as a laborious, explicit, and slow ‘System 2’-exercise, rather than the intuitive leaps of ‘System 1’.

Indeed, it couldn’t be otherwise: how should we know whether a definition is accurate, if we didn’t have a grasp on the concept beyond that definition? Read more »