December 2008
How Do We Learn Math?
Mathematician Keith Devlin is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute at Stanford University

“God made the integers; all else is the work of man.” Probably one of the most famous mathematical quotations of all time. Its author was the German mathematician Leopold Kronecker (1823-1891). Though sometimes interpreted (erroneously) as a theological claim, Kronecker was articulating an intellectual thrust that dominated a lot of mathematics through the second half of the nineteenth century, to reduce the real number system first to whole numbers and ultimately to formal logic. Motivated in large part by a desire to place the infinitesimal calculus on a “sound, logical footing,” it took many years to achieve this goal. The final key step, from the perspective of number systems, was the formulation by the Italian mathematician Giuseppe Peano (1858-1932) of a set of axioms (more precisely – and imprecise formulation turned out to be a dangerous rock on which many a promising advance floundered – an infinite axiom schema) that determines the additive structure of the positive whole numbers. (The rest of the reduction process showed how numbers can be defined within abstract set theory, which in turn can be reduced to formal logic.)

Looked at as a whole, it’s an impressive piece of work, one of humankind’s greatest intellectual achievements many would say. I am one such; indeed, it was that work as much as anything that led me to do my doctoral work – and much of my professional research thereafter – in mathematical logic, with a particular emphasis on set theory.

Mathematical logic and set theory are two of a small group of subjects that generally go under the name “Foundations of Mathematics.” When I started out on my postgraduate work, the mathematical world had just undergone another of a whole series of “crises in the foundations,” in that case Paul Cohen’s 1963 discovery that there were specific questions about numbers that provably could not be answered (on the basis of the currently accepted axioms).

…Now, Lakoff and Nunez do not claim that these metaphors – mappings from one domain to another – are deliberate or conscious, though some may be. Rather, they seek to describe a mechanism whereby the brain, as a physical organ, extends its domain of activity. My problem, and that of others I talked to, was that the process they described, while plausible (and perhaps correct) for the way we learn elementary arithmetic and possibly other more basis parts of mathematics, does not at all resemble the way (some? many? most? all?) professional mathematicians learn a new advanced field of abstract mathematics.

Rather, a mathematician (at least me and others I’ve asked) learns new math the way people learn to play chess. We first learn the rules of chess. Those rules don’t relate to anything in our everyday experience. They don’t make sense. They are just the rules of chess. To play chess, you don’t have to understand the rules or know where they came from or what they “mean”. You simply have to follow them. In our first few attempts at playing chess, we follow the rules blindly, without any insight or understanding what we are doing. And, unless we are playing another beginner, we get beat. But then, after we’ve played a few games, the rules begin to make sense to us – we start to understand them. Not in terms of anything in the real world or in our prior experience, but in terms of the game itself. Eventually, after we have played many games, the rules are forgotten. We just play chess. And it really does make sense to us. The moves do have meaning (in terms of the game). But this is not a process of constructing a metaphor. Rather it is one of cognitive bootstrapping (my term), where we make use of the fact that, through conscious effort, the brain can learn to follow arbitrary and meaningless rules, and then, after our brain has sufficient experience working with those rules, it starts to make sense of them and they acquire meaning for us. (At least it does if those rules are formulated and put together in a way that has a structure that enables this.)

Here is the full article.

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