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.
Math is much easier than chess.
Just wondering, is it possible for you to put the author of the article on top? I have thought you were the one who wrote the article.
God made PI irrational.
Jack: Your site crashed Firefox on my Red Hat Linux network installation, because it serves a Flash movie automatically. This may be a problem for any user with a “dumb terminal” that is served over a network—as opposed to users with their own PCs. It’s not something you realize when you have your own PC, but it’s considered good netiquette not to serve any videos or animations without visitors to your site clicking first. (ESPN.com breaks this rule, and it’s annoying.)
Being on a dumb terminal behind a network also is useful to judge load times for large webpages. Sometimes Susan’s main page has a several-second slowdown, as some other commenters have reported, but () this never occurs on my home PC, and () has been infrequent and never severe enough to recommend any change.
Finally, my “real” followup to Jack: Interestingly enough, we still do not know whether God made PI normal!
As a professional mathematician/computer scientist as well as chess IM, I can give another side to Keith Devlin’s observations. I started out as a logician, from Oxford no less, and initially subscribed to the rule-based view called Formalism which this article seems at first to be expounding.
However, Devlin’s actual views, as represented by his book The Math Gene, are that math is a science of patterns. This is my experience both in chess and in my research. My research is in a field, Computational Complexity Theory, which is neither inherently numerical nor symbolic, but rather is based on composing and measuring the efficacy of tasks, goals, and “chunks” of algorithmic thinking. Thus compared to other mathematical fields, “design patterns” take precedence over calculation or manipulations of symbolic rules. Well, this actually is what Devlin is saying here, in math and in chess: Once we get beyond “the horsey jumps 2 squares up and 1 square over”, we think in patterns such as “Sac Bxh7+, then Knight to g5 and Queen up wins”—patterns that are second-nature to strong players even when they don’t work! Similarly, in two cases my professional obligation to write up “journal versions” of my theorems in conference papers has been delayed because I see tangible improvements of those results, but haven’t yet gotten the “rules” in the proofs to work! (Many would say, OK then I don’t have the results, but my point is that in my experience, vision precedes calculation with all the successful stuff too. This is so with tactical players who make speculative sacrifices as well as with positional players.)
The difference is meaningful because in math we have “math engines” like chess engines, and we speculate whether they will similarly surpass us in generating theorems. This has already happened in some formal-intensive areas like polynomial lattice theory. In 1994 I asked Professor Robin Milner, who is eminent in computer theorem proving, how long it would take to convey the basic undergraduate course in my field to a theorem-proving system. His immediate answer: “Not in 50 years”. Moreover, some philosophical interpretations that go with our field’s Millennium Problem “P Versus NP” support Emil Post’s position that “The Logical Process is essentially Creative.” I have to stop commenting now and get on with some overdue referee reports—which is where I wish I had a computerized “math engine”! 🙂
Ah, from the whole article I see Devlin even has an earlier book titled The Science of Patterns. And here is an interesting excerpt from it regarding what I wrote:
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Here is a simple test…Consider the “doubling function” y = 2x (or, if you prefer more sophisticated notation, f(x) = 2x.) Question: When you start with a number, what does this function do to it?
If you answered, “It doubles it,” you are wrong…[and] your underlying concept of a function is wrong. Functions, as defined and used all the time in mathematics, don’t do anything to anything. They are not processes. They relate things. The “doubling function” relates the number 14 to the number 7, but it doesn’t do anything to 7. Functions are not processes but objects in the mathematical realm. A student who has not fully grasped and internalized that, whose underlying concept of a function is a process, will have difficulty in calculus, where functions are very definitely treated as objects that you do things to…
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However, in computational complexity, one must also see a function as a process. Indeed the whole ballgame is to ascertain how information must be aggregated and sifted during a computation. It is known that methods that look only at inputs and outputs must “relativize” and hence are incapable of resolving the P vs. NP question. (And if you look at the second bullet in that link and click natural proof, you come to a Wikipedia article which I revised.)
Mathematicians are usually useless at chess.
KW — Sorry about the Firefox crash. I use Firefox under XP w/o problems.
I do have Fedora on a system here at work, and I’ll check it out. However, it is not MY site — a friend, Joe Budzinski, just lets me write on his blog. Still, I will do what I can.