Rigor, Math, and Numbers
Quickly researching the work of the two Nobel Prize winners Monday morning has given me more than the usual amount of thinking to blog on. I came across an interesting thought in the classic 1962 Gale/Shapley article that, as you’ll see, it did not make sense to put in my article in the Wall Street Journal. But it was insightful and well-stated. It reminded me of why I had become a math major at the University of Winnipeg, although I admit that my math skills (though not my algebra skills, Bryan) have atrophied.
Here’s the short verbal but mathematical proof from David Gale and Lloyd Shapley that led to the insightful quote to come:
We assert that this set of marriages is stable. Namely, suppose John and Mary are not married to each other but John prefers Mary to his own wife. Then John must have proposed to Mary at some stage and subsequently been rejected in favor of someone that Mary liked better. It is now clear that Mary must prefer her husband to John and there is no instability.
Now the quote about math:
Finally, we call attention to one additional aspect of the preceding analysis which may be of interest to teachers of mathematics. This is the fact that our result provides a handy counterexample to some of the stereotypes which non-mathematicians believe mathematics to be concerned with.
Most mathematicians at one time or another have probably found themselves in the position of trying to refute the notion that they are people with “a head for figures.” or that they “know a lot of formulas.” At such times it may be convenient to have an illustration at hand to show that mathematics need not be concerned with figures, either numerical or geometrical. For this purpose we recommend the statement and proof of our Theorem 1. The argument is carried out not in mathematical symbols but in ordinary English; there are no obscure or technical terms. Knowledge of calculus is not presupposed. In fact, one hardly needs to know how to count. Yet any mathematician will immediately recognize the argument as mathematical, while people without mathematical training will probably find difficulty in following the argument, though not because of unfamiliarity with the subject matter.
What, then, to raise the old question once more, is mathematics? The answer, it appears, is that any argument which is carried out with sufficient precision is mathematical, and the reason that your friends and ours cannot understand mathematics is not because they have no head for figures, but because they are unable [or unwilling, DRH] to achieve the degree of concentration required to follow a moderately involved sequence of inferences. This observation will hardly be news to those engaged in the teaching of mathematics, but it may not be so readily accepted by people outside of the profession. For them the foregoing may serve as a useful illustration.