Originally Posted: 10 February 2010
Here is a quote from an article I read last week.
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.The article is a classic in economics by Gale and Shapley. The title? College Admissions and the Stability of Marriage. If you "have a head for inferences," I suggest you read it [here]. It is an interesting and influential paper, and it is only 7 pages long.
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 your friends and ours cannot understand mathematics is not because they have no head for figures, but because they are unable to achieve the degree of concentration required to follow a moderately involved sequences of inferences. This observation will hardly be news to those engaged in teaching 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.