Towards the end of his celebrated autobiography that was published in 1976, mathematician Stanislaw Ulam makes a striking remark about the way mathematics is presented:
‘(…) This was more agreeable than the present style of the research papers or books which have so much symbolism and formulae on every page. I am turned off when I see only formulas and symbols, and little text. It is too laborious for me to look at such pages not knowing what to concentrate on. I wonder how many other mathematicians really read them in detail and enjoy them.’
To wit, these are the words of someone who really has enjoyed mathematics and has been engaged in the highest ranks of the subject for almost all of his life.
For me this is quite a relevant statement, since I started studying mathematics at the University of Leiden (The Netherlands) in the year 1975. And for me it was like Ulam describes. Lectures in mathematics almost entirely involved the stating of theorems and the subsequent proofing of them. Little was said about the meaning of what was proofed, why it would be interesting, or even what the essential idea of a proof was; most of the time no background or context of any kind was given. A semester of Lebesgue integration theory was given without even referring to the problems that had arisen with more basic forms of integration like the Riemann-Stieltjes Integral. It made a lot of the matter less exiting than it could have been. And to be honest, most of the proofs stayed quite unintelligible: one could follow the details but kept missing the big picture.
The point however is, that it only now becomes clear to me that I have been a fashion victim, that what I perceived as the way mathematics was done period, was only a relatively new style of writing and teaching, a fashion that had been en vogue for only a few decades yet.
This reflection of Stanislaw Ulam is confirmed by Davis & Hersh in their 1981 book The Mathematical Experience. In a section on the philosophy of mathematics they remark:
‘The formalist style gradually penetrated downward into undergraduate mathematics teaching and, finally, in the name of “the new math”, even invaded kindergarten (…)’. (p.344)
And they continue with the observation that the formalist style might have had its longest time. Actually I’m not sure that such a thing will happen. At least some of the formalism seems to me related to a certain machismo between mathematicians; the shorter and the less intuitive the proof, the better the mathematician.
In their section on Teaching and Leaning Math, Davis & Hersh give an example of the contrast between a short formal proof and a more elaborate and a more intuitive one. It is about ‘the two-pancake problem’, the problem of cutting two pancakes in halves by cutting only once in a straight line. And the pancakes aren’t on top of each other. The example of ‘the two-pancake problem’ is put in the context of the contrast between what is called ‘the logic of scientific discovery’ and that of ‘the logic of scientific justification’. The latter being a streamlined version of the former, a logically tight presentation with all hurdles and frustrations left out. It is a linear ‘success only’ story, told in a highly stylized language, ideally that of formal logic.
Now such a linear success story has only one goal, and that is to bring home the message of success. Formal proofs do that, but with the same price paid as with other success stories: because of the lack of drama it is difficult to get engaged by them and the insights that the storyteller gained in his struggles are not the focal point of the story, only the message of success is.
So I think there’s something to say for a math education (if not math itself) where insights from storytelling are used to bring home the insights of the great mathematicians.
Do you have a story to tell?