Using History to Teach Mathematics | Dijkstra's Rallying Cry for Generalization

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Dated:<br>5 September 2022

In my previous post I maintained that reasoning like a mathematician helps in order to grasp the history of mathematics. Here I shall support a complementary claim which is less obvious: knowing important developments in the history of mathematics facilitates mathematical comprehension. Specifically, I engage with fellow scholars and I use Henri Poincaré as an historical actor in an attempt to explain why teaching mathematics with a historical dimension is desirable. Finally, in the last two paragraphs I briefly mention my own agenda on how to combine history and maths.

Henri Poincaré wrote on mathematical education. First he described mathematical heterogeneity. Quoting from [1, p.120-1] with my numbering:

(1) Many children are incapable of becoming mathematicians who must none the less be taught mathematics ...

(2) [M]athematicians themselves are not all cast in the same mould. We have only to read their works to distinguish among them two kinds of minds—logicians like Weierstrass, for instance, and intuitionists like Riemann.

(3) There is the same difference among our students. Some prefer to treat their problems "by analysis," as they say, others "by geometry."

(4) Since the word understand has several meanings, the definitions that will be best understood by some are not those that will be best suited to others.

Second, Poincaré addressed the evolution from mathematics which was mostly "devoid of exactness" to the formal rigor of David Hilbert et al. — an evolution which came with a "sacrifice" [1, p.124]:

(5) What [the science of mathematics] has gained in exactness it has lost in objectivity. It is by withdrawing from reality that it has acquired this perfect purity.

Here we see Poincaré scrutinize the work of the modern logician, a topic which I have addressed repeatedly on this blog, e.g., in this recent post. My current understanding is that Poincaré was an ontological dualist and in a similar way to Einstein [2], as the last sentence in the following quote suggests:

(6) We used to possess a vague notion, formed of incongruous elements, some a priori and others derived from more or less digested experiences, and we imagined we knew its principal properties by intuition. Today we reject the empirical element and preserve only the a priori ones. One of the properties serves as definition, and all the others are deduced from it by exact reasoning. This is very well, but it still remains to prove that this property, which has become a definition, belongs to the real objects taught us by experience, from which we had drawn our vague intuitive notion. In order to prove it we shall certainly have to appeal to experience or make an effort of intuition; and if we cannot prove it, our theorems will be perfectly exact but perfectly useless . [1, p.125, my emphasis]

Lost in Logic — makes a great title for a book; the preface would go like this:

(7) When the logician has resolved each demonstration into a host of elementary operations, all of them correct, he will not yet be in possession of the whole reality; that indefinable something that constitutes the unity of the demonstration will still escape him completely [to the extent that the lecturer does not even realize this: read, e.g., my 2021 article].

What good is it to admire the mason's work in the edifices erected by great architects, if we cannot understand the general plan of the master? Now pure logic cannot give us this view of the whole ; it is to intuition we must look for it. [1, p.126, my emphasis]

(My oral histories with Peter Naur and Michael A. Jackson convey a similar view w.r.t. computer programming.) The crux is that students need case studies and lots of intuition in order to appreciate mathematical definitions , let alone theorems and proofs.

Third, how then did Poincaré propose to teach both the intuition and the rigor pertaining to Mathematics? By resorting to the history of mathematics. In his words:

(8) Zoologists declare that the embryonic development of an animal repeats in a very short period of time the whole history of its ancestors of the geological ages. It seems to be the same with the development of minds. The educator must make the child pass through all that his fathers have passed through, more rapidly, but without missing a stage. On this account, the history of any science must be our first guide . [1, p.127, my emphasis]

Mathematical education and history come together splendidly. And since I'm interested in both topics I shall quote Poincaré in full:

(9) Our fathers imagined they knew what a fraction was, or continuity, or the area of a curved surface; it is we who have realized that they did not. In the same way our pupils imagine that they know it when they begin to study mathematics seriously. If, without any other preparation, I come and say to them: "No, you do not...