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 , 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 know it; you do not understand what you imagine you understand; I must demonstrate to you what appears to you evident;" and if, in the demonstration, I rely on premises that seem to them less evident than the conclusion, what will the wretched pupils think? They will think that the science of mathematics is nothing but an arbitrary aggregation of useless subtleties; ...
Later on, on the contrary, when the pupil's mind has been familiarized with mathematical reasoning and ripened by this long intimacy, doubts will spring up of their own accord, and then your demonstration will be welcome. It will arouse new doubts, and questions will present themselves successively to the child, as they presented themselves successively to our fathers, until they reach a point when only perfect exactness will satisfy them. It is not enough to feel doubts about everything; we must know why we doubt. [1, p.128, my emphasis]
Not totally unrelated to Poincaré's narrative is Carlo Rovelli's account of what science (in general) entails: "it's the awareness of our ignorance that gives science its reliability" [3, p.230-1]. Likewise, Evangelos N. Panagiotou's 2011 article, entitled Using History to Teach Mathematics: The Case of Logarithms , not only provides seversal reasons why students can benefit from historical awareness, it also explains how the history of mathematics can be used as a didactical tool. For instance, engaging with Freudenthal , Panagiotou writes:
The presentation in the form Definition-Theorem-Proof-Corollary can be elegant and can save time but the students remain with the query: How did the idea for these definitions and theorems come [about]? According to Freudenthal [5, p.107]: "[T]he basic definitions should not appear in the beginning of an exploration, because in order to define something one should know what this is and also in what it is useful." [4, p.28]
Several references to the literature are provided in Panagiotou's article. In contrast to most of these references however, my methodological preference is to address a particular episode (in the history of mathematics) from the vantage points of various historical actors. So, instead of focusing on a detailed chronology (which is useful), I want to convey, say, three very different receptions of Cantor's diagonal argument: receptions by Georg Cantor himself, Henri Poincaré, and Ernest Hobson — as I shall expound in a forthcoming blog post. To borrow the terminology of my previous post: Cantor was an actualist (embracing an actual infinity and Platonism, as we say today), Poincaré was a potentialist (eschewing completed infinities), and Hobson was an operational actualist (reasoning operationally with a completed infinity). Subsequently, I will connect each of these historical actors to a present-day mathematician or computer scientist.
Each of Poincaré's concerns (listed above) will come to the fore in my actor-dependent account of Cantor's diagonal argument. First, mathematical heterogeneity will be illustrated by comparing the intellectual positions of different actors. Second, the contrast between a modern logician or a set theorist (on the one hand) and an intuitionist or a constructivist (on the other hand) will become apparent due to my specific choice of historical actors: Cantor versus Poincaré (although the latter was more a semi-intuitionist than a Brouwerian intuitionist). Third, the doubts cast by each actor onto the writings of his contemporaries will allow students to see different developments of mathematical minds without me having to repeat the whole history of its ancestors. In this sense, then, my proposal is more practical and, at any rate, quite different both from Poincaré's position conveyed in (8) above and from Panagiotou's educational case study of logarithms .
[Last update: 15 September 2022]
- Henri Poincaré, "Mathematical Definitions and Education" in Science and Method, Thomas Nelson and Sons, 1913.
- Thomas Ryckman, Einstein, Routledge, 2017.
- Carlo Rovelli, Reality Is Not What It Seems, Penguin Books, 2016.
- Evangelos N. Panagiotou, Using History to Teach Mathematics: The Case of Logarithms, Science & Education (2011) 20:1-35.
- H. Freudenthal (1973). What groups mean in mathematics and what they should mean in mathematical education. In A. G. Howson (Ed.), Developments in mathematical education (pp. 101-114). Cambridge University Press.