4
$\begingroup$

Lakatos, in his Proofs and Refutations, rejects the Euclidean methodology and exposition of mathematics: where axioms and definitions precede the proofs. In other words, a Euclidean mathematician makes clear the axioms and definitions, then proves novel theorems in an altogether linear fashion. This rigid methodology is not present in mathematical practice, Lakatos contests. “This led to" his "dialectical methodology, a ‘heuristic style’ that reveals the struggle and adventure of mathematical creation" [Paolo Mancosu’s Preface in Proofs and Refutations].

Lakatos seeks to convince us of this through case studies -- the historical contexts behind uniform convergence, bounded variation, and the Carathéodory definition of a measurable set, for example.

Are the cases that Lakatos discusses special cases? Or does most of post-Euclidean mathematics progress in this heuristic, dialectical style? Examples of some recent discoveries and how they came about would be interesting.

$\endgroup$
  • 2
    $\begingroup$ I believe that Lakatos' general description of the scientific method and the way in which scientific knowledge progresses is the most up-to-date and accepted work in that field. I'm not aware of any significant developments since. $\endgroup$ – Steve Jan 3 '18 at 13:13
  • 2
    $\begingroup$ That mathematical practice does not follow the "methodology" of deriving theorems from "arbitrary" definitions and axioms (which is a caricature of even Hilbert, who explicitly rejected it) is not particularly controversial, this is not specific to Lakatos. And Euclid's order of exposition certainly does not follow the historical order either. For a recent analysis of Euclid's approach (as opposed to Hilbert's axiomatic method) and its relation to modern practice see Rodin's Doing and Showing. $\endgroup$ – Conifold Jan 3 '18 at 23:46
12
$\begingroup$

I do not agree with the assumptions made in the title and in main text of this question, namely that Lakatos rejects the Euclidean methodology and exposition of mathematics.

In the way I read it, Lakatos is putting his fingers on a rather different topic.

Many people, while describing mathematics as a science, tend to confuse the Euclidean exposition with what mathematics is. In passing let me say that I doubt any mathematician ever in the world thought that there was something like an Euclidean methodology: a way of doing mathematics in the linear and terse style of an axiomatic theory. So Lakatos is pointing us towards understanding that math in practice is very different from its final outcome and therefore, from a philosophical point of view we cannot describe it merely as an activity aiming at proving theorems, which requires that you have a theorem to start with and a path to prove it afterwards.

Mathematics progresses in a heuristic, dialectical style, yes. But this was not Lakatos' thought. Just read first hand reports on Bourbaki method of work (I guess the Bourbaki team can be rightfully considered as exponent of contemporary Euclidean exposition) and you will see that Cartan, Weil, Dieudonné, Grothendieck all knew very well that new math does not flow in the form of Definition-Theorem-Proof like in their books (and no surprise that Bourbaki's books have very interesting historical notes - something Lakatos would definitely have advocated). This was wide clear to many.

Lakatos put his finger on one problem. Describing math from a philosophical viewpoint in those years it was common to forget this fact and to identify math as it was written with math as it was done. And this is not the case.

$\endgroup$
  • 1
    $\begingroup$ Very good. Indeed, the "writing of mathematics" is not "mathematics itself", and also, slightly more subtly, is not the "doing of mathematics". $\endgroup$ – paul garrett Jan 19 '18 at 1:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.