How is mathematics really done, and — once done — how should it be presented? Imre Lakatos had some very strong opinions about this. The current book, based on his Ph.D. work under George Polya, is a classic book on the subject. It is often characterized as a work in the philosophy of mathematics, and it is that — and more. The argument, presented in several forms, is that mathematical philosophy should address the way that mathematics is done, not just the way it is often packaged for delivery.

The longest and most polished part of the book comes from Chapter 1 of his thesis. This part alone was previously published in *The British Journal for the Philosophy of Science. *It is a dialogue between teacher and students about polyhedra and, in particular, the relationship connecting the number of vertices V, edges E, and faces F. The students suggest that V – E + F = 2, and that this holds for all polyhedra. That was Euler’s original conjecture. The conversation continues from there through a series of counterexamples, conjectures, proofs, concept adjustment, progress and retreat, more conjectures, more proofs, monster examples and hidden lemmas. In many respects the dialogue appears to parallel the historical path of Euler’s conjecture.

On my first reading several years ago, I took Lakatos’s discussions as a fairly straightforward description of how mathematics is done. First, there are the raw ideas; then clearer definitions, conjectures and failed proofs, backing up, limiting, re-stating, proving again, looking to generalize and so on. No one had to convince me that mathematicians do not proceed flawlessly from axioms and definitions to polished theorems. This previous view seems a bit naïve to me now, and I think that there are more subtle issues involved. There is a sizable contingent of mathematicians who clearly believe that the path to discovery need not (or should not) be reflected in the path to justification and exposition.

This disagreement, of course, has some consequences for the way mathematics is taught. In some sense it echoes the mathematical culture wars of the past several years. Lakatos takes up the teaching and presentation of mathematics in Appendix 2 of this book. He lands clearly on the side of heuristics in teaching and learning. He has very little patience for “deductivists” and what he calls their authoritarian choice to “hide the story”.

Some have argued the polyhedral example of the dialogue is not a fair example because it is “empirical” and counterexamples could be visualized more easily. It is curious to note in retrospect that Euler (who started all of this) thought of polyhedra as solid things he could cut, while Cauchy saw them as wire frames. Poincaré’s conception was more abstract and topological; his proof of the Euler formula (which appears in Chapter 2 of the book) embodies that abstraction. What can we learn from this except that our proofs proceed from our own conceptions of the objects we are studying?

To reinforce his argument with a less “empirical” example Lakatos describes the development of the notion of uniform convergence. He does this in more conventional exposition (i.e., no dialogue) in Appendix 1. Nonetheless, it has its own characters; here they are Cauchy, Fourier and Abel, with an important walk-on part for Seidel. At the time of Cauchy and Fourier it was commonly believed that any convergent series of continuous functions converges to a continuous function. Cauchy prepared a proof using his new, more rigorous definition of continuity. Abel suggested that some Fourier series gave rather blatant counterexamples. Eventually — years later — Seidel identified the hidden assumption that led to the error in Cauchy’s proof and introduced the notion of uniform convergence as what Lakatos calls a proof-driven concept. Lakatos argues that this rather messy development owed a good deal to Cauchy’s aspirations as a formalist and “deductivist”, and his devotion to the “Euclidean methodology”. That’s rather unfair, I think; it casts Cauchy as a villain of the piece and offers a rather biased interpretation of how the concept of continuity evolved. Lakatos has a surprising tendency to see issues in black and white, and he seems at times to miss the subtle interplay of heuristics, deduction, and rigor.

This is a charming, sometimes annoying, thoroughly thought-provoking book. Its highlight is clearly the strikingly original dialogues of the first part. Lakatos died before the book was prepared for publication, and he never had an opportunity to polish, edit and revise. The editors chose to do little with the text except to add some introductory material, as well as comments or corrections in footnotes. The latter parts of the book are sometimes rough, but the whole thing is a delight to read.

See also our review of a previous edition.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.