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Conjecture and Proof

Miklós Laczkovich
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
[Reviewed by
Michael Berg
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I love this book! It's a very well designed, problems-driven course on the nuts-and-bolts of doing proofs, done in the splendid Hungarian tradition. Aimed at smart and enthusiastic novices, at beginners who aim to become true practitioners of the art, the material in this compact paperback is intrinsically interesting and challenging as well as elegant. Miklós Laczkovich's Conjecture and Proof (a phrase due to Erdös) is by far the best book I've seen for the purposes of teaching a gifted rookie the rudiments of theorem proving (and conjecturing, if only en passant) and exposing him or her to some very beautiful things at the same time, e.g. the irrationality of pi, the transcendence of e, the irrationality of the square root of 2 done five times, Liouville numbers, the Banach-Tarski paradox, the Peano curve: all marvellous stuff!

We generally have to take some care not to intimidate and alienate our students in "introduction to proofs" courses with approaches which are Darwinian if not draconian. So we generally settle for a somewhat prosaic approach, perhaps pitched rather lower than we might want — and then (quite naturally) find our better students properly bored by it all. The ever-growing isomorphism class of mainstream texts serves a clear but ultimately modest purpose: to impart to the average student a certain collection of tools to be used in later courses, e.g. mathematical induction, indirect proofs, etc., etc. — we all know the menu all too well. But, since they are generally heavily steeped in pedagogical "paint-by-numbers" ploys (pardon the pun), these texts don't serve well as a vehicle to convey the unique excitement of "doing" mathematics. Ipso facto we opt to train a large set of majors rather than educate the small subset of potential colleagues, so to speak. (Doubtless I am just voicing a frustration that I myself suffer.)

Laczkovich's book, which naturally fits in at the sophomore level, is markedly distinct from the aforementioned texts. Laczkovich tacitly assumes his audience to be already far better motivated toward having real adventures in mathematics than holds true for the average mathematics major. So Conjecture and Proof will resonate properly only with those who are already committed to the cause. But students who use this book — and who really must work every single problem to get the full effect and benefit — will be initiated into the Hungarian tradition which by now has attained legendary status. Indeed, the book "is an elaborate version of the lecture notes for a one-semester course of the Budapest Semesters in Mathematics... for American and Canadian students. [And the] program was designed and initiated by Paul Erdös, Lásló Lovász, Vera Sós, and Lásló Babai with the intention to offer undergraduate courses conveying the tradition of Hungarian mathematics." It doesn't get much better than that.

Michael Berg ( is professor of mathematics at Loyola Marymount University in Los Angeles, CA. His research interests are algebraic number theory and non-archimedian Fourier analysis.