The only way to learn mathematics is to do mathematics. - Paul Halmos

Most books about uni-level mathematics follow a strict scheme of giving you the content and letting you practice with it with some problems at the end of each chapter. Often boring. In "The Lady, or The Tiger?", Smullyan blasts the reader with (at first) seemingly unrelated logic puzzles that lead the reader to understand a complex concept such as Gödel's theorems.

I love Smullyan's taste for teaching math and I'm an unconditional follower of the Socratic method (maieutics). So I was wondering: are math books that combine both to teach other stuff? I'm thinking of something like the following.

  • On the outside, it looks like a regular problem compendium. Inside, it's a lot more.
  • The book's entry level is some first-year math but can deepen a lot. The subject can be anything, from general topology to combinatorial geometry.
  • It essentially drives the reader's own discovery of the subject by asking the right questions in the right order. Never does it prove a statement nor a theorem; instead, it poses it as a problem to the reader. It's allowed to give definitions, but ideally only when the reader has a proper grasp of the concept and feels an uncomfortable urge to name the object they've had in mind for some time already. (Of course this can never be timed exactly, but the author should care to make this happen on average at the right time.)
  • If the author really wants to state some fact, they should turn it into an easy question.
  • Some parallel narrative is most welcome!

If this book requires one thing, is for the reader to know when they know (and when they don't!) since it gives zero proofs.

Does anything like this exist?

  • 4
    $\begingroup$ Halmos is correct. The same applies to music or any other discipline. $\endgroup$
    – John Douma
    Commented Jan 17 at 0:00
  • 1
    $\begingroup$ There is a highly recommended book like that, Proofs and Refutations by Imre Lakatos. $\endgroup$
    – Dan Asimov
    Commented Jan 17 at 1:05
  • 2
    $\begingroup$ @DanAsimov Proofs and Refutations. An amasing book, but I am afraid not exactly what OP is after. $\endgroup$
    – user58697
    Commented Jan 17 at 5:26
  • $\begingroup$ Arnold's Problems (V.I. Arnold, ed., Springer 2005). $\endgroup$
    – Kurt G.
    Commented Jan 17 at 9:38
  • 1
    $\begingroup$ This sounds like a great book! $\endgroup$
    – user1279061
    Commented Jan 17 at 9:49

1 Answer 1


A Problem Seminar by D.J Newman

Fits almost all of the requirements.

More generally, I would pickup anything from Series 714, especially the older titles.

Albeit not related but you might enjoy Problems In Applied Mathematics by Murray Klamkin.


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