Recently I finished How to Bake π by Eugenia Cheng. It is nice to pick up a bit of mathematics popularization every now and again and look for new ways of explaining the essential nature of our subject with the uninitiated. Since Cheng is a category theorist, and I don’t know any category theory, this gave me some hope for a new perspective or two. These hopes were reinforced by the fact that she has chosen to use making dessert as the analogy to describe ideas!
The book is enjoyable, and Cheng does an admirable job of trying to explain what mathematics is and how one does it. Of course, she then moves on to focus on what the essential nature of category theory is (her answer: “the mathematics of mathematics”). I found a few nuggets of new analogies and viewpoints to share with my students who are struggling to write their first proofs and find their first conjectures.
The part I most enjoyed was in the last few pages. Cheng describes three different viewpoints on truth:
I really like that these are clearly differentiated. Most importantly, she describes the process of mathematical communication really well, and with a useful diagram. (I suppose that a useful diagram is to be expected from a category theorist.) Here is the diagram.
I think what is truly great here is that the picture is a narrow ravine (she points out that it is difficult and dangerous to jump across). And her accompanying description of mathematical communication is as a process by which you (1) somehow pack your understanding into a proof, and then (2) share that rigorous proof. This plays up the role of a axiomatic proof as the kind of thing which can be done without (or maybe just with a lot less) ambiguity, while recognizing that it might obscure the understanding a little bit. But it also highlights that there is a job for the reader, which is really an inverse to the job of the writer. The reader must (2′) confirm receipt of and agreement with the logical argument, but then (1′) unpack it to construct their own understanding of the ideas.
This is a model I can share with my new proof-writers to help them do their jobs better.