# Byers’ How Mathematicians Think

W Byers How Mathematicians Think: Using ambiguity, contradiction, and paradox to create mathematics 2007

I thought that this might have been a book to help non-mathematicians work with mathematicians, but it seems to be aimed at educationalists. It starts by seeming to assume that its readers will think that mathematics is mostly mechanical and about algorithms, which my colleagues do not. But it does have some interesting insights, pointing out that mathematics has its ‘deep’ aspects, and that ambiguity is key as are complexity and the limits of logic as pointed out by Gödel.

Less helpfully, Byers emphasises the role of immediate (psychological) certainty in mathematics, and presents mathematics as if it were justified as being abstracted from reality. In contrast, I think creativity, insight, coping with ambiguity and the ‘aha’ of ‘immediate certainty’ are common to many activities. It seems to me a mistake for mathematicians to claim that their ‘immediate certainty’ is in any sense better than anyone else’s. Instead, it seems to me that what distinguishes mathematics (along with logic) is that its products involve formal constructs and proofs that are maximally open to criticism by anyone who has learnt the technical language. What distinguishes mathematics from other areas of logic is the nature of its language.

Byers seems to think that mathematicians only think in mathematical language, but this seems no more true than that poets only think in poetry: I prefer to think of the formal language of mathematics as the ‘target’ language, much as verse is, with mathematicians being unconstrained in how they think, as long as they can represent the results mathematically.

A frequent source of confusion throughout the book is that, for example, Geometry is talked about as if the mathematical points and lines were necessarily abstractions of physically real points and lines. But mathematically, Geometry stands or falls on its logical correctness: labels such as ‘points’ and ‘lines’ are aids to intuition, but the correctness of any correspondance to reality is the subject of Physics, not mathematics. Mathematics is not folk physics or folk psychology, even though mathematics is often valued for its supposed applicability.

Given the sub-title, I had hoped that Byers might have supplied a critique of probability theory, but he does not. Let me supply that omission: there are a range of formal theories of probability and decision-making that are perfectly valid as mathematics, but as mathematics they have nothing at all to say about real-world ambiguity, uncertainty or decisions, much less about ‘homo economicus’. Any beliefs about the application of mathematical theories are the province of some theory, such as physics, psychology or economics. If mathematicians claim Byers immediate certainty about the nature of probability or decision-making, their mathematical qualifications give no more grounds for believing them than do those of anyone else, or than in thinking that the axioms of Geometry are a true abstraction from the real world. Read Byers, but with caution.