For my birthday my love gave me John Stillwell’s Roads to Infinity: The Mathematics of Truth and Proof. It was a wonderful read. More, it’s the sort of read that gets me excited about a subject.
The subject in this case is mathematical logic, and specifically the sections of it which describe infinitely large sets, and the provability of theorems. That these are entwined subjects may seem superficially odd. Stillwell explains well how the insights developed in talking about infinitely large sets develops the tools to study whether logical systems are complete and decidable.
At least it explains it well to me. I know I’m not a typical reader. I’m not certain if I would have understood the book as well as I did if I hadn’t had a senior-level course in mathematical logic. And that was a long time ago, but it was also the only mathematics course which described approaches to killing the Hydra. Stillwell’s book talks about it too and I admit I appreciate the refresher. (Yeah, this is not a literal magical all-but-immortal multi-headed beast mathematicians deal with. It’s also not the little sea creature. What mathematicians mean by a ‘hydra’ is a branching graph which looks kind of like a grape vine, and by ‘slaying’ it we mean removing branches according to particular rules that make it not obvious that we’ll ever get to finish.)
I appreciate also — maybe as much as I liked the logic — the historical context. The development of how mathematicians understand infinity and decidability is the sort of human tale that people don’t realize even exists. One of my favorite sections mentioned a sequence in which great minds, Gödel among them, took turns not understanding the reasoning behind some new important and now-generally-accepted breakthroughs.
So I’m left feeling I want to recommend the book, although I’m not sure who to. It’s obviously a book that scouts out mathematical logic in ways that make sense if you aren’t a logician. But it uses — as it must — the notation and conventions and common concepts of mathematical logic. My love, a philosopher by trade, would probably have no trouble understanding any particular argument, and would probably pick up symbols as they’re introduced. But there’d have to be a lot of double-checking notes about definitions. And the easy familiarity with non-commutative multiplication is a mathematics-major thing, and to a lesser extent a physics-major thing. Someone without that background would fairly worry something weird was going on other than the weirdness that was going on.
Anyway, the book spoke to a particular kind of mathematics I’d loved and never had the chance to do much with. If this is a field you feel some love for, and have some training in, then it may be right for you.