This book covers a range of science and mathematics topics by focusing on paradoxes — surprising results, or seeming contradictions. This is an effective way to quickly give a taste, and a few deeper results, across a range of areas. (Chapter list included below.)
Such a range of topics in a single book often means that some are fumbled, sometimes badly. Keeping each to a single chapter obviously mandates omissions and simplifications but, pleasingly, I didn't notice any screw-ups. (Although perhaps that's the bliss of ignorance since I'm not an expert in any of the subjects.)
(At the end one learns that the chapters on physics were written by other authors, who are experts in those areas. That's probably a wise choice and it explained some stylistic discontinuities.)
Personally I was familiar with probably half of the paradoxes covered, which did make some of the chapters a little skippable. But I found that the unfamiliar ones were useful enough to justify hunting down. The chapter that left the strongest impression on me was actually the first one, on infinities. I think I've actually come across all of the paradoxes in that chapter — Hilbert's hotel is pretty famous — but having them all laid out in series left me wondering whether infinity wasn't a mistake:
Mathematics could have left the question of "how many natural numbers are there?" as undefined, like dividing by zero. There are as many as you want, but it's not an expressible quantity. Instead we gave it a name and explored what followed from it. That's not a bad idea in general: giving sqrt(-1) a name and seeing where that led turned out really well! But, as far as I know, the structure behind infinity is bereft, useless, and full of nonsense. Having the paradoxes written out, one after another, drove that home. Although perhaps I'm wrong, perhaps there's a rich and useful study of infinities that I'm not aware of, but that chapter left me wondering that maybe we shouldn't have bothered.
The chapter on voting (“Social Choice”) was the topic that I was least-well versed in, and was the one chapter that I went back and read again once finished. But voting systems are within the bailiwick of LessWrong so I'm not suggesting that most readers here would find it valuable. But, perhaps, it's worth reading the chapters that one is very familiar with as an exploration of how paradoxes might work for explaining things to unfamiliar readers. It might be a useful explanatory tactic in one's own writing.
One hiccup in the book that did distract me:
There are obvious names that you expect to appear in a book like this: Einstein in the physics chapters, Gödel on formal systems, and so on. But the name that comes up more than any other is … Ayn Rand? And for no very obvious reason? I suspected that the author lost a drunken bet to mention Ayn Rand in every chapter or something, and I think a sharper editor would have cut it out. But maybe objectivism has profound and useful things to say in this space and I'm too biased by having slogged through both The Fountainhead and Atlas Shrugged many years ago to try and find what people saw in them to see it.
But I wouldn't want to end on that note. It's an impressive book that I enjoyed. The format makes random access effective and the main downside is that you might already know a lot of it.
Chapter list:
- Infinity
- Zeno's paradoxes of motion
- Supertasks (paradoxes resulting in physically impossible systems)
- Probability
- Social choice (voting)
- Game theory
- Self-reference
- Induction (more worthwhile than it sounds)
- Geometry
- Operations (basically “miscellaneous”)
- Classical physics
- Special relativity
- Quantum mechanics
- A poem by 3Blue1Brown.
I mean it in contrast to, for example, sqrt(-1). There was clearly a “hole” in polynomial equations: equations that couldn't be solved. Cardano decided to just define the thing that would fit in that hole and explored the structures that resulted. That turned out fantastically! The structure of complex numbers is incredibly rich and, with 20th century physics, turns out to be arguably more fundamental than the reals.
People tried to repeat that with higher-dimensional numbers. Quaternions have some uses but, as you go up the dimensions, you lose structure, and they become less and less useful.
Infinity strikes me as the same kind of trick as i: there was a hole (“how many natural numbers are there?”) and an object was defined by the shape of that hole. But the results seem to be more like the sedenions (16-dimensional numbers) than complex numbers, and not really worth the bother.
I agree that non-mathematicians can trip over infinities and believe they have found contradictions. This book is very clear that it is defining “paradox” as a surprising result, not a contradiction, and it gives a resolution for each paradox. But having all the results around infinity laid out one after lead me to wonder, what else did infinity give us? Maybe there is something useful that I don't know about! But I was left with the feeling that such a common concept was actually a dead end.