I don’t really intend to go on for too long about Everything & More, D.F.W.’s book about the history and philosophy behind infinity and maths. Suffice it to say that you’ll either like it or not depending primarily on two factors:
- Like: If you are a D.F.W. fan (which counts me in)
- Dislike: If you know a lot about the subject matter (i.e., you’re a set theorist–mathematician yourself, which counts me out)
This is assuming, also, that you’re the type of person who likes to read either unique works of literature and/or popular science–type things. Without a fairly good working knowledge of maths, be prepared to work hard to follow along with a good proportion of the book that’s dealing with technical content; it’s not impossible, however: the friend who lent the book to me managed to get through without much background knowledge.
Apparently there are a certain number of technical inaccuracies, or elements of the story that are simply wrong. I’m not one to judge these; (the only error I ever saw was a basic oversight-style typing mistake in a big list of examples of the application of differential equations (or something like that) in which was written F = m dx/dt) but it’s easy to find numerous critiques of the book in which the technical content is rather heavily denounced. However: It doesn’t matter. First of all, in some cases I’m sure D.F.W. was aware of some of the technical problems his popular descriptions required. He admits as much quite early on. But secondly, this is not a book to learn set theory in a mathematical sense. Just as you read Gödel, Escher, Bach to get a taste of some rather meaty mathematics in the context of a much broader philosophical discussion, Everything & More does a truly excellent job elucidating how all the things we learnt in school & university about abstract ideas such as irrational numbers and limits tending to infinity really were huge mathematical/philosophical problems back in the day and we should do better than take them for granted.
The best example of this is the way anyone’s who’s done a little university-level maths can reject Zeno’s paradox by their being taught about convergence of infinite series. How can you cross the road if you first have to reach half-way, and before you reach half-way you must reach the quarter-way mark, and before that get to one-eighths of the way, and before that one-sixteenths, … , ad infinitum? Anyone who’s studied the maths “intuitively” knows that this particular infinite sum equals one, q.e.d., but this result requires them to have already abstracted in their head the very idea of an infinite sum itself as something that is actually possible. It’s not exactly easy to explain how this works without using terms like “tends towards infinity” that a priori assume that infinity is an abstract concept that can be used to explain infinite sums. The formulation of a rigorous (explanation of a) proof is sort-of the main goal of the book (plus fleshing out a good amount of material about how this happened historically, and a number of consequences of what effects this had on the mathematics of the time leading into the current era).
This is a book about how some certain results were discovered, with a sufficient explanation of those results to expand your mind a little or a lot. And like all of D.F.W.’s writing, even just going along for the literary ride is well worth the effort.