The best way– really, the only way– to sum up David Foster Wallace’s Everything and More: A Brief History of ∞ is by quoting a bit from it. This comes from the middle part of the book, after a discussion of Fourier series, in one of the “If You’re Interested” digressions from the main discussion:
(IYI There was a similar problem involving Fourier Integrals about which all we have to know is that they’re special kinds of ‘closed-form’ solutions to partial differential equations which, again, Fourier claims work for any arbitrary functions and which do indeed seem to– work, that is– being especially good for physics problems. But neither Fourier nor anyone else in the early 1820s can prove that Fourier Integrals work for all f(x)‘s, in part because there’s still deep confusion in math about how to define the integral… but anyway, the reason we’re even mentioning the F. I. problem is that A.-L. Cauchy’s work on it leads him to most of the quote-unquote rigorizing of analysis he gets credit for, some of which rigor involves defining the integral as ‘the limit of a sum’ but most (= most of the rigor) concerns the convergence problems mentioned in (b) and its little Q.E.I. in the —Differential Equations part of E.G.II, specifically as those problems pertain to Fourier Series.)
There’s a little footnote just before the closing parenthesis, which reads:
There’s really nothing to be done about the preceding sentence except apologize.
That’s the book in a nutshell. It’s a breathless survey of several thousand years of mathematical history, replete with footnotes, asides, and quirky little abbreviations (“Q.E.I.” is a “Quick Embedded Interpolation,” and “E.G.II” is “Emergency Glossary II”). The quoted paragraph is admittedly an extreme example, but if that style makes you want to run screaming, don’t pick this book up.
On the other hand, if it makes you say, “Hmmmm…. That’s a unique approach to a math text…,” then get this and read it, because the whole thing is like that, only better.
The book (or “booklet,” as he refers to it throughout, which I suppose he’s entitled to do, as he’s best known as a writer of thousand-page novels) is a really interesting stylistic exercise. It’s a densely argued survey of mathematics, full of forward and backward references (“as we will see in §7” and “recall from §3(f),” respectively), but the entire thing is written in a headlong sort of rush to suggest that it’s being improvised in one lengthy typing session. There are even little asides containing phrases like “if we haven’t already mentioned it, this would be a good place to note that…” It’s a remarkable piece of work, and does a good job of conveying a sense of excitement regarding some pretty abstruse mathematical issues.
The other fascinating thing about it, for a popular science work, is just how much it focusses on the math. There’s a three-page (or so) biographical interlude about Georg Cantor, and there are a smattering of references to the more melodramatic aspect’s of Cantor’s career, but those remain firmly in the background. This is in stark contrast Richard Reeves’s book on Rutherford, part of the same Great Discoveries series of books, in which the scientific aspects are subordinate to the biography.
This is a very math-y book, and quite daunting in some places. If you can handle Wallace’s writing style, though (personally, I love it), the math shouldn’t be too much of a challenge. And the discussion of the math of the infinite is really outstanding.
This isn’t a book that will suit all tastes– far from it– but if you’ve read and liked other things by Wallace, it’s worth a read. You’ll never look at pure math the same way again.