Everything and More: A Compact History of Infinity (Great Discoveries) (Hardcover)

Wallace starts his good-humored and sympathetic tone from the beginning: His "Small But Necessary Foreword" begins, "Unfortunately, this is a Foreword you have to read." There are plenty of footnotes, but half of them are marked "IYI": "If You're Interested," as are many of the paragraphs in the main text (along with "Semi-IYI"). In a history composed of increasing mathematical rigor, Wallace jokes and uses slang. Much of the history has to do with trying to solve the paradoxes of Zeno, like the one about how you can ever get to the other side of the street when you first have to go halfway, then half of the rest of the way, then half of that, and so on. The paradoxes were curious, but when supremely useful calculus came along, the infinitesimals used had never been rigorously defined. It was not until Georg Cantor showed how to deal with infinities as real mathematical entities that calculus had a mathematical foundation. He showed that infinities could be compared, and some infinities were larger than others. The proofs of these ideas (about one of which a mathematician said, "I see it, but I don't believe it!") gave calculus roots, but also gave rise to questions that eventually shook all of mathematics to its foundations.

Wallace doesn't get much into Gödel and his eventual Incompleteness Theorem, and it is just as well. There is enough excitement here, at least exciting for anyone who finds paradoxes a charming way to make the neurons spin. Admittedly, he has included equations and propositions full of Greek letters and advanced functions that only math buffs will absorb. He says of Weierstrass's demonstration of continuity, for instance: "There's a reason this all looks so hideously abstract: it _is_ hideously abstract." The substance is tough going, but there is enough style here, in jokes, curious illustrations, and piquant asides, to make this a fascinating show of intellectual prowess.

I was, by turns, frustrated with this lack of rigour, and appreciative of it. I can't put it better than Wallace does in a footnote on pp.220-221, "Rhetoricwise, let's concede one more time that if we were after technical rigor rather than general appreciation, all these sort of connections would be fully traced out/discussed, though of course then this whole booklet would be much longer and harder and the readerly-background-and-patience bar set a great deal higher. So, it's all a continuous series of tradeoffs." - Informed readers take note of his use of the term "continuous series" here!

Thus, Wallace does the best that I think any writer could in walking the tightrope between over-the-top technical mare's nests which only a few members of the faculty at Mathematics departments (and a few autodidacts) could grasp, and what he derides as the "Pop" accounts of such things as the development of Set Theory.-So, nobody, including Wallace, and myself, is going to be completely satisfied. While not a complete technical purist, I do wish he'd chosen to be more technical in some parts and less so in others. As a former student who has always wished his "formal" training in Mathematics went further that first year college Calculus (though I later worked my way through more advanced textbooks on my own), I was genuinely interested in the technical illuminations this book might provide. On the other hand, as an appreciator of fine writing, I know the two do not go hand in glove.

So, in the end, I should say that this book is as good a "tradeoff" as you're going to find. I was pleased to see that Wallace's wit and style haven't suffered from the subject matter. He rather resembles, in this respect, another writer who is more often quoted herein than any other for, as Wallace terms it alliteratively, his "pellucid prose": to wit, Bertrand Russell, a mathematician of first order, whose renegade life and pixie wit served him well throughout his (as Wallace puts it, wryly, in the penultimate footnote of the "booklet") long, distinguished life. Let's hope Wallace's life and output are equally as long and energetic.