Everything and more
"A gripping guide to the modern taming of the infinite."—The New York Times. With a new introduction by Neal Stephenson. Is infinity a valid mathematical property or a meaningless abstraction? David Foster Wallace brings his intellectual ambition and characteristic bravura style to the story of how mathematicians have struggled to understand the infinite, from the ancient Greeks to the nineteenth-century mathematical genius Georg Cantor's counterintuitive discovery that there was more than one kind of infinity. Smart, challenging, and thoroughly rewarding, Wallace's tour de force brings immediate and high-profile recognition to the bizarre and fascinating world of higher mathematics.
Stats for nerds
30recommend
45mentions
35people
2008–2026years
84% recommend13% neutral2% critical
Mentions per year
What Hacker News says
- I don’t know the answers to all of your questions - but I believe you’d benefit from some mathematical history books around the formalization of the real analysis; I’m not the best person to give you that history.
A couple comments, though - first, all mathematics is linguistics and arguably it is all sleight of hand - that said the word “gaps” that you’ve rightly pointed out is vague is a journalists word standing in for a variety of concepts at different times.
The existence of the irrationals themselves were a secret in ancient greece - and hence known for thousands of years, but the structure of the irrationals has not been well understood until quite recently.
To talk precisely about these gaps, if you’re not a mathematical historian, you have to borrow terminology from the tools that were used to describe and formalize the irrationals -> if former concepts about the lines sound hand-wavy to you, it is because they WERE handwavy. And this handwaviness is about infinity as well, the two are intimately connected. In modern terms, the measure of the rationals across any subset of the (real) number line is zero - that is the meaning of the “gaps”. There is, between any two rationals, a great unending sea where if you were to choose a point completely at random, the odds of that point being another rational is zero.
EDIT: for a light but engaging read about topics like this, David Foster Wallace’s Everything and More is excellent.
- Personally I don't agree with that review at all, and I loved Everything and More. I'm even somewhat of a "finitist" myself mathematically, but I think you can just view the whole hierarchy of infinities as a finite mode of description, and get utility from it without worrying about the "reality" of the whole thing. But that's a bit of an aside.
The review-writer clearly hates the book before he even gets to the mathematics. The whole book is striking a middle ground between expressing every step of mathematical rigor, and describing in a more literary way how to feel about these concepts. All of these "errors" are just the author passing over a point in less detail than you would want from a text on set theory. I thought it was great, and for anyone who enjoys both set theory and Infinite Jest, I would strongly recommend the book.
I understand that some people don't like it, but it should be viewed as "controversial" rather than "everyone agrees that it's riddled with errors".
- When I took real analysis, I didn't get an intuitive sense of limits and all that delta/epsilon stuff. It was too abstract. Strangely enough, when I read David Foster Wallace's "Everything and More" on the history of infinity it all made more sense, because he described all the paradoxes and dead ends that mathematicians had run into before Cauchy and others brought rigor to the problem. Wallace was a postmodern novelist. I don't know why he described infinity so clearly; I didn't enjoy his other work.
- Not a textbook, but Everything and More - A Compact History of Infinity by David Foster Wallace is a fun read about infinity. It’s technical but not quite as technical as a textbook that would be used in a college course. I have a PhD in math and I liked Everything and More a lot (but then again I also just love David Foster Wallace).
- Strangely enough, I took a whole degree in mathematics, and never felt comfortable with the epsilon-delta proofs for limits. It was only by reading Everything and More a Compact History of Infinity, by, of all people, David Foster Wallace that I understood the motivation behind it!
He explains how the ideas of infinity were developed, and which paradoxes and absurdities forced 19th century mathematicians like Cauchy to put the whole thing on a more rigorous footing.
- Yeah, this was a VERY controversial subject for a long time. Maybe even still today!
David Foster Wallace's Everything and More was an enjoyable history of the topic.
I think the rebuttal to the author's "write all the primes down, you can't" would be something like "tell me how many numbers there are between 0 and 1." I'd die before I'd finish writing out all the primes, but the author or that professor referenced in the article could just never even have a way to give me an answer to my request. (EDIT: If they say that's not "real-world" enough, how many possible pieces of wood of distinct length could they cut that would be between 0 and 1 meters long? Unless someone wanted to try to brute-force it based on like molecule size... maybe it's not truly infinite? But could you even measure that difference? Which brings you back to infinity from a precision aspect...)
