Regarding usefulness of infinities in physics: the uncertainty principle requires literally infinite number of eigenstates for both position and momentum operators, and cannot be reproduced in the limit of a finite number of states taken to infinity.
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
Yes , it has something to say:
There are no contradictions in reality, but there can appear to be.
Cook keeps mentioning Rand because he is trying to confirm something she held to be true.
In more detail, Objectivists hold that
logic and maths are about empirical reality , not independent magisteria
Rand has certain apriori knowledge that there are no contradictions in reality
therefore, there should be no contradiction-implying paradoxes in logic or maths.
I don't understand what you mean by "the structure behind infinity". Do you mean things like the difference between countable infinity and uncountable infinity?
[W]e gave [infinity] a name and explored what followed from it.
The paradoxes with infinity tend to be created by non-mathematicians. When mathematicians use infinity, they must prove the concept of infinity makes contextual sense before they are allowed use it. Reckless philosophers start by assuming infinity makes contextual sense, and then trip over their own words when it doesn't.
This book seems to misrepresent how infinity is used in formal mathematics. In my experience, the most common use of infinity isn't as an object of study. "Infinity" is usually just an intuitive shorthand for particular limits and sets. It might be that the author doesn't have a rigorous mathematical background. That would explain why he had to outsource the physics sections. Checking his biography…he's an economist.
I don't understand what you mean by "the structure behind infinity"
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.
Goodstein's theorem is a theorem about finite numbers that can be proven using infinite ordinals, and can't be proven using just Peano Arithmetic. I take it that's the kind of thing you're asking about? Though I couldn't tell you how useful that in turn is.
I understand clearly what you mean now. From the group theory perspective (what you call "structure"), infinity is less interesting than . From the notational perspective, isn't really a number and is therefore not relevant to this discussion.
I think the most interesting thing to come out of the study of infinity is the difference between countable and uncountable infinities. A countable infinity is the order of a set that can be put into one-to-one correspondence with the natural numbers. The rational numbers are an example of a countable infinity. An uncountable infinity is the order of a set too large to be put into one-to-one correspondence with the natural numbers. The irrational numbers are an example of a countable infinity.
Infinity is one of those concepts which seems fundamental but isn't. Infinity isn't fundamental because infinity isn't really a single concept. It is an bundle of several related ideas. In mathematics, there are many ideas adjacent to infinity like "calculus" and "conditional convergence" which are important, but it is true they do not come from treating infinity as a number. Rather, they come from not treating infinity as a number.
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.
OK.
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:
The paradox in Hilbert's hotel is that infinite quantities dont work like finite quantities ... which is only surprising if you have the expectation that they should. So why is infinity itself the mistake ? Why not solve the paradox by dropping the expectation that infinitty works like finity? (And how does Cook solve the paradox?)
Why not solve the paradox by dropping the expectation that infinitty works like finity? (And how does Cook solve the paradox?)
The book "solves" the paradox by stating that, yes, you can add an infinite number of guests to Hilbert's hotel, even when it was full to begin with. Again, it's only stating surprising results and if Hilbert considered it sufficiently surprising to articulate then I'm not going to argue!
It's not that infinity doesn't work, it's that it struck me that it's barren of interesting structure. Yes, infinity + infinity is still infinity. And there's an unlimited number of infinities that are sufficiently ill-behaved that they don't even form a set. It seems like a concept that has very little to offer.
It's interesting that you choose dividing by zero as your comparison to infinity, because there are infinite possible solutions to x/0.
It seems to me that by introducing infinites and infinitesimals to mathematics, mathematicians did something similar to how algebra made addition and multiplication "live together" despite their incompatability. By giving definition to something that sometimes can and sometimes can't work with other parts of math, mathematicians brought the outside in, and fenced the universe.
I also find myself wondering if anyone thinks giving zero a name was a mistake. Zero is the reason there's an x/0 asymptote.
As someone who read the book, you can answer this question: how often was zero (or nothingness) included in the paradoxes in the book? Without having read it, I'm guessing all of them hinge on some weirdness of 1 (unity), zero (null) or infinity.
It's interesting that you choose dividing by zero as your comparison to infinity, because there are infinite possible solutions to x/0.
I think if you ask a mathematician what x/0 is, they'll say "undefined" or "that's not a valid question". But if you ask how many natural numbers there are they'll say "infinity" (or ℵ-zero). But we could have defined x/0 as "foo" to see what resulted, like sqrt(-1) is i. But I think not much results and so people don't bother, and maybe we shouldn't have bothered with infinity either.
(I don't think the same about infinitesimals though! Analysis is a valid field of study!)
how often was zero (or nothingness) included in the paradoxes in the book?
There's one of the silly 1==2 tricks where a divide-by-zero is obfuscated. There's a number that involve infinite series, or infinite processes. The chapters on formal systems, voting, physics, etc don't involve such things though, so I wouldn't say that they're all based on it.
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: