See page six of the paper for the authors dealing with this point. It’s certainly a potential explanation, but the map of obesity in the US does seem to suggest that being, say, at the mouth of the Mississippi basin is much worse than being on the west coast, despite them both being at sea level.
Bob Baker(Bob Baker)
Karma: 123
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.
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.
I think that personal choices about morality are unaffected by the fact that significantly different cultures exist. Perhaps they call for a soupçon more humility, but your moral intuitions remain axiomatic for you.
Rather I think the adjustment needed in some cases is a greater weight on the idea that your moral intuitions are significantly shaped by the culture that you found yourself in, and that the scope of possibilities is wide.
Perhaps this has little practical impact because, though your axioms might be more arbitrary than supposed, you have little choice but to use them. But there will exist people shaped by very different cultures, who formed different rules, and it’s not clear that there’s necessarily any ground for debate; the desire for universal morality might be hopeless.
(Or perhaps communications technology will cause Earth to tend towards a singular culture, giving grounds for a morality universal to all, at least until aliens or disconnected space colonies.)
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.