4 ms·

I think it's more arbitrary than that, quoting his comment: > Why do I believe these things? For no better reason than the crude generalized-intuition argument

by j2kun 12y ago

I think it's more arbitrary than that, quoting his comment: > Why do I believe these things? For no better reason than the crude generalized-intuition arguments like above. However, I also think its important to remember how baseless these beliefs are. Its like how I can root for a football team, but still remember that my allegiance is due primarily to proximity and nothing more meaningful. I think that on average people who deny the axiom of choice are the people who approach set theory and probability theory with intuition, and then rather than discard their intuition as being incorrect, they discard the axiom of choice as not lining up with their intuition (and then go on to do great math in other fields). But seeing as the author is an algebraic geometer, and published this post early in graduate school, I'm willing to bet he switched his ideology. I don't think AoC is something you can deny while actively working in his field, since you need to make tons of assumptions about algebraic closures to even get started.

wbl 12y ago
So if you are willing to go full Grothendieck from the get-go you can dodge the need to have algebraic closures. Even making an algebraic closure doesn't require the axiom of choice, as you can adjoin the roots of all polynomials with coefficients in your field without invoking it. The problem is that a lot of categorical constructions involve sets that get big. This isn't too much of a problem, but if you want to show that they are nonempty, choice comes in in a big way. Even worse, ZFC isn't enough: you need to play tricks to formalize category theory inside of set theory (or use proper classes instead).
judk 12y ago
If a mathematician would throw away his mathematical understandings in order to prove more results, I worry for his soul. A theorem in ZFC-but-not-ZF is much less interesting than a theorem in ZF, except as a topic in model theory.
- j2kun 12y ago
  Mathematical understanding quite often contradicts intuition. Part of becoming a mathematician is learning when to trust your intuition, and when you start your mathematical career your intuition is pretty unreliable. The thing about AoC and measure theory is that it's notorious for challenging your intuition after you think you've built up a lot of good intuition. Also, from my experience the vast majority of mathematicians don't care whether their theorems are ZF-compliant or need ZFC, unless it's marketed to logicians or specifically deals with AoC in a certain field.

by j2kun 12y ago

wbl 12y ago

So if you are willing to go full Grothendieck from the get-go you can dodge the need to have algebraic closures. Even making an algebraic closure doesn't require the axiom of choice, as you can adjoin the roots of all polynomials with coefficients in your field without invoking it. The problem is that a lot of categorical constructions involve sets that get big. This isn't too much of a problem, but if you want to show that they are nonempty, choice comes in in a big way. Even worse, ZFC isn't enough: you need to play tricks to formalize category theory inside of set theory (or use proper classes instead).

judk 12y ago

If a mathematician would throw away his mathematical understandings in order to prove more results, I worry for his soul. A theorem in ZFC-but-not-ZF is much less interesting than a theorem in ZF, except as a topic in model theory.

j2kun 12y ago
Mathematical understanding quite often contradicts intuition. Part of becoming a mathematician is learning when to trust your intuition, and when you start your mathematical career your intuition is pretty unreliable. The thing about AoC and measure theory is that it's notorious for challenging your intuition after you think you've built up a lot of good intuition. Also, from my experience the vast majority of mathematicians don't care whether their theorems are ZF-compliant or need ZFC, unless it's marketed to logicians or specifically deals with AoC in a certain field.