I think your title “The Axiom of Choice is Not Controversial” is literally false, because “controversial” is a social property referring to how much disageement there is, and not whether something is true. Rejecting the AoC is a minorinty view but it is not that minority. It’s also a respected alternative view, and texts often go out of their way to mention when they are acceping the AoC. It may if anything be the most controversial thing in math.
I think among working mathematicians it is that much of a minority. E.g. it is much less controversial than something like many-worlds among physicists, or something like heritability of intelligence among geneticists. It is broadly incredibly accepted.
EDIT: But I do agree it’s a respected alternative view, and I do think it should stay that way because it is interesting to investigate. I just think people get the wrong idea about its degree of controversy among working mathematicians.
I tried to search for surveys of mathematicians on the axiom of choice, but couldn’t find any. I did find one survey of philosophers, but that’s a very different population, asked whether they believed AC/The Continuum Hypothesis has an answer rather than what the answer is: https://thephilosophyforum.com/discussion/13670/the-2020-philpapers-survey
My subjective impression is that my Mathematician friends would mostly say that asking whether AC is true or not is not really an interesting question, while asking what statements depend on it is.
I think your title “The Axiom of Choice is Not Controversial” is literally false, because “controversial” is a social property referring to how much disageement there is, and not whether something is true. Rejecting the AoC is a minorinty view but it is not that minority. It’s also a respected alternative view, and texts often go out of their way to mention when they are acceping the AoC. It may if anything be the most controversial thing in math.
I think among working mathematicians it is that much of a minority. E.g. it is much less controversial than something like many-worlds among physicists, or something like heritability of intelligence among geneticists. It is broadly incredibly accepted.
EDIT: But I do agree it’s a respected alternative view, and I do think it should stay that way because it is interesting to investigate. I just think people get the wrong idea about its degree of controversy among working mathematicians.
I tried to search for surveys of mathematicians on the axiom of choice, but couldn’t find any. I did find one survey of philosophers, but that’s a very different population, asked whether they believed AC/The Continuum Hypothesis has an answer rather than what the answer is: https://thephilosophyforum.com/discussion/13670/the-2020-philpapers-survey
My subjective impression is that my Mathematician friends would mostly say that asking whether AC is true or not is not really an interesting question, while asking what statements depend on it is.
Yes sorry to be clear I’m not talking about whether it is true, I am talking about whether they would use it or not in proving ‘standard’ results.