These are the answers by logician Michael Beeson to the ten questions about intuitionism:
Mu.
Well, "it is impossible" means that we could derive a contradiction from the assumption of such a total function, so few will agree with that, but "we do not know how to (constructively) define a total..." is a more sensible question. And if you put the word "constructively" in there, again it becomes a matter of fact that people won't need to argue about.
Mu.
If you only know the values of the function to an arbitrary precision, you cannot calculate a zero to an arbitrary precision.Well, your comment states the relevant fact. Everyone can agree on that. Now, that same fact can be expressed this way: "The intermediate value theorem does not hold constructively the way it is usually stated".
Since that just rephrases the fact, nobody would disagree with that either. So why try to get people to argue when in reality there's nothing to argue about? Or if there is something to argue about, it is this: what kinds of theorems and proofs are worth expending time and energy on? Are constructive proofs more interesting than non-constructive ones? f so how much more interesting and are non-constructive proofs of no interest at all? The days are a century gone when people took an all-or-nothing attitude to these issues. Nowadays most mathematicians would prefer a proof that gives an algorithm, if they can find such a proof, but if not, they'll take the proof they can get.
Mu.
Here the real question is whether it's legitimate to treat power sets as objects in their own right. I think even the later Brouwer changed his mind about this, and produced a "theory of species", where species are sets of sets (of sets of sets, etc.)
Mu.
Indeed there's some disagreement about this, but not particularly along constructive lines, I don't think. Feferman, for instance, would not agree, but he doesn't consider himself a constructivist.
Mu.
The continuum hypothesis talks about subsets of the line, and Freiling has shown how to "disprove" it using plausible probabilistic reasoning. However, the big cardinal axioms talk about sets that are so big that they are completely irrelevant for all "normal" mathematics. Many people therefore consider the "big cardinal axioms" to be meaningless.They do? Then can you mention some of those "many people"? (I assume you are talking about mathematicians, not about a random sampling of people in the street.)
Mu.
See the next question.
Mu.
Questions 6 and 7 are again, unlike the set theory questions, just a matter of specifying the meaning of the words, where we do know two precise meanings of the words to choose from (unlike in set theory).
Mu.
Here you could get arguments about it depending on what proof you're talking about.
Yes.
No.
I don't like opinion polls in general and particular ones where the questions are little semantic traps, whose answers depend a lot on how the listener interprets certain words. It seems to try to create or at least emphasize disagreement, rather than formulate matters unambiguously so that people see what's really being stated.
Most mathematicians now have a glimmer of a concept that "constructive proof" means something, but their concept is just that "a constructive proof is one that provides a way to compute the things proved to exists". They still don't realize the connection between logical reasoning, e.g. the excluded third, and computational information in the proof. For that reason, they are unaware of the distinction between various versions of the same classical theorem that are constructively inequivalent, such as the many versions of the intermediate value theorem.