These are the answers by computer scientist Rob Arthan to the ten questions about intuitionism:
No.
No.
For both 1 and 2, I say no because I think something along classical lines gives the abstraction that best deserves the name "real numbers". I would give the same answer if you asked about the least upper bound property. Constructive and non-standard reals should always be qualified as such - they are interesting and useful, but not, in my view, the right abstraction of our geometrical intuitions.
Mu.
I don't really believe in the existence of any mathematical entities except as mental abstractions.
Mu.
I believe that all statements in the language of set theory are meaningful. However, I don't see any reason to believe that there is some single canonical model of that language. Consequently CH for me has just the same status as x^2 = 1 in group theory - true in some models not in others.
Mu.
My position on this is the same as my position on CH.
Mu.
There are perfectly good mathematical questions that depend on value judgments. E.g., it can be perfectly good mathematical discourse to discuss the elegance or conceptual utility of a theory. Question 8 below provides an excellent example.
No.
Mu.
My "Mu" would be "yes" if it said "information" rather than "insight". The extra information may or may not add insight. E.g., a constructive proof of the mutilated chessboard problem would be an algorithm that given a set of dominoes finds one or more of, (i), two dominos that overlap, (ii), a square on the mutilated board that is not tiled or, (iii), a square off the mutilated board that is tiled. That's an interesting thing to think about, but the proof based on counting gives a very nice insight.
Note also, that different proofs often yield different insights with no one proof most insightful in any sense.
Yes.
When you have complete information, then I suppose all reasoning is just classical logic. However, "doing mathematics" is not just writing down proofs. Mathematical investigation involves a vast diversity of reasoning methods.
No.
Truth is truth.