These are the answers by computer scientist Jakob Grue Simonsen to the ten questions about intuitionism:
Mu.
Mu.
Mu.
Mu.
Mu.
Mu.
Mu.
Mu.
Yes.
Mu.
The large amount of mus reflects my "it depends"-attitude to the questions. Indeed, the only positive answer I've given (question 9: ... mathematics can be done using different kinds of reasoning ...) reflects this.
Ultimately, answering a definite 'yes' or 'no' to some
of these questions begs the hard question: "What does it mean
that
In the practise of doing mathematics, this question usually remains unanswered. When defining a total function from the reals to the reals, there are different rules to the game one wishes to play: Can I describe a discontinuous function which makes sense to other people? Sure. Can I expect to do exact arithmetic with this function on a machine (with infinite memory)? Hell, no. Do I wish to prevent myself from defining such a function by working within a constructive framework like Intuitionism? Sure, sometimes.
It seems to me that a definite answer to any of the questions on the list (except question 9) is either (1) like wanting to play only one of games with wildly different rules that mathematics has to offer, or (2) like implicitly stating "the universe works thus!" by appealing to philosophy, not observables. Neither of those appeals to me, but I'm willing to concede that I've just constructed a false dilemma ;-)
Question 3 is interesting, btw, since certain rabid finitists will answer 'no', as will most people in classical mathematics, though their reasons will be highly different :-)