These are the answers by computer scientist Jan van Oort to the ten questions about intuitionism:
No.
Mu.
No.
There are as many infinite cardinalities as we can think up. When someone comes with, say, the third infinite cardinality, then I can admit the existence of a fourth infinite cardinality ℵ_{N} which is "greater" than the third. The fact that I am not immediately able to construct it, or to provide a proof for its existence, does not imply that ℵ_{N} does not exist; this I will only accept if and only if one shows me the proof of the non-existence of a proof for its existence, something no one showed me, yet.
Mu.
The continuum hypothesis has been proven to be independent of Zermelo-Fraenkel set theory. (This is the set theory I tend to use in my everyday professional life. ) Therefore we may either believe it to be true or false i.e. to have a definite truth value with this having an impact upon the validity of the set theory we work with. For sure, in both cases, we tacitly admit that we know what it is. If we do not know what it is, then the last statement of Wittgenstein's Tractatus logico-philosophicus applies: What we cannot speak about, we must pass over in silence. This silence is best rendered by MU.
Yes.
For the argumentation, see question #3.
No.
Some mathematical questions are undecidable. Said more simply: There are mathematical questions that are meaningless statements dressed up as questions, or that are based upon assumptions which are meaningless statements. Also, there are paradoxes, a very special class of meaningless statements. Paradoxes make it rather uneasy to build such a machine.
No.
Given any two statements p and q, it is possible to devise a logic in which neither ( p ⇒ q ) nor ( q ⇒ p ), and in which both ( p ⇒ q ) and ( q ⇒ p ) can be disproved.
Yes.
Only in the following sense, however: under condition that the classical proof already exists, the constructive proof may increase the amount or depth of insight we already had. If only a constructive proof is available or known to us, then the question #8 is most probably undecided.
No.
Even if we use different logics, e.g. multi-valued logics instead of the classical two-valued logic, reason remains the same: human reason is one. Without this (metamathematical) assumption, mathematics becomes impossible. BTW: as a Zen buddhist, I recognize the assumption of the unity of human reason to be part of the Zen experience of satori or awakening, one I only very partially had myself.
No.
This is the most "tricky" or trap-like of the 10 questions, as it is not strictly a mathematical question, and of all the 10 questions seems to exhibit most strongly a metamathematical character. My answer is still No: Not within the current state of mathematics, based upon and a result of 2500 years of two-valued logic. True is simply true, false is only false. There are meaningless statements, e.g. the continuum hypothesis as discussed in question #4; this more exactly means these statements are mathematically meaningless, and will never make it to mathematical truth, just as much they will never make it to mathematical falsity.