These are the answers by logician Herman Geuvers to the ten questions about intuitionism:
No.
A non-continuous total function from the reals to the reals may not be computable, but it is a well-defined function.
No.
Just like 1: the computational version of IVT does no hold, but IVT itself does.
No.
There are more cardinalities, where I take "cardinality" as a formal concept. I don't believe that they really "exist", but the statement doesn't speak about existence in the real world. These cardinalities do exist in set theory.
Yes.
The continuum hypothesis states that there is no cardinality between the cardinality of the naturals and the reals. Both these cardinalities have a "real" existence, so it's a meaningful statement that has a definite truth.
No.
Inaccessible cardinals are a product of set theory that don't exist in reality.
No.
"Building a machine" implies for me that you can point at it and tell that it's the one. Building two machines is a nice trick, but I don't buy it.
Yes.
If the statements have a meaning, they are either true or false.
Yes.
The statement is put in a very general way, so it is easy to reject it on that basis, but I won't take that easy way out. A constructive proof usually gives more information and insight because it constructs elements or decides between cases.
Yes.
Within a specific domain, a specific type of reasoning may apply.
No.
A statement is true or false. One proof may be more informative then another, though.
A general comment: some of these questions are hard to answer, so putting a Mu is tempting, but I tried hard not to. While answering these questions, I noticed that I am more of a Platonist then I thought, while I don't believe in set theory as the systems for describing the Platonist reality.