These are the answers by logician Herman Geuvers to the ten questions about intuitionism:

  1. Do you agree that it is impossible to define a total function from the reals to the reals which is not continuous?

    No.

    A non-continuous total function from the reals to the reals may not be computable, but it is a well-defined function.

  2. Do you agree that the intermediate value theorem does not hold the way that it is normally stated?

    No.

    Just like 1: the computational version of IVT does no hold, but IVT itself does.

  3. Do you agree that there are only three infinite cardinalities?

    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.

  4. Do you agree that the continuum hypothesis is a meaningful statement that has a definite truth value, even if we do not know what it is?

    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.

  5. Do you agree that the axiom which states the existence of an inaccessible cardinal is a meaningful statement that has a definite truth value, even if we do not know what it is?

    No.

    Inaccessible cardinals are a product of set theory that don't exist in reality.

  6. Do you agree that for any mathematical question it is easy to build a machine with two lights, yes and no, where the light marked yes will be on if it is true and the light marked no will be on if it is false?

    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.

  7. Do you agree that for any two statements the first implies the second or the second implies the first?

    Yes.

    If the statements have a meaning, they are either true or false.

  8. Do you agree that a constructive proof of a theorem gives more insight than a classical proof?

    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.

  9. Do you agree that mathematics can be done using different kinds of reasoning, and that depending on the situation different kinds of reasoning are appropriate?

    Yes.

    Within a specific domain, a specific type of reasoning may apply.

  10. Do you agree that all mathematical truths are true, but that some mathematical truths are more true than other mathematical truths?

    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.