These are the answers by logician Henk Barendregt 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?

    Mu.

    No, I do not necessarily agree. It depends on what you mean by define. If you mean that no continuous function can be specified, then I disagree. If you mean that no continuous computable total function on R can be defined, then I agree.

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

    Mu.

    Same proviso.

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

    Mu.

    This I do not know.

  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?

    No.

    I disagree.

  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.

    I disagree.

  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?

    Mu.

    Similar provisa as in 1. If I am allowed to build two machines and I do not know which one it is, then the answer is yes. Otherwise no.

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

    Mu.

    Similar proviso as in 1.

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

    Yes.

  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.

    Namely the state of our knowledge. Sometimes we do not have a constructive argument, then classical proofs do give some information.

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

    Yes.

    Oh yes: some existence statements are only clasically true, while others are constructively true. But the difference is related to our knowledge.