These are the answers by philosopher Mark van Atten to the ten questions about intuitionism:

Like others, I have noticed a certain degree of indeterminateness in some of the questions; I have chosen to think of this as intentional on Freek's part, and have taken the liberty to interpret the questions in what I think of as a Brouwerian way.

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

    Yes.

    Brouwer has shown this.

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

    Yes.

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

    Yes.

  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.

    I take it that you mean the continuum hypothesis in the classical sense. Applying Russell's analysis of propositions about non-existing entities, one arrives at the conclusion that the hypothesis is false.

  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?

    Yes.

    And it is false, see 4.

  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.

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

    No.

    I do not believe that Ex Falso holds generally.

  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.

    This depends on the domain: what kinds of reasoning are cogent depends on the types of the objects under discussion, and whether there are finitely or infinitely many of them.

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

    No.

    Less true is not true at all. Of course one will make certain idealizations, for example regarding what counts as constructible, but once these have been fixed, then all truths are equally true.