These are the answers by logician Henrik Nordmark 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.

    The function that is 1 for x ≥ 0 and 0 for x < 0 is total and is not continuous.

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

    No.

    I believe the intermediate value theorem does hold in the usual way it is stated.

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

    No.

    I believe there are infinitely many infinite cardinalities.

  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 believe that CH is indeed a meaningful statement.

    However, I do not believe it has a definite truth value. The truth value depends on what axioms we choose to use.

  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 believe that large cardinal axioms are meaningful statements.

    However, I do not believe they have a definite truth value. Whether we choose to accept a large cardinal axiom as being true is a pragmatic choice. If it is interesting for set theorists to assume the existence of large cardinals because it is fruitful for research, then let them assume it to be true.

  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?

    Yes.

    Build two machines, one with the "yes" light on and one with the "no" light on. Then you can be certain that you have built at least one machine that answers the question.

    Remark:
    There are some mathematical questions that do not get determined by our choice of axioms. Thus, if you wanted to build one and only one machine, it would have no chance of determining the truth value of an arbitrary mathematical statement.

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

    Mu.

    This depends on the semantics of "implies". If we are thinking about implication in the sense of classical propositional logic, then yes. If we have a notion of implication closer to the natural language notion of implication, then no.

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

    Mu.

    Constructive proofs provide a lot of insight in terms of how to explicitly construct certain mathematical objects. However, this type of constructive insight comes at a price and it may sometimes be easier to follow and understand a classical non-constructive proof than a very detailed constructive proof of the same theorem.

  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.

    For example, classical mathematics and constructive mathematics are different ways of doing mathematics with somewhat different rules about inference. They each constitute a different language game to be played.

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

    No.

    There are mathematical truths that are more widely accepted than others. For example, "2 = 2" is more widely accepted than "there exists a large cardinal". However, this does not make the former more true than the latter.