These are the answers by philospher Andrei Rodin 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?


    Except if definability is properly restricted. I assume that such restrictions can make perfect sense but I don't assume that they are always necessary.

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


    With similar reservations concerning assumed logical setting.

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


    With a similar reservation. A interpret the above questions as posed about any possible setting, and I assume that there are different acceptable settings. Hence reservations (which don't mean uncertainty about given answers).

  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?


    Again if the question concerns any possible setting. In some settings this might appear to be true.

  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?


    Not in any setting.

  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?


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


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


    But quite often so.

  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?


    And I also think that a "kind of reasoning" in a sense relevant to mathematics could be itself a mathematical matter. Which doesn't exclude the possibility of universal (kind of) mathematical reasoning (to leave aside universal reasoning in a wider sense).

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


    But I think this wording could be given good sense, in particular when one does mathematics in a topos. If one doesn't like it for some philosophical reason, one could perhaps avoid such wordings even working in a topos. I rather like. An example is found in the end of: Conceptual Mathematics by Lawvere and Schanuel. In any event I don't believe that only bi-valuated (or only bi-valuatable) mathematical statements are meaningful.