These are the answers by computer scientist Rob Arthan 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.

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

    No.

    For both 1 and 2, I say no because I think something along classical lines gives the abstraction that best deserves the name "real numbers". I would give the same answer if you asked about the least upper bound property. Constructive and non-standard reals should always be qualified as such - they are interesting and useful, but not, in my view, the right abstraction of our geometrical intuitions.

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

    Mu.

    I don't really believe in the existence of any mathematical entities except as mental abstractions.

  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?

    Mu.

    I believe that all statements in the language of set theory are meaningful. However, I don't see any reason to believe that there is some single canonical model of that language. Consequently CH for me has just the same status as x^2 = 1 in group theory - true in some models not in others.

  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?

    Mu.

    My position on this is the same as my position on CH.

  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.

    There are perfectly good mathematical questions that depend on value judgments. E.g., it can be perfectly good mathematical discourse to discuss the elegance or conceptual utility of a theory. Question 8 below provides an excellent example.

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

    No.

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

    Mu.

    My "Mu" would be "yes" if it said "information" rather than "insight". The extra information may or may not add insight. E.g., a constructive proof of the mutilated chessboard problem would be an algorithm that given a set of dominoes finds one or more of, (i), two dominos that overlap, (ii), a square on the mutilated board that is not tiled or, (iii), a square off the mutilated board that is tiled. That's an interesting thing to think about, but the proof based on counting gives a very nice insight.

    Note also, that different proofs often yield different insights with no one proof most insightful in any sense.

  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.

    When you have complete information, then I suppose all reasoning is just classical logic. However, "doing mathematics" is not just writing down proofs. Mathematical investigation involves a vast diversity of reasoning methods.

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

    No.

    Truth is truth.