These are the answers by mathematician Toby Bartels 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.

    It's possible to define such a function by making appropriate assumptions, such as the assumption that equality in the real line is decidable. One problem in mathematics is that people don't realise very well that this really is a significant assumption, and that it's optional. Nevertheless, it is a possible and reasonable assumption to make.

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

    Yes.

    As normally stated, it doesn't include all of the assumptions. So this is the other side of my response to #1.

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

    No.

    Maybe I don't understand what Brouwer was getting at, but I don't see how anybody at all can maintain this! Even for Brouwer, since he would answer Yes to #1, surely the sets R(the real line), R- {0} and R- {0,1} are all different cardinalities (no bijections between them). So there are at least 3 different uncountable cardinalities to him; and I can't think of any other (actually used) form of mathematics that has 3 different cardinalities but not 4 different cardinalities. What am I missing?

  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.

    The last thing that constructivists should be doing is to cede the word "truth value" to the excluded middlers. In intuitionist logic, every meaningful statement has a truth value, not just those statements that are true or false. Since I accept this statement as meaningful, it has a truth value, and furthermore I know what it is: simply "the truth value of the continuum hypothesis".

  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.

    You can make a stronger case that this statement is meaningless. But I accept it as meaningful, so it too has a truth value.

  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.

    The machine with both lights on is an elegant response. But barring that, this is a futile exercise.

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

    No.

    I know what excluded middlers mean when they prove this. But like the Intermediate Value Theorem as normally stated, it's phrased too absolutely to be strictly correct.

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

    Yes.

    Of course, sometimes the constructive proof gives no more insight (such as when the obvious classical proof is already constructive!). But sometimes is does give more insight, and it never gives less (assuming both proofs do exist).

  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 is really the most important point of all! Classical reasoning has its place, despite Brouwer's polemics. And constructive reasoning really is useful, despite Hilbert's. Can't we all just get along?

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

    Mu.

    Is this supposed to make the same basic point as #9? It doesn't seem to me to be the right way to put it. The formulation of #9 is much better.

  11. Once more, for emphasis:

    Mathematics can be done using different kinds of reasoning, and depending on the situation different kinds of reasoning are appropriate.

    Amen!