Ten Answers from Jakob Grue Simonsen

These are the answers by computer scientist Jakob Grue Simonsen to the ten questions about intuitionism:

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

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

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

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

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

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

Mu.

The large amount of mus reflects my "it depends"-attitude to the questions. Indeed, the only positive answer I've given (question 9: ... mathematics can be done using different kinds of reasoning ...) reflects this.

Ultimately, answering a definite 'yes' or 'no' to some of these questions begs the hard question: "What does it mean that exist?"

In the practise of doing mathematics, this question usually remains unanswered. When defining a total function from the reals to the reals, there are different rules to the game one wishes to play: Can I describe a discontinuous function which makes sense to other people? Sure. Can I expect to do exact arithmetic with this function on a machine (with infinite memory)? Hell, no. Do I wish to prevent myself from defining such a function by working within a constructive framework like Intuitionism? Sure, sometimes.

It seems to me that a definite answer to any of the questions on the list (except question 9) is either (1) like wanting to play only one of games with wildly different rules that mathematics has to offer, or (2) like implicitly stating "the universe works thus!" by appealing to philosophy, not observables. Neither of those appeals to me, but I'm willing to concede that I've just constructed a false dilemma ;-)

Question 3 is interesting, btw, since certain rabid finitists will answer 'no', as will most people in classical mathematics, though their reasons will be highly different :-)