These are the answers by logician
Jesper Carlström to the
ten questions about intuitionism:
I must object a little. You use the principle of
excluded middle too much in the questions. For instance: if intuitionists doubt
the principle "there are discontinuous functions", they need not agree that
"there are no discontinuous functions", because they don't hold that there
either are or not.
So it would have been more proper to formulate the questions in the positive
way. Instead of asking
Do you agree that it is impossible to define a total function from the reals to
the reals which is not continuous?
one should ask
Do you agree that it is possible to define a total function from the reals to
the reals which is not continuous?
For the next time, perhaps.

Do you agree that it is impossible to define a total
function from the reals to the reals which is not
continuous?
No.
(But I have seen no such definition and I don't expect one; nor do I expect a proof that it is impossible.)

Do you agree that the intermediate value theorem does not
hold the way that it is normally stated?
No.
Only that we have no proof of it, nor a disproof.

Do you agree that there are only three infinite
cardinalities?
No.

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.
This is two questions. CH is meaningful, but I don't know that it "has a
definite truth value, even if we do not know what it is".

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.
Same answer as for CH.

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.
(Only for mathematical questions whose answers are yes or no.)

Do you agree that for any two statements the first
implies the second or the second implies the first?
No.

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

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.
(Isn't this the opinion of every mathematician?)

Do you agree that all mathematical truths are true,
but that some mathematical truths are more true than
other mathematical truths?
No.