These are the answers by philosopher
Robert Black 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
Do you agree that the intermediate value theorem does not
hold the way that it is normally stated?
Do you agree that there are only three infinite
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?
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?
It's (hopefully) true in some models and false in others. It
might be false in all models, but nobody thinks this.
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?
Do you agree that for any two statements the first
implies the second or the second implies the first?
(True if "implies" means truth of the material conditional and
the statements have truth values.)
Do you agree that a constructive proof of a theorem
gives more insight than a classical proof?
(Obviously sometimes yes, but nonconstructive proofs can
generalize when constructive proofs don't.)
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?
(Let a hundred flowers bloom!)
Do you agree that all mathematical truths are true,
but that some mathematical truths are more true than
other mathematical truths?