These are the answers by logician
Henk Barendregt 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
No, I do not necessarily agree. It depends on what you mean by define.
If you mean that no continuous function can be specified, then I disagree.
If you mean that no continuous computable total function on R can be
defined, then I agree.
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
This I do not know.
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?
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?
Similar provisa as in 1.
If I am allowed to build two machines
and I do not know which one it is,
then the answer is yes. Otherwise no.
Do you agree that for any two statements the first
implies the second or the second implies the first?
Similar proviso as in 1.
Do you agree that a constructive proof of a theorem
gives more insight than a classical proof?
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?
Namely the state of our knowledge.
Sometimes we do not have a constructive argument,
then classical proofs do give some information.
Do you agree that all mathematical truths are true,
but that some mathematical truths are more true than
other mathematical truths?
Oh yes: some existence statements are only clasically true,
while others are constructively true. But the difference is related to our