These are the answers by logician
Leon Horsten 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
It has been done.
Do you agree that the intermediate value theorem does not
hold the way that it is normally stated?
It has been proved.
Do you agree that there are only three infinite
There are infinitely many.
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?
But it may be very very very hard to know whether
your "machine" does the job. I think this is not fair at all. Many
readers are going to read this question in a way that is different
from what it literally says, and it is intended by the questioner
that readers misread this question: it is formulated in such a way
that misinterpretation is encouraged (by the word "machine", of course).
Do you agree that for any two statements the first
implies the second or the second implies the first?
(And classical mathematics is by no means committed to it.)
Do you agree that a constructive proof of a theorem
gives more insight than a classical proof?
It gives a different kind of insight.
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?
Classical logic is applicable to all mathematical problems.
Do you agree that all mathematical truths are true,
but that some mathematical truths are more true than
other mathematical truths?