These are the answers by computer scientist
John Harrison 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?
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?
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?
Do you agree that all mathematical truths are true,
but that some mathematical truths are more true than
other mathematical truths?
The pattern behind the answers is that I feel quite sure that statements
like "for all N" or "there exists N" are definitely objectively true or
false, but I feel no such confidence in statements about higher
The real numbers are the difficult middle ground where I'm
not sure what I believe.
But (1) has to be false for my mental picture of
the reals to be at all satisfying.
And (7) is really just a question of
definition of "implication".
I'd rather qualify (8) by saying that a
constructive proof may give more insight, but "yes" is closer to my
feelings than "no".