These are the answers by philosopher
Mark van Atten to the
ten questions about intuitionism:
Like others, I have noticed a certain degree of indeterminateness in
some of the questions; I have chosen to think of this as intentional on
Freek's part, and have taken the liberty to interpret the questions
in what I think of as a Brouwerian way.
Do you agree that it is impossible to define a total
function from the reals to the reals which is not
Brouwer has shown this.
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?
I take it that you mean the continuum hypothesis in the classical
sense. Applying Russell's analysis of propositions about non-existing
entities, one arrives at the conclusion that the hypothesis is false.
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?
And it is false, see 4.
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?
I do not believe that Ex Falso holds generally.
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?
This depends on the domain: what kinds of reasoning are cogent
depends on the types of the objects under discussion, and whether there
finitely or infinitely many of them.
Do you agree that all mathematical truths are true,
but that some mathematical truths are more true than
other mathematical truths?
Less true is not true at all.
Of course one will make certain
idealizations, for example regarding what counts as constructible, but
once these have been fixed, then all truths are equally true.