These are the answers by philospher
Andrei Rodin 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
Except if definability is properly restricted.
I assume that such
restrictions can make perfect sense but I don't assume that they are always
Do you agree that the intermediate value theorem does not
hold the way that it is normally stated?
With similar reservations concerning assumed logical setting.
Do you agree that there are only three infinite
With a similar reservation.
A interpret the above questions as posed
about any possible setting, and I assume that there are different acceptable
Hence reservations (which don't mean uncertainty about given
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?
Again if the question concerns any possible setting.
In some settings
this might appear to be true.
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?
Not in any setting.
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?
But quite often so.
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?
And I also think that a "kind of reasoning" in a sense relevant to
mathematics could be itself a mathematical matter.
Which doesn't exclude the
possibility of universal (kind of) mathematical reasoning (to leave aside
universal reasoning in a wider sense).
Do you agree that all mathematical truths are true,
but that some mathematical truths are more true than
other mathematical truths?
But I think this wording could be given good sense, in particular when
one does mathematics in a topos. If one doesn't like it for some
philosophical reason, one could perhaps avoid such wordings even working in
a topos. I rather like. An example is found in the end of: Conceptual
Mathematics by Lawvere and Schanuel. In any event I don't believe that only
bi-valuated (or only bi-valuatable) mathematical statements are meaningful.