FOM: 'constructivism' as 'minimalistic platonism'
richman at fau.edu
Tue Jun 6 09:51:34 EDT 2000
Jeffrey Ketland wrote in a reply to Peter Schuster:
> The working mathematician, like normal people, in using "A or ~A" actually
> means "either A is true or ~A is true, but I don't know which".
Lots of people on both sides of the aisle try to explain what working
mathematicians mean by "A or B". Constructivists like to say that they
mean "not( not A and not B ). From my experience, that seems
ridiculous. On the other hand, I can't believe that they mean "but I
don't know which". Such a thought would never occur to them.
How do mathematicians use such statements? At some point in a proof
they might have an integer n > 1. This will not in general be a
specific integer, like 88001, about which they might conceivably say,
"88001 is either prime or composite, but I don't know which". They
might say, "either n is a prime, or n = ab for integers a,b > 1", then
proceed to argue in each of those two cases. The idea that they do not
know which alternative holds would make no sense to them in this
situation. It's not like the case of 88001. Here they don't even know
what n is, so how would they know if it were prime or composite?
I have the feeling that we all mean the same thing by "A or B".
> The intuitionist rejects the introduction (during a proof) of "A or B"
> until there is already either a proof of A or a proof of B. But I do
> not see why we need to make this *extremely restrictive* assumption.
Only if you make this assumption do your theorems acquire
computational interpretations. The intuitionist views a proof as a
program. The statement "A or B" is typically used as a branch point in
that program. If the program can't figure out which branch to take,
it's not going to be able to proceed.
More information about the FOM