FOM: human well-being; constructivism; anti-foundation
richman at fau.edu
Thu Jun 1 09:26:43 EDT 2000
On Wed 31 May 2000 Stephen G. Simpson wrote:
> Against my view, several people noted that one can "be interested
> in" or "work on" intuitionistic systems, without actually
> "believing in" the underlying philosophical ideas.
> I concede this point, but I say that it has nothing to do with
> the philosophical/foundational issue. Intuitionistic systems of
> mathematics were originally introduced in service of a Kantian or
> subjectivist philosophy. If these formal systems take on a life of
> their own, that does not erase the philosophical issues that gave
> rise to them. In particular, if intuitionistic logic and type
> theory are convenient for computer-aided algebra or computer-aided
> proof systems such as Nuprl, that has no necessary connection to
> the philosophical issue, which remains vital for f.o.m.
My impulse is to disagree with the last paragraph. It seems to me that
the philosophical discussion of intuitionistic systems should focus on
how they are currently viewed and used, not on some "original intent".
If intuitionistic logic is undergoing a renaissance because of
computer algebra, then that is what needs to be clarified, not
Brouwer's thoughts on what mathematics is.
I'm not quite sure what philosophical issues are. Many years ago, when
I worked one summer in a lab, I was told that there were two
philosophies about the machine shop: you could give the work to the
people there to do, or you could do it there yourself.
Is the dichotomy between hard and soft analysis a philosophical issue?
This can be viewed as a controversy over the proper way to do
analysis. There is always the cop out that we can and should do both,
course, is true, but doesn't address the fact that these are, in some
sense, competing world views.
The underlying metaphysics (?) of category theory might be summarized
by the statement that whenever you study mathematical objects, it is
essential that you also study the maps between them. If that isn't a
philosophical issue, what is it?
The question as to whether classical logic or intuitionistic logic is
more appropriate for doing mathematics must be a philosophical issue,
but I doubt if an analysis of subjectivity and objectivity is going to
be relevant to its resolution. More likely we would need to develop
criteria for judging mathematical works both wholesale and retail.
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