FOM: ReplyToThayer/Halpern

[Harvey Friedman]

Thayer writes:

>This is possible true, but there is also a long history of the notion that
>foundational/fundamental/radical study of any field is aimed at
>clarification, not reduction.  It may be that reduction is possible, but the
>hierarchy that Steve posits may look like some non-founded set.

It is a gross oversimplification of the traditional f.o.m. that it is
reductionist. There are reductionist developments and there are
clarification oriented developments. Even the development of axiomatic set
theory can be looked at as both. Everything gets reduced to sets, yet the
axioms clarify the concept of set.  The independence results are not
reductionist. They clarify and illustrate the limitations of mathematical
practice. Another example is the work on algorithms which clarifies the
informal concept of algorithm. The MRDP theorem is also not reductionist.

As far as the hierarchy of concepts is concerned: I think it impossible to
have a productive intellectual life at the highest level without resorting
to some at least implied hierarchy of concepts. It may change over time,
but it is something that needs to be constantly referred to in order to
formulate and evaluate research programs.

>Let me try to rephrase what I think Anand (and some others ) are complaining
>about.;
>As to (1):
>The definition is 'cultish' because it smacks of Objectivism, and
>Objectivism is thought by some to be a cult.

I can't speak for Anand, but I don't think he had Objectivism in mind.

>There seems to be some historical evidence
>that implicit definition is as useful as explicit definition.  The former is
>typical of the sturcturalist view; the latter of the reductionist view.

Traditional f.o.m. is largely based on implicit definition. One can view
the axioms of set theory as a kind of implicit definition - although I
don't quite know what you mean by implicit versus explicit definition.

>My personal offering would be that something is of foundaitonal interst when
>it shows that something which has always been thought of in one way, can
>also be concieved in a totally different way.  This seems to square with
>many of Anand's examples and may be what he is getting at.

You may have always viewed your wife as a mother, and when the kids are
grown, you may then conceive of her as a companion. Is this an example of
fooundations?

>Even is Steve is ultimately
>unable to show that his fundamental concepts are not replaceable by others
>which serve equally well, the attempt is highly useful as a reductio
>argument.

I don't think that nonreplaceability is an essential feature of Steve's
view in the following sense. If it is eventually seen that one has
overlooked equally fundamental concepts, then the discovery of such is
itself an important piece of foundations. So Steve cannot lose this game.

Halpern's posting suggests the following sharper formulation of what he is
concerned with:

How can we account for the remarkably accurate (though of course flawed)
intuition that we have about mathematical objects and conjectures when we
have not yet backed them up by proofs? This is an interesting, though
exceedingly difficult question. Another striking context in which this
rears its head is in chess. A grandmaster has a remarkable intuition about
the positions that defies current formalization. We know that the
grandmaster's are not just fooling each other because they are so
successful against brute force (represented by computers), especially at
very slow time controls such as correspondence chess.

>It's a way of suggesting a
>change of focus toward more consideration of how mathematics is practiced as
>opposed to the collection of theorems that result.

I still want to state emphatically that we have only scratched the surface
of understanding what can be proved or cannot be proved from what. Reverse
mathematics and also Concrete Independence Results are both in early stages
of development and show no signs of slowing down. In a way, of course
Reverse Mathematics does take into consideration how mathematics is
practiced in that it is concerned with the fundamental axioms that are
actually used in proofs of basic classical mathematics.