FOM: MRDP theorem; Mazur; Quinean holism; what are basic concepts?

[Stephen G Simpson]

1. Number theory:

Lou van den Dries writes:
 > I fail to understand what has been suggested by Steve (I am not
 > sure about Harvey) that number theorists have not given enough
 > attention to MRDP and similar issues of undecidability or
 > independence.

I don't think I said that exactly.  What I said was that we might
reasonably expect number theory to develop in different, possibly more
interesting directions, if number theorists paid more attention to
foundational issues.  An example is Barry Mazur's paper in the JSL,
where he investigates number-theoretic problems suggested in part by
MRDP.  I don't think there are many number theorists who would be
capable of writing such an interesting paper.

This example came up in answer to questions of Neil Tennant.  Neil was
asking whether most pure mathematicians (a) are interested in
foundational issues, (b) would do things differently if they were.  My
answers to these questions were (a) no, (b) yes.  I used number theory
and MRDP as an example illustrating both points.

 > Much of their work is so far "below" where MRDP is likely to be
 > relevant that they work almost in a different mental universe.

Yes, I recognize this.  And I would go farther: Most number theorists
(and most pure mathematicians, for that matter) are in a mental
universe or orbit which is very far away from foundational issues.

2. Quinean holism:

I want to follow up on Neil's postings about Quinean holism (QH for
short).  Neil described QH as an epistemological theory which says
that no concepts are `privileged'.  By `priviliged' Neil appeared to
mean, more basic than other concepts.  In other words, according to
Neil, QH says that no concept is more basic than any other.  Obviously
I could never agree with this form of QH.  Indeed, it's hard to
imagine how any mathematician could agree with this form of QH.  For
example, most mathematicians would certainly regard the concept `real
number' as necessarily more basic than the concept `symplectic
manifold', since the latter is defined in terms of the former.

But perhaps I misunderstood Neil's description of QH.  Perhaps the
real QH is less radical than what I described above.  Perhaps the real
QH merely says that all concepts must be woven together in a
consistent, mutually reinforcing structure.  This sounds much more
acceptable to me.  Indeed, it sounds like the hierarchy of concepts,
which I definitely believe in (see
www.math.psu.edu/simpson/Hierarchy.html).

Perhaps Neil ought to clarify what he means by Quinean holism.

And perhaps I ought to clarify what I mean by insisting that concepts
are arranged in a hierarchy with some concepts being more basic than
others.  I don't mean that there is one such hierarchy valid for all
eternity.  What I do mean is that the hierarchical method of
organizing knowledge is essential to the human method of cognition.
Thus, at any stage in the history of human knowledge, a conceptual
hierarchy exists, but over time parts of the hierarchy may be revised
and reorganized as new insights accumulate.  For instance, a long
historical development led to the current orthodox Bourbaki/ZFC
foundational scheme for mathematics, in which the concept `set' is
considered basic and all other mathematical concepts are defined in
terms of this one alone.  And it may take another long historical
development to overthrow the Bourbaki/ZFC scheme and replace it with
something better, more in tune with applications.

-- Steve