FOM: quasi-empiricism and anti-foundationalism

[Stephen Ferguson]

On Tue, 15 Sep 1998, Stephen G Simpson wrote: (in reply to Hersh)

> You seem to be missing my point.
> 
> Your original point in 11 Sep 1998 13:22:13 was that, according to
> you, there is a historical consensus against indubitability.  I
> replied in 12 Sep 1998 10:57:25 that, on the contrary, there is
> actually a massive consensus in favor of indubitability,
> i.e. mathematical rigor.  I elaborated on that reply.  You failed to
> acknowledge or answer that reply.
> 
> Let me ask you again: Do you deny the obvious fact that a large
> majority of mathematicians accept and take pride in the current high
> standards of rigor (definition-theorem-proof)?  And that this
> consensus has been in place for most of the 20th century?
> 
> If you accept these obvious facts, then how do you square them with
> your claim of a consensus against indubitability?

What is it for something to be "indubitable" -- to be beyond doubt. If you
look back in history, at the times when mathematics was taken to be
indubitable, standards of rigour were poorer than today. Why do we have
such high standards today -- precisely because we acknowledge that the
work that we do *CAN* be wrong, and that the standards of the profession
are in place in order to make sure (at least, as sure as we can be) that
results are correct.

But, if what Lakatos calls the "Euclidean" view is taken of mathematics,
there would be no point in all this rigour, because we would be certain,
right from the start that we were correct. On the Euclidean view,
mathematical knowledge is infallible knowledge. Today, *because* we
recognise that our work in mathematics is fallible, we have such high
standards of rigour.

Now, with that in place, can we finally on this list begin to see what was
so odious about ninteenth century foundationalism? If so, then can I draw
some final points.

The 'mathematical' discipline of F.O.M. is in one sense the heir of
ninteenth century foundationalism: it grows out of the work of
mathematicians come philosophers such as Frege, Hilbert, Brouwer, Cantor,
etc. but carries on the formal side of their work, without regard (or
without primary/ substantial regard) for the philosophical motivations
behind that earlier work. If you understand that, then you can see why
Hersh did not include Friedman et al in his sweeping condemnation of
foundationalists.

Ninteenth century foundationlaism is an epistemological movement -- it
seeks to find out what it is that grounds mathematical knowledge. The
Euclidean answer is that it is all based on clear and self evident truths,
with rules for generating new truths from the old ones. Logicism,
formalism and intuitionism utilise just this kind of Euclidean
epistemological strategy. John Mayberry wrote that there was one sense in
which there is a clear sense of foundations, which were one to deny that,
would be to deny mathematics. Sure -- but it is not a Euclidean
conception of foundations that he is talking about.

About the time of Euclid, there was another Greek, Strato, called "the
Physicist". Call a non-Euclidean approach to mathematical knowledge
Stratoan if it is based on the thought that we have mathematical
knowledge, and that the important epistemological question to answer in
philosophy of mathematics is: how does that mathematical knowledge grow?
Or -- What accounts of mathematical progress? Part of the investigation
of a Stratoan strategy *MUST* include an investigation into what are the
concepts upon which all mathematicians accept, and which they rely without
further proof/detail/ etc.: but this is precisely what Mayberry is calling
for.

Finally, while there must be room in a Stratoran account for a detailed
investigation of foundational concepts, such an investigation is not the
main thrust of the enquiry, neither will it form the core of suchan
enquiry: but it will be the acid test of a theory to pass as a credible
Stratoan account of mathematical knowledge.

So, what will a Stratoan account look like? Well, it should give an
account of the methods that mathematicians use to generalise old results,
to show that there are exceptions to well established results, and explain
how it is that by showing a result previously accepted as
correct mathematical result to be 'wrong', you invariably show that it is
a special case of a more general result: i.e. you show that it is correct
only for a range of fairly normal and easily delineable cases. Of course,
on those cases, the result still holds: so it is still correct. (And
anyone who has read _Proofs and Refutations_ will recognise that Lakatos
was giving a first stab at such an account)

Giving such a Stratoan account will require philosophers to learn more
case histories -- to delve into the history of mathematics and do some
serious trend spotting, and liaising with mathematicians as to how new
breakthroughs occur. But to do that, there will need to be a much greater
awareness to the fact that mathematicians and philosophers use different
vocabularies, which have some overlap, but also some 'false friends',
words that sound alike but have differnt connotations.

All the best,

Stephen

 --------------------------------------------------------------
Stephen Ferguson MA, MSc, PhD	Logic and Metaphysics, 
				The University, 
				St Andrews, KY16 9AL
				tel:(01334) 462484
				fax:(01334) 462485

http://www.st-and.ac.uk/~www_spa/STUDENTS/srf1
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