FOM: anti-foundationalism

[Stephen Ferguson]

On Mon, 29 Mar 1999, Stephen G Simpson wrote:

> You [Stewart Shapiro] say that `foundationalism
> ... has few proponents today, and for good reason' (page vi) and that
> mathematics is `a house built on sand' (page 26).  You briefly mention
> two foundationalist programs, logicism and Hilbert's program, only to
> remark that `both these programmes failed to achieve the
> foundationalist goal and, for various reasons, few people seriously
> hold out hope for repairs' (page 29).  

> Could you please explain why you yourself (not Quine or Weyl or other
> authorities) are so vehemently opposed to foundationalism?

In any discourse, foundationalism about that discourse is the thought that
study of the foundations of that discipline is motivated by philosophical
rather than subject specific concerns. For example, what motivated the
work carried out by Frege, Brouwer, Hilbert etc in the foundations of
maths? If they thought that studying the foundations of maths was a
philosophical endevour, then they were foundationalists. If they thought
they were doing math pure and simple, then they were not foundationalists.

The move in the phil of math taken by Putnam, maybe Wittgenstein, and
certainly the late Tymozco, was that studying the foundations could have
no *philosophical* privelidge over any other area of maths, in fact, study
of foundations in that light would be misleading.

Of course, this says *nothing* about the benefits of studing foundations
from a mathematical point of view, nor does it deny that there are
foundations. 

If I read him right, Stewart in his book 'F w/o Foundationalism', argues
that the main motivations for taking logic to be FOL, are philosophical
reasons rather than mathematical reasons, and so only someone committed to
foundationalism would be convinced of those reasons for prefering FOL
over SOL.
The softening of his line since then has been the result of various
genuine/mathematical reasons for opting for FOL rather than SOL.

The philosophical insight supposedly gained by studying foundations is
epistemological -- there are non-foundationalists who think that there are
important lessons to be learned for philosophy of maths, from studying
foundations, but that they do not think these lessons relate to
epistemology; rather they think that they uncover something about ontology
or the genuine logical structure of mathematical statements. 

Philosophers, on the whole, gave up foundationlaism about any discourse,
after it became apparant that to judge the epistemological status of the
foundations of any discourse, involved getting so close to the standards
with which we evaluate evidence, that there is no difference between the
tool used to evaluate and the items to be evaluated. Maybe that looks
obvious after a while when considering sense-perception, where the
philosophical debate was certaily in crisis during the last few days when
the foundationalistparadigm reigned: it is less obvious in the case of
maths, where there are things that have a foundational role and studying
them does seem to lead to genuine insight: it is very easy to confuse that
mathematical insight gained with the philosophical insight that teh
founding fathers were seeking. Hence foundationalism in phil of math
remained long after the rest of the phil community had given up
foundationalism.

Hope this maybe clarifies some of the points that have been exchanged
between Shapiro and Simpson,

All the best

Stephen


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Stephen Ferguson MA, MSc, PhD	
The Philosophical Quarterly		Department of Philosophy, 
The University, 			University of Dundee, 
St Andrews, KY16 9AL			Dundee, DD1 4HN
tel:(01334) 462484			tel: (01382) 344517
fax:(01334) 462485

home:(01334) 476784
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