Stephen G Simpson
simpson at math.psu.edu
Thu Apr 15 21:54:39 EDT 1999
There is a striking contrast between (1) the stridency of Stewart
Shapiro's anti-foundationalism in his book ``Foundations Without
Foundationalism'', (2) the timidity of Shapiro's anti-foundationalism
in his PM article and here on the FOM list. I have been trying to
draw Shapiro out, but apparently Shapiro is unwilling or unable to
discuss his views in a public forum.
I am a passionate pro-foundationalist. The reason I am so interested
in Shapiro's anti-foundationalism is that I detect a strong current of
anti-foundationalism among academic philosophers generally, and I want
to get to the bottom of why that is.
Acting as a Shapiro surrogate, Stephen Ferguson 6 Apr 1999 12:27:55
> 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.
This is interesting. Some people might say it is weird. Here you
are, a philosophy student, saying that philosophically motivated
investigations in any specific subject are ``foundationalist'' (a
pejorative in your lingo) and therefore automatically invalid.
Why? Why do you think philosophically motivated concerns are less
valid than other kinds of concerns? Are academic philosophers
afflicted with some sort of inferiority complex?
> 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.
Obviously Frege, Russell, Hilbert, Brouwer, Weyl, G"odel, et al *were*
foundationalists in this sense. They certainly *did not* think of
themselves as doing math pure and simple. They clearly saw themselves
as engaged in far-reaching philosophical/mathematical investigations
which would not only (a) shed light on any number of philosophically
motivated questions about mathematics, but also (b) change forever the
way mathematics would be viewed within the broad structure of human
knowledge as a whole.
What I want to know is: If great thinkers like Frege et al considered
this a worthy endeavor, why don't academic philosophers like you and
Shapiro consider it a worthy endeavor? Why do you dismiss it out of
> The move in the phil of math taken by Putnam, maybe Wittgenstein,
> and certainly the late Tymozco,
Which phase of Putnam was that?
> was that studying the foundations could have could have no
> *philosophical* privelidge over any other area of maths
First of all, this statement implicitly assumes that f.o.m. is just
another area of mathematics, like algebraic topology. This is a
mistake. F.o.m. isn't mathematics. It has a very different
And anyway, even if you want to insist that f.o.m. is just another
branch of mathematics, why would you make the further assumption that
no branch of mathematics is more philosophically interesting than any
other? Instead of invoking Putnam, could you please summarize the
reasons for such a stand?
> 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.
Yes, I think that's what Shapiro is saying. But why did he dismiss
these alleged philosophical motivations and reasons out of hand,
without a hearing? Why should a philosopher go around saying that
philosophical concerns are automatically worthless?
It seems there is an easy way to turn Shapiro against anything.
Namely, in order to convince Shapiro that X is worthless, all you have
to do is convince Shapiro that X is connected to philosophy.
Yet Shapiro is himself an academic philosopher. So this attitude
seems very strange. On the other hand, Shapiro isn't unique in this
respect. A lot of academic philosophers seem to have this same
attitude. My question is: Why? I want to understand this phenomenon.
Can you help me? Is there a paper of Putnam that explains it?
> The softening of his line since then has been the result of various
> genuine/mathematical reasons for opting for FOL rather than SOL.
Your phrase ``genuine/mathematical'' is interesting and perhaps
revealing. You seem to be assuming that the only *genuine* reasons
for doing anything are subject-specific (in this case mathematical).
In other words, according to you, broad philosophical concerns are to
be automatically dismissed as phony. Again I ask, why?
> 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.
Huh? This passage has so many qualifications and provisos that I'm
not sure I have untangled it correctly.
I *think* you are trying to change your earlier definition of
foundationalism. You now seem to be saying that foundationalism is
not concerned with *all* philosophical issues but only with
Is that correct? If so, why? Why is it that epistemological concerns
are foundationalist but ontological concerns are not?
However, this seems to be a side issue. The main issue is, what do
you guys have against foundationalism?
> 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.
OK, now we seem to be getting somewhere. But I still don't
understand. Could you please explain this more clearly?
Let me tell you where I am coming from. In my understanding,
epistemology is the branch of philosophy that deals with the nature of
knowledge and how we humans acquire it. Part of this is standards of
evidence. So, standards of evidence are of epistemological interest.
Furthermore, within any particular discipline (e.g. mathematics),
questions of evidence (i.e. mathematical evidence, in this case) are
of epistemological interest with respect to that discipline. Right?
OK, so why did philosophers give up on this? You are simply saying
that they gave up, but you haven't told me *why* they gave up. Please
keep talking, and explain this to me.
What is your point about ``tools used to evaluate'' versus ``items to
be evaluated''? Maybe you are trying to say that philosophy is only
concerned with standards of evidence in general, not standards of
evidence that are specific to any particular scientific discipline.
Is that your point? It might be an interesting point to discuss. I
might even agree with some of it. But first let's establish that this
is indeed what you are trying to say.
> 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.
Again, I don't understand, but please keep talking.
By ``the founding fathers'' I assume you mean Frege et al, i.e. the
founding fathers of modern f.o.m. research. You seem to be saying
that modern f.o.m. research has yielded some mathematical insight but
no philosophical insight, and the poor stupid foundationalists are too
benighted to know the difference.
Could you please elaborate? Don't you think that modern
f.o.m. research is of philosophical value? I certainly do ....
For example: (1) The development of the predicate calculus shows that
there is a simple system of inference rules which are defined in a
purely formal syntactic way and which adequately explicate a huge
variety of valid inferences, certainly sufficient for mathematics and
probably sufficient for other sciences. (2) The G"odel incompleteness
theorem shows that arithmetic will never be a ``closed'' system, in a
certain precise and relevant sense. (3) The set-theoretic
independence phenomenon brings ZFC-style set-theoretic foundations
into question. Etc etc. Just to name a few.
Are you saying that these insights of modern f.o.m. research are of no
> Hence foundationalism in phil of math remained long after the rest
> of the phil community had given up foundationalism.
Well, I guess you academic philosophers will just have to re-educate
us poor f.o.m. researchers and bring us up to speed. You can help us
to put aside our benighted foundationalism and trade it in for --
what? Sociology of science? Historicism? Literary criticism?
Sorry, I was only joking. But seriously, you want us to give up
foundationalism, but you are not saying how. Why don't you propose a
role model? According to you, which parts of the philosophy community
have reaped intellectual benefits by taking the appropriate
All of this is very interesting to me, because I tend to see things in
precisely the opposite way. In my view, foundationalism in philosophy
of mathematics has been extremely successful, so successful that other
branches of philosophy ought to emulate it ....
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