FOM: anti-foundationalism

[Volker Halbach]

Simpson:
>For those of us who are not academic philosophers, what textbook are
>you referring to here?  A textbook-style statement of the
>anti-foundationalist position would certainly be helpful.

Keith Lehrer, "Theory of Knowledge", 1990. Lehrer introduces the notion of
foundationalism in the following way: "According to foundationalists,
knowledge and justification are based on some sort of foundation, the first
premises of justification. These premises provide us with basic beliefs
that are justified in themselves, or self-justified beliefs, upon which the
justification for all other beliefs rest." Similar characterizations of
foundationalism may be found in almost any textbook of epistemology, e.g.
Laurence BonJour: "The Structure of of Empirical Knowledge", or John
Pollock: "Contemporary Theories of Knowledge". 

Obviously the basic beliefs play a prominent role in foundationalism. As
above basic beliefs are supposed to be self-justifying, but at least most
authors do not think that this implies that basic beliefs cannot be
revised. There are forms of foundationalism (e.g. Roderick Chisholm) where
it is explicitly claimed that basic beliefs are fallible. So on this
terminology, foundationalism does not rule out that there is no certainty
in mathematics.

In a paper "Foundationalism and Foundations of Mathematics" that has
appeared in "Proof and Knowledge" (Mic Detlefsen, ed.) Shapiro
distinguishes two kinds of foundationalism. The weaker one is the
following: "Define moderate-foundationalism for P [a body of knowledge,
like mathematics, V.H.] to be the view that is is possible to provide at
least one foundation for P, again either absolutely secure or as good as
possible".

Simpson:
>Am I wrong in thinking there's a common denominator here? 

I think that Shapiro's definition of foundationalism is different from the
standard one in epistemology. Perhaps both definitions are equivalent over
some additional assumptions, but I don't see this. 

> > Do you believe in the axioms of ZFC and claim that they are
> > self-justifying or self-evident and not in need of any
> > justification by appeal to some other principles?
>
>These are good f.o.m. questions.  By asking these questions you are
>acting like one of the (dreaded?) foundationalists.  In other words,
>you are engaged in f.o.m. inquiry.  A typical f.o.m. activity is to
>study these questions as well as similar questions regarding
>alternative foundational schemes other than ZFC.  

When asking these questions I am not necessarily a foundationalist on the
*usual* definition of foundationalism in epistemology (e.g. Lehrer's). The
foundationalist would claim that you sometime arrive at basic beliefs and
then you cannot go on asking how they are justified, because they are
self-justifiying and thus not in need of any further justification. Shapiro
might call this already foundationalism.

There are, however, anti-foundationalist positions that are
anti-foundationalist on Shapiro's and the usual account, namely e.g. if
somebody claims that we are justified in believing the ZFC axioms, because
they "fit" or "cohere" with our other knowledge or because we need ZFC
doing sciene. Seligman has already mentioned Quine. Nowadays, I think, many
philosophers believe in the so-called Quine-Putnam indispensability
argument which roughly goes as follows: We need ZFC for doing science (and
perhaps ZFC coheres with our reamining body of knowledge), so we are
justified in believing the ZFC axioms." I think this is a very common
position and might provide an explanation why people think that a
foundational programe like Hilbert's is not required. Somehow Quine's view
may have been plausible when Quine argued this way for set theory, but I do
not have to explain you that we now know that we can do science in much
weaker systems that ZFC. Notably Quine after his own (nominalist) programe
had failed. This is an illustration to Seligman's claim: 

>In summary, I'd say that the "anti-foundationalism" common among academic
>philosophers is a reaction to the failure of purely foundationalist
>approaches
>to epistemology that seeks to find self-justifying basic beliefs and reliable
>mechanisms of inference.

----------------------

> > I am not anti-foundationalist, but I also find it very
> > unsatisfying to accept axioms without having any reason to do so.
>
>Good.  That means you have the f.o.m. spirit.  You are not willing to
>blindly accept a foundational doctrine, for instance the ZFC
>consensus, without further examination.  That's a good thing.  That's
>foundationalism.

That's foundationalism in Shapiro's sense perhaps, but it does not mean
that I have to believe in basic beliefs. 

>Shapiro tries to marry them in his book.  But I really think that's a
>side issue.  The really important issue raised by Shapiro is, is
>foundationalism (i.e. f.o.m.) a good thing or not?  I say it is.  What
>do you say?

Foundational programes are necessary. Even if one wants to spell out
allegedly anti-foundationalist programes ("ZFC is required for science) you
easily find yourself involved in *proving* that you need all axioms for
showing certain theorem and you are involved in a foundationalist programe.
So I am foundationalist in Shapiro's sense, I guess. But I do not know
basic beliefs that can serve as foundations for mathematics, but perhaps
there are such basic beliefs. So I cannot answer your last question, if
foundationalism is understood in the usual epistemological sense.

Seligman:
>Hilbert's program, loosely (and anachronistically) interpreted, and the
>modern variant discussed by Steve Simpson on FOM, fit into this category.
>The familiar idea is to isolate a fragment of mathematics (the "finitary"
>part) and use this to demonstrate that the rest is at least consistent, and
>so epistemologically harmless, if we regard the non-finitary part as merely
>a useful device to gain real (finitary) mathematical knowledge. In the
>modern version, the task is not to justify all of ZFC, but just enough of
>it to justify "ordinary mathematics" - the theorems proved in mathematics
>journals.  The spilt between real (finitary) and ideal (instrumental)
>mathematics marks a departure from the original foundational program
>because only the real part is shown to be genuine *knowledge*.  Criticisms
>of this line of attack are well-known.  Those unaware of the modern version
>tend to focus on the limitations to Hilbert's program imposed by the second
>incompleteness theorem. But there is also the question of how to justify
>the finitary part.
>
>Perhaps this is worth emphasising. To complete the foundational project,
>a modern-day Hilbert has to justify our belief in the finitary part of
>mathematics. What could count as such a justification?  Is there a further
>foundation on which it is based?  To resist the question with the claim
>that no sane person would question the truth of *this* part of mathematics,
>is to reject foundationalism.

If someone said that finitary mathematics, perhaps identified as the axioms
of PRA, are self-evident and not in need of further justification, then
this would be foundationalist in Shapiro's sense and in the usual
epistemological sense. If one said that also finitary mathematics has to be
justified by appealing to further principles acting as foundations for
finitary mathematics and that one never relies on basic self-justifying
beliefes, then one would be foundationalist with respect to maths in
Shapiro's sense, but anti-foundationalist in the other.

Volker Halbach
***************************
Volker Halbach
Universitaet Konstanz
Fachgruppe Philosophie
Postfach 5560
78434 Konstanz Germany
Office phone: 07531 88 3524
Fax: 07531 88 4121
Home phone: 07732 970863
***************************