FOM: anti-foundationalism

[Jerry Seligman]

Volher Halbach did a good job of explaining "foundationalism", as used by
academic philosophers. I want to add a few points that may help to
clarify the matter further. (Given the need for a brief presentation of
a host of complex ideas, I beg forgiveness for the many subtleties that
will not be addressed in what follows.)

There are two parts to a foundational project. We have to show that
the superstructure is supported by the foundations, and that the
foundations are secure.  For example, to justify ZFC as an epistemological
foundation for mathematics, we must

(1) show that mathematical knowledge can be justified on the assumption that
we are justified in believing the theorems of ZFC, and
(2) show that we are justified in believing the theorems of ZFC.

It is generally accepted that part (1) has been completed successfully.
Part (2) is more difficult.  There are various options, some more
foundationalist in
spirit than others.

The first, thoroughly foundationalist, option is to show that the axioms of
ZFC are justified, and that following the rules of inference (predicate
calculus) is a reliable method of generating true beliefs from true
beliefs.

The issue of the reliability of predicate calculus takes us quickly into
deep waters in the philosophy of language. The rules are justified because
of the meanings of the symbols, but different theories of meaning will
(arguably) justify different rules.  The is Michael Dummett territory, and
home turf to
some FOM philosophers. Philosophers not specifically concerned with this
debate do not like to prejudge its outcome (if any), and so tend to regard
it as a problematic part of the foundational project, but are usually
willing to give foundationalists the benefit of the doubt.  Another reason
for their
tolerant attitude is that there are significant problems elsewhere.

Even if we are satisfied that predicate calculus is reliable, we have also
to justify the axioms of ZFC. It is not at all clear where to start.  Some
philosophers have tried to justify the axioms in terms of a pretheoretic
conception of the iteration; others have tried to identify sets with
conglomerations of physical objects, possible syntactic constructions, and
so on.  Many of these attempts will be familiar to members of FOM.
Unsurprisingly, perhaps, none of them are generally regarded as successful.

The fumbling for a justification for basic beliefs is not limited to f.o.m.
All foundationalist epistemologies have met a similar fate: for example, the
idea that our knowledge of ordinary physical objects can be justified by
means of a foundational project whose basics beliefs are beliefs in simple
sensory
experiences ("I see a red patch now").

The widespread failure to find justifications of basic beliefs led
philosophers to try something new. The more conservative
"anti-foundational" approaches keep the preliminary foundational analysis,
and merely seek to carry out step (2) in another way. They aim for a
justification for the theorems of ZFC that does not depend on a separate
justification of the axioms and rules of inference.

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.

Here's a further anti-foundationalist thought: perhaps these questions are
only difficult to answer because the right answers do not serve the
foundational project.  Following Quine, we could say that ZFC is justified
just because it succeeds in formalising mathematics. This "holistic"
approach turns (1) on its head to give an answer to (2). Epistemic
practices, such as mathematics, are taken to be self-justifying, and the
"foundational" theory (ZFC) is justified because it does not conflict with
the practice. The part of the foundational project that shows how
mathematics can be based on ZFC is seen to have a systematising,
regularising role, rather than a purely epistemological one.

Another anti-foundationalist move comes from latter-day logicists, such as
Stuart Shapiro. There are a number of subtleties to this approach, but the
basic
idea is to eliminate the need for justifying basic beliefs (the axioms of
ZFC) by having an inferential mechanisms that can do without them.
First-order logic is not up to the task, but second-order logic is.
Moreover,  justifying the rules and axioms of second-order logic is no more
or less of a problem than justifying the rules and axioms of first-order
logic, and---here is the main anti-foundationalist move---there is no
justification to be found.  A milder anti-foundational sentiment is that we
just don't know how to justify logical rules, or that the issue is complex
and connected with deep, intractable problems in the philosophy of
language.

(The main task for philosophers in this camp is to argue that justifying
the rules and axioms of second-order logic is no more of a problem than
justifying the rules and axioms of first-order logic. This depends a lot on
what could count as a justification of logical rules, and so gets us back
in deep water. I don't think this point came out clearly in last month's
FOM debate, although I
suspect that it was lurking beneath the surface.)


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.  It is not an opposition to ZFC, or to the
importance of a foundational theory in the epistemology of mathematics,
although some positive proposals may suggest alternative foundational
theories or alternative roles for ZFC in the epistemology of mathematics.
Some go further, and claim that no systematisation of knowledge is possible
or desirable, but that is
not the common view.


One more thing. The hostility to foundationalism is linked to a distaste for
reductionism, resulting from disatisfaction with various failed
reductionist strategies in philosophy.  There is a sense that any approach
that has to code up knowledge of one kind as knowledge of another kind is
likely to be missing the point. It might work, technically, but it is not
getting to the heart of the matter. I suspect that those attracted to
category-theoretic foundations
are motivated in part by considerations of this kind.  One could argue, in
Quinean fashion, that CT plays as good, if not better, a systematising and
regularising role as ZFC, without claiming that it has an epistemologically
foundational role.   (I am not suggesting that defenders of CT foundations
agree with Quine, only that this is one way of interpreting the claim that
CT is foundational.)



Jeremy M. Seligman
Department of Philosophy,
The University of Auckland, Private Bag 92019, Auckland, New Zealand
Tel: +64-9-373-7599 xtn. 7992, Fax:  +64-9-373-7408, Time Zone: GMT +13 hours