FOM: on foundations etc.

[Neil Tennant]

This email is on foundations of mathematics, so please read on! It
starts by talking about foundations of other areas first.

What properties does a field X of human intellectual endeavour need to
have in order for there to be an identifiable sub-area or associated
area to be known as 'foundations of X'?  Is the phrase 'foundations
of' different from the prefix 'meta'?

Is foundations of physics different from metaphysics? My guess is
not--or at least, that many a metaphysician would not accept the
identity claim in question. For metaphysics is concerned with, among
other matters, questions such as personal identity, the existence of
free will, the relation between mind and body, the difference between
abstract and concrete existence,...; and these are topics that the
worker in the foundations of physics would not even address (with the
possible exception of the possibility, from quantum mechanics, that
consciousness and matter are deeply interconnected, or involved in some
complex interplay). 

Could there be such a discipline as 'foundations of biology', or such
a discipline as 'metabiology'? If so, are they the same? If not, why not?
Ditto for 'economics' in place of 'biology'.

When we talk of 'foundations of ethics' or 'foundations of morality',
are we referring to the field known now as metaethics? Maybe not.

There is a journal (hence presumbaly a recognizable sub-area of
philosophy) called 'Metaphilosophy'. But somehow the phrase
'foundations of philosophy' would jar. Philosophy is already supposed
to embrace the foundations of everything, including itself! (this last
being a metaphilosophical, hence also philosophical--but not, I think,
*foundational*--remark).

Now, what about metamathematics and foundations of mathematics?  Let's
not assume without argument that these are the same.  Proposal:
metamathematics is that part of foundations of mathematics that is
prosecuted in a completely mathematical way. Suggested implication:
there could be more to foundations of mathematics than is amenable to
metamathematical treatment. Alternative proposal: only some, but not
all, of metamathematics would count as foundational. A necessary
condition for the metamathematics to count as foundational is that is
have some recognizable philosophical significance.

Question: on either proposal, how is that extra part of foundations of
mathematics that cannot itself be mathematized to be made
intellectually accessible, arresting and worthwhile to the working
mathematician? Is it "merely" philosophy of mathematics? Does the WM
have any intellectual responsibility to consider it? Would the
ordinary work of the WM be enhanced by paying any attention to it?
Could it guide and shape, or help set the agenda, of the community of
WMs?

I note with interest that many of the mathematicians who have
contributed to the list so far write as though the discussion of
foundations of mathematics cannot proceed without a precise technical
definition of its methods and concerns. But this may be to misconstrue
the precise nature of the enterprise. (I say this with all due respect
to all the mathematicians involved!) If, however, one were to accede
to their demands for precise technical definition, one might end up
'defining away' (i.e. definitionally excluding or occluding) the very
essence of the 'non-metamathematical foundational', and be unable even
to consider its possible impact.

The very phrase 'foundations of...' carries with it the implication
that the field in question can be built, or reconstructed, on some
solid base on which the foundationalist can focus. It is a Cartesian
conception, according to which all knowledge in the area in question
is well-founded by some (possibly unique) relation of justification.
(Actually, it should be called the 'Euclidean' conception as far as
mathematics is concerned; but what Descartes did was construe *all* of
human knowledge on this model.)

Now of course mathematics is the primary candidate area of human
knowledge inviting construal of this kind. (Axioms; lemmas; theorems;
corollaries...deductive accumulation...axioms being 'simple and
evident and certain'...theorems being 'deep and difficult', but, in
the end--thanks to proof--being made just as certain as the
axioms...I'm sure the reader gets the picture.)

The Quinean holist rejects such a picture for knowledge in general.
According to the holist, all knowledge is interdependent, none of it
privileged, every 'known' proposition enjoying that status only
because of how it lends support to, and finds support from, other
propositions in the system. Every proposition is in principle
revisable (as believed/disbelieved/neither); no claim is absolutely
privileged. This is a compressed and perhaps overly crude statement of
the holist's doctrine, but if you have never encountered it before,
you ought nevertheless to get the idea.

Now despite all the Quinean moves during the last half-century in the
philosophy of logic and language, and in epistemology, I am pretty
sure that the Cartesian conception of the justificatory orderability of
mathematical knowledge still holds sway among mathematicians; and
indeed, could well be the prime counterexample, in general, to the
sweeping claims of the epistemological holist.

Second question, which maybe the mathematicians on the list could
address: do developments in mathematics in this century leave the
Cartesian conception intact? I am genuinely interested in the
collective consensus (as a piece of 'philosophical sociology' about
the discipline). I had two colleagues in the mathematics department in
Edinburgh who used to alternate with a course in functional analysis.
They joked that what each took as axioms the other took as the main
theorems to be proved. But is such 'arbitrary re-axiomatizability'
really a viable picture for mathematics in general?--especially if
there is to be any emphasis on the epistemic accessibility of the
axioms? 

Knowledge is best thought of as a system of propositions, or
sentences, or thoughts, depending on one's philosophical tastes about
'truth-bearers'. And the latter are made up (composed out of)
concepts. Further question: what would foundationalism about
mathematical knowledge have to do with foundationalism about grasp of
concepts? Sometimes concepts that are difficult to grasp are imparted
and acquired by laying down (as evident/certain/...?) a set of axioms
involving them (the method of implicit definition). In other cases,
the concepts are offered as abbreviations of necessary and sufficient
conditions involving other concepts already available (the method of
explicit definition). Other times one believes oneself to have a clear
grasp of the concept, but then finds oneself struggling to articulate
an optimal set of axioms governing it (witness the evolution of the
axioms of set theory after the crisis of Russell's paradox). Would one
respectable 'foundational' enterprise be a definitive articulation of
the hierarchy of *definitions* of concepts in all the main fields of
mathematics?

Finally, echoing Hilary Putnam, I would be most interested to know
(again from the mathematicians) whether or not they are still
collectively exercised by what is to count as valid proof. As a
starter: do they think of the meanings of mathematical terms as
determined by the rules of inference (including those expressive of
definitions) governing them?---or do they think of those meanings aas
having some other source, to which those rules of inference are to be
held responsible when judging of their correctness?



Neil Tennant