FOM: set-theoretic and other foundations

[Neil Tennant]

I don't quite understand the dialectical situation reached with Robert
Tragesser's opposition to set-theoretic reductionism.  He calls the
position "hand-waving" and thinks that those who espouse it speak in
"snotty" tones. The most charitable explanation for any imagined
snottiness would perhaps be that with all the hand-waving allegedly
going on, the hand-wavers cannot find time to blow their noses!

But I am sure neither Pen nor I intended to come across in such tones;
indeed, I think this is a case of someone mistakenly reading into
email ciphers on the screen an intonation that simply was not in the
writer's mind. 

I did, though, pose some challenges to Robert, asking him to back up
some of his claims. Perhaps it would be helpful to explain further
exactly what my stand is on the issue of set-theoretic reductionism,
so that if I remain a moving target Robert will at least know in what
direction I am moving. 

First, I think that 

(1) Set-theoretic reductionism does go through.

That is, it is possible to re-cast mathematics in set-theoretic
terms. As I think Pen pointed out earlier, this is an empirical claim,
based on the extraordinary successes since the late 19th
century---successes of definitional analysis, increased logical
rigour, and axiomatic organization. Thus far, I believe, the available
evidence strongly supports (1). When Robert *denies* (1), I think he
owes us an impossibility proof---that is, an account of some branch of
mathematics that provably defies reduction to set-theoretic terms.
Whatever one's attitude to (1), I think this much is held in common:
*If* (1) is true, it's pretty amazing.  I for one would not feel
any embarrassment at expressing my wonder at such a truth. Robert tried
to explain *why* (1) might hold, by appeal to "symmetry breaking"---an
explanation which, alas, I did not understand, and therefore asked him
to clarify.

Secondly, I think that with proper sensitivity to the "ur-math"
involved,

(2) The necessary reductive definitions can be chosen in such a way
that the concepts that yield insight and that guide the formulation of
conjectures and the search for proofs remain available at a suitably
"compiled" level in the set-theoretic version of the branch of ur-math
in question.

Note that implicit in (2) is the recognition that the ur-math (and not
its set-theoretic version) flourishes from the wielding of its own
indigeneous, unreduced concepts.

Thirdly,

(3) It is perfectly coherent to acknowledge that set-theoretic
reduction is possible for all of mathematics (i.e. acknowledge (1)
above) while not advocating such reduction---neither for the
furtherance of the branch of mathematics in question, nor for its
"increased foundational security", and certainly not for its
pedagogical clarification.

Indeed,

(4) It is coherent to acknowledge (1) and even (2), and yet prefer an
*alternative* brand of foundations altogether.

And I don't necessarily mean that category theory would be an adequate
alternative!

For my own part, I much prefer an analytic-cum-synthetic approach that
aims to cast light first on the "natural logic" of the branch of
mathematics in question, and secondly on its existential
presuppositions. The fashionable view about the trend towards
set-theoretic foundationalism is that "anschauliche Gewissheit" was a
will o' wisp, thoroughly discredited after Kant's Euclidean dogmatism
had been undermined by the development of non-Euclidean
geometries. The move was made supposedly towards greater *conceptual*
clarity and certainty. For a while the logicists had the upper hand
over those who extolled "intuition" as a source of insight and
certainty. But the crisis in foundations precipitated by the paradoxes
showed that the "conceptual" resources being resorted to carried no
greater guarantee of certainty or consistency than the "intuitive"
resources they had displaced.

Set theory in the wake of the responses to the paradoxes seems to me
to be a melange of the intuitive and the conceptual. The purely
logical notion of set (class) as the extension of a concept had to
give way to the hybrid notion of sets as mathematical objects produced
by an iterative procedure, but also obeying certain laws of
abstraction. As a reducing theory, ZFC and its extensions do not
dictate any "style" that now clearly ought to be mimicked by any other
attempt at foundations.

An alternative but very different style, it seems to me, would be the
formulation of basic rules of inference governing single occurrences
of the salient primitive mathematical operators or predicates---by
analogy with the way Genten analyzed logical operators in terms of
introduction and elimination rules. One can in this way distinguish
the "analytic" part of the mathematics from the "synthetic" part. One
also thereby respects the need to prosecute the mathematics in its own
intrinsic terms. This is very much in the spirit of Hilbert's FoG, to
which Robert refers; although it would ideally be rule-theoretic
rather than axiomatic.

Such a program can be carried out for Peano-Dedekind arithmetic; for
the *logic* of sets (as opposed to the theory of sets, which deals
with what sets actually exist); and for projective geometry. It treats
each branch of mathematics as sui generis, and seeks within the
behaviour of its own primitive concepts the epistemological security
otherwise sought by recourse to a more "general" carrier theory such
as set theory. (Note that insofar as set theory is now a branch of
mathematics in its own right, quite apart from its potential role as a
reducing theory, it would have to be accommodated in the new
perspective here suggested; and indeed it can.) It remains to be seen
how much further this style of foundations can be extended.

My own preference is for such an alternative foundation. It offers, at
least, an interesting *program*, even if it is nowhere near as fully
carried out as, say, the program of set-theoretic foundations has
already been. Unlike Robert, however, I do not think that
set-theoretic foundations is so self-evidently inadequate as a vehicle
for the uniform re-expression of mathematics.

Let a thousand blossoms bloom!---unless they irritate the sinuses...

Neil Tennant