FOM: SET VooDoo Re:More on Maddy on method
Robert Tragesser
RTragesser at compuserve.com
Mon Feb 23 15:07:50 EST 1998
Neither Maddy nor Tennant are facing up to the fundamental
task of deeply understanding the reductive powers of
SET and its limitations (what gets lost in translation). For example,
it could be that set
theory "works" because it is so rich in symmetry (so close to paradox)
that most any bit of conceptual mathematics can break the symmetry (but
not completely reduce to the asymmetry!).
Any other "system" as madly rich in symmetry could do the job of
creating an impression of posessing like modelling power. (Indeed,
dare I whisper: category theory?)
(Perhaps a less fancy -- but less suggestive of workable analogy
for
exact development -- metaphor might be: set theory is
a clay which can assume mathematical shapes all too readily,
and so therefore all too uninterestingly, uninstructively.)
I present this metaphor as something which could possibly
be made exact.
Neil Tennant wrote:
[[["But is it not still mind-boggling and astonishing that every single
deductive twist and turn in any branch of mathematics, with however
highly compiled concepts might be thus employed, can in principle be
reduced, by appropriate modelling definitions, to trains of reasoning
that concern *only* sets and the membership relation? The mystery is
even more wonderful then we remark further that it can all be done in
*pure* set theory, describing this dyadic tissue of things teetering
on the empty set."]]]
Wonderful, but so what?!
Or: maybe "wonderful" because one
doesn't yet see how the trick is done.
Set theory constructs as VooDoo dolls -- so resonant, so lifelike,
so _sympathetic-, if you stick them, will mathematics bleed?
[1] In fact the reductions or translations from conceptual mathematics into
set theory generally do not go over so nicely.-- Both mathematical concepts
and proofs are truncated and distorted. Here are a number of lite
considerations, examples.
[i] as an exercise just try to so reduce the proofs of the
first 20 theorems say of Hilbert's FoG as they occur there -- you are
in serious trouble right off because a number of them -- or if one is very
sensitive, none of them -- go over into a predicate calculus derivation.
[ii] Notations and the development of notations that think for
themselves is an exceedingly important mathematical enterprise. . .as
well as understanding a notation is an important mathematical enterprise.
As for example the theory of Dynkin diagrams in Lie algebra. Rota's
work on the umbral calculi. or the various mathematical constructions
that have emerged from interpreting dx? (Rather than suppressing it a la
Cauchy-Weierstrass), e.g., theory of differential forms, exterior
algebras.
[2] Instead of making google eyes at set theories reducing power (like an
18th century creationist at the wonders of nature), shouldn't one try to
understand why, make a serious problem of, the reduction of so much of
mathematics to set theory.
THIS IS WHAT I DO NOT UNDERSTAND: it's an old trick to dismiss
what you can't examine by the tools of which you are a master to call
that illusion or psychology. But the fact that NT finds something
to be amazed about suggests also that live, practiced, unreduced
mathematics
has about it a TRACTABLE content that is lost in set theory. MUCH BETTER:
to come to grips with that content (rather than to rather snottily -- to
capture in a word the tone I am getting from Tennant and Maddy, and by
that
I mean using rhetoric to avoid facing up to a hard and important problem. .
.
rather than waving a hand or turning up a nose to dismiss it)
in an exacting to explain why that content can in such surprising measure
go over into set theory, and to delimit what of the content can't, and
to decisively evaluate the importance of what can't.
[3] The Bourbaki had the sense that set theory could be replaced by some
other
"foundational theory".-- That set theory is _not entailed_ by the
mathematical
structures they set out.
An interesting suggestion is that: set theory works because it is
so excessively rich in symmetry;-- conceptual mathematics "breaks" those
symmetries.
Any other "theory" rich in symmetry might work as well, though
differently.
There does seem to be some sense to this metaphor. It is
definitely
deflationary -- lowering the philosophical worth of "fundamental theories"
which work through "symmetry breaking".
In general, one system X is weakly/insignificantly reducible to
another
system Y when (a) X does not entail Y, and (b) some Z distinct from Y
would
equally support X. (Examples: regular crystals, autopoietic(living)
systems. . .
are thus not significantly reducible to fundamental physics...)
robert tragesser
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