FOM: Boolos'Conjecture; Generation ZF+Xer foundations.
Robert Tragesser
RTragesser at compuserve.com
Thu Feb 26 07:44:53 EST 1998
Abstract: no parity whatsoever between
fundamental physics and SET.
Fruitful interaction of mathematical
domains as no justfication for SET as
fundamental (indeed, more as grounds
for being suspcious of SET). The
need for an explanation of the empirical
"fact" of near ubiquitous reducibility
to SET. The need for
Reverse Reverse [sic.] Mathematics (i.e.,
separation of equivalences). Boolos's Conjecture
as the only promising avenue for investigat-
ing the foundational significance of SET.
[[First, apologies. The curious phenomenon
of my e-mail provider, Compuserve, crossing
two "Tragesser" accts. has meant that I've
not been receiving postings, and so not
acknowledging, responding, or responding
appropriately. Don't know if I now have
secured all e-mail.]]
My rhetoric -- well mocked by Tennant --
was not so much responding to Maddy/Tennant
as to a kind of dogmatic/ideological barrier
I've long experienced from Generation ZF+Xer
foundationalists. The seemingly endless
back/forth on FOM over CAT/TOP vs SET increasingly
suggested that only in some complex of pragmatic
senses, considerations of expedience and convenience,
is ZF+X taken as foundational. So I wanted
to question this.
I want below to give the only sensible
defense I've heard for the fundamental character
of ZF+X; I heard it from George Boolos. But
first, some explanations, "responding" to
(what I have from) Tennant, Maddy, Shipman
[Mayberry requires a separate treatment].
Tennant has made it clear that he
is not a Generation ZF+Xer [and has
formulated an independent foundational
program which has appeal]. He does give the
impression that he thinks that set theory
does not have a deeply coherent over-arching
sense. Tennant wrote:
_[Tennant]_
[[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...]]
_[EndTennant]_
That set theory does not have a strong
and coherent sense, that it's basically a
product of tinkering, and that it has not
been re-produced from a clear and distinct
idea that allows us to see why it is so fundamental,
does suggest that the Generation ZF+Xer
foundationalists are (were?--are there any?)
driven not by serious thought but
by some combination of ideology and convenience. The
"picture" of the cumulative hierarchy is so blurry,
so mathematically, logically and philosophically
problematic,that one cannot imagine "it" having
(a right to) foundational authority over the clearer,
significantly less problematic, "picture"
of the finite ordinal numbers. (I mean, if one
wants to trade on pictures -- I generally do not).
Even if it is thought that the cumulative
hierarchy is an expansion on "set of" as
something like a (Neo-)Platonic Form, and
that the cumulative hierarchy picture is defective
simply because we have not sufficiently
plumbed this form, we are still owed an
EXPLANATION of why this Form ("set-of") is
fundamental to mathematics, always and forever
(or at all).
I am not in the least moved -- and deeply
wonder why anyone is -- by the reputed "empirical"
fact that every piece of mathematics known can
be coded in ZF+X (for some well-considered X's).
I do not find this "marvellous"; rather I
find it suspicious. This "empirical law" needs
an accounting, an explanation: What is there
about mathematics that they can be buried
in SET, and what is it about SET that makes
them their Earth (Dust)? Are there really
no mathematically flavored explanations (theorems)
here? Could it be that the standards of
mathematicians are so much
lower than those of theoretical physicists?
[1] There is no parity between fundamental physics
and SET.
I dare say that for a rather great
range of mathematics, mathematicians gain nothing
by modelling their conceptions in SET; worse,
they likely lose touch with their subject. Consider
instead the relation of say Botany to, say,
quantum theory. (I take the example of Botany
because of the way Pen Maddy used it.) There
is no serious Botanist whose imagination and
understanding is not going to be fruitfully
furthered by learning as much quantum mechanics
as they can manage, and not just "in principle"
or out of physicalist/reductionist convictions,
but because quantum mechanical facts/principles
have immediate cash value for understanding a
thousand processes they encounter daily. I
see no evidence of this kind of turn around wrt
principles and facts of set theory (sufficient to
take set theory as fundamental).
[2] As Shipman pointed out, perhaps the
most powerful source of understanding and
creativity in mathematics comes from the
"interaction" of mathematical domains,
that is, of modelling items of one domain
in another, etc. But I do not see here
any justification [Shipman seems to] for
set theory in its role as the great unifier
of domains, as undwrwriting or even as
explaining these fruitful "interactions".
As I think Saunders MacLane has most
elaborately pointed out, mathematics
is extraordinarily "symmetric" among
its various parts. Items in one part can
be widely modelled, reflected, or approximated
by items in other parts.
This happens far in excess of the
reflections, etc. being mathematically
fruitful/significant!
Just how amazingly reflective/symmetric
among its parts mathematics is, is disclosed
by the results of Feferman, Simpson et al
{{as Surveyed in Feferman's essay "What rests
on what?" and Simpson's "Partial Realizations
of Hilbert's Program"]] as also the amazing
"equivalence classes" of seemingly very
distinct theorems in Reverse Mathematics.
Does SET really afford us the explanation
of such amazing contractions? (This
is a real question, not a rhetorical question.)
As plant metabolisms, signalling,
tropisms, etc. can be INSTRUCTIVELY
reflected down into quantum mechanics [WHICH IS
VERY DIFFERENT FROM SAYING THAT SUCH
ASPECTS OF PLANTS CAN BE EXHAUSTIVELY
UNDERSTOOD BY SUCH REFLECTION...THEY
CANNOT BE, NOT EVEN IN PRINCIPLE...
at the very least because as Gould,
Margulis, and others put it, they
have an historical dimension], as
geometry can be instructively
reflected in analysis/algebra (and
vice versa, surely), so in general
mathematical conceptions ARE NOT
instructively reflected in SET.
Surely there is a missing "theorem"
or conjecture here, something analogous to
Fourier's theorem/conjecture which
goes beyond the empirical of the representa-
bility of functions by "their"
Fourier series?
This theorem would constitute an
explanation of the empirical fact of the
translatability of arbitrary pieces
of mathemtics into SET, and therefore
show us exactly what those translations'
are worth to any arbitrary piece of
mathematics.
I have by the way an analogous
problem with the sorts of compression results
of Feferman and Simpson: exactly
what are we learning about mathematics?
What is the mathematical signifiance of such
results?
I guess that I am haunted by Emily
Noether's aphorism: "It is cheating
to prove that a=b bu proving that a is
lessthanoequal b and b is lessthanorequal a,
for one has not then gotten down to essentially
why a = b."
When I think of important mathematics,
I think of mathematics that doesn't stop
with a proof of a theorem but goes on
to try to understand the theorem more
profoundly. But independently of this
prejudice of mine, most any research
program in mathematics where the
aim is not just to secure "truths", but
to understanding something (e.g., Charles
Feferman in pursuit of "uncertainty"
phenomena dropping out of Fourier analysis)
would likely not be furthered, but be
hampered by (so that its fundamental "facts"
are not adequately coded or represented by,
but rather distorted by)
such PRA, PR, ZF reductions.
Perhaps what is bothering me in
all these reductive and condensing enterprises
(ZF-X foundations, Reverse mathematics...)is
that they lose what strikes me as
fundamentally important in mathematics:
That is, yes, it is surely important to know
the conditions under which, say,
the Bolzano-Weierstrass theorem, Ramsey's
Theorem(s) for coloring, the theorem that
every countable field of characteristic 0
has a transcendence base.. But it does
seem that once one knows this, then it
becomes deeply important to know (but it is
deeply important anyhow) how
these theorems are inequivalent,--
the exact senses in which they cannot
be substituted for one another salve
veritate.
I'll have to save Boolos' Conjecture
for another occasion; but it was this
(told to me in conversations circa 1990):
"An intuitive proof [=a proof wanting
in some respects in formal-logical rigor]
of a theorem is [based on an] an intuition
that it is demonstrable in ZF+X."
I want to discuss how employing
some details of Goran Sundholm's study
of proof one can make good sense of this.
Indeed, although I do not believe Boolos'
Conjecture, I think that working
out why it is plausible
gives one the best philosophical shot at
explaining the foundational authority of
ZF+ .
Above all it forces one to study
what is too little or never studied,
how one backs up from an "intuitive"
(= not fully logically rigorous proof)
proof toward a logically rigorous proof,
learning along the way what the hazards are,
what is lost, what might be the point of
doing it, and of not doing it.
robert tragesser
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