FOM: Past Issues:Set techniques, natural, transcenndental, acioms, postmoder
Robert S Tragesser
RTragesser at compuserve.com
Sat Dec 13 09:32:56 EST 1997
I've begun reading through all the previous postings and have just
completed the first series. Clearly my posting would have been better
informed if I'd done this first, but at the same time those posting would
not have been as significant for me.
Here are some inter-related past issues from past issues (which
might have been treated in subsequent postings).(Sorry there is so much
material that I haven't worked out how to provide references.)
NATURAL VS. SET-THEORETICAL. There seemed to be the accusation that
RevMath was at bottom cunningly utilizing set theoretical techniques to
give a natural appearance to results in RevMath, but a natural appearance
does not entail naturalness.
Many postings later--more toward October 2-- (without there having
been a response to this), Steve Simpson remarked on the naturalistc
Aristotelian flavor of Reverse Mathematics. He also speaks of wanting to
explore "natural" axiomatizations of mathematical subject matters. In one
strong sense, "natural" means: before formal-logical
reduction/representation/reconstruction. Informal rather than formal
One can embrace RevMath as finding a sense in which Aristotelian
genos-divided, autonmous, self-contained (self-causing) is possible, and
perhaps even exploring the extent to which mathematical subject is
essentially non self-contained (self-causing). But wouldn't the actual
practice require informal contents (natural meanings in this sense)?
 TRANSCENDENTAL METHODS? Harvey Friedman spoke of the seminal FOMists
as thinking that their view of mathematics would be changed in a radical
way if theorems obtained by transcendental methos cannot be obtained by
Do nontranscnendetal methods = "elementary" methods, "natural"
WARNING ON THE HISTORICAL ROOTS OF THE MEANING TO "TRANSCENDENTAL".
In its proper and original sense it does not be "higher" but rather
"other". It was used in Scholastic Philosophy to mean: not among the
categories of Aristotle, thus (for example) FINITE and INFINITE were
"transcendental" because they were not among the categories of Aristotle.
But, for example, MOOD, is also transcendental because it is a category
of being not among the categories of Aristotle.
Transcnendetal then went through a number of shifts of meaning.
For example, by virtue of its association with the inifinite,
transcendental came to mean: higher or sublime.
Along another axis, transcendental can to me any being or relation
which was irreducible to the material-natural, so that it could be studied
via natural science. (Thus its use in Kasnt and Husserl.)
Taking the primary sense of transcedentalal as other, then we can
regard not only transfinite techniques in algebra or whaever as
transcendental, but also geometric technikques in algerbra as
transcendental, and vice versa. CAN WE THEN SAY TRANSCENDENTAL IS
OPPOSED TO ELEMenTARY/Natural?
 POSTMODERNISM. Among philosophers of mathematics, this has its
origin in the later Wittgenstein. In particular, his view that
mathematics consids of a burlap sack of proof techniques, and
logical/metalogical techniques being just down there in the sack jumbled
with all the other techniques, and certainly occupying no priveged
position, in fact of rather dubious utility [N.B. the discussions of
mahemticians shying away from set theoretic techniques as N-B-G, "No
 Steve Simpson somehow puts the unification of the sciences together
with Aristotelianism. But aren't these diametrically opposed? The
history of modern science is a history of attempts to move toward Unity
away from Aristotelian individualism. [See Funkenstein.]
 AXIOMS. Only a brief remark since I suspect my question gets
answered in later postings.
In his memoriam for Emily Noether, Hermann Weyl portrays her as
mothering the use of the axiomatic method in working mathematics (rather
than foundational mathematics). He admits of being initially critical of
this but later appreciated its power -- of isolating the mathematically
fruitful and salient traits by brooding over a mess of theorems. By
understanding such axioms and their use, one automatically sees deeply
into the subject (Banach Algebra, say). Clearly, neither axioms of
Peano Arithmetic nor axioms of ZFC are of this character?? When I taught
Probabilty, the text placed great stress on using the given axioms in
order to fly over the net of having to retain the limp notion "equally
likely", to equiprobable spaces (and those as a special case of
probability spaces). But also probability theory was not developed there
qua strict deductions from those axioms.
Maybe it's Noetherian axciomatics which have gone out of fashion?\
I also have this vague idea that Bourbaki was inspired by, but
transformed the sense of, Noetherian axiomatics.
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