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john.kadvany@us.pwcglobal.com
john.kadvany at us.pwcglobal.com
Tue Mar 2 18:38:10 EST 1999
Following is my (John Kadvany?s) reply to Steve Simpson?s FOM/Wider
Cultural Consequences posting of March 1, 1999, 23:35:34 in which he
discusses my article on Godel, skepticism, and postmodernism.
1. My take on pomo is that the debates thoroughly confuse normative
and descriptive ideas. There is also a lot of bad writing and hype
associated with the subject, but that's true of lots of intellectual
topics. Pomo also has many senses and uses, like marxism, historicism,
and other isms. My descriptive use is that lots of areas of
knowledge, culture, and science are fragmented, not unified by a
global paradigm, and without evident criteria of progress that look
anything like traditional standards, or any standards worth embracing.
This should be an uncontroversial observation/definition. In my Godel
paper I wanted to provoke the reader by the descriptive claim that
foundational work in logic now has this character: we have many
foundations, and the whole problem of "foundations" is very much a
mathematical and no longer a philosophical problem of great weight.
Two questions arise: what general aspects of mathematical theory may
underly this condition? Second, for those who don"t like a normative
pomo slant, which here would mean just reveling in every kind of
foundational topic you can pick, regardless of some kind of progress
or contradictions with the next guy"s "foundations", what kind of
"antitdote" is possible, but without invoking a traditional dogmatic
and na=EFve search for foundations? My answer, in short, is an
historical understanding of mathematical theorem-proving and
concept-formation. Here"s how I get to that position.
2. The answer to the descriptive condition of multiple, competing
foundations is that foundational practices are characterized in many
ways by the methods of ancient skeptics, who e.g., can be credited
with the discovery of the informal idea of undecidability in what is
called isostheneia. As described in the paper, the heuristic
structure of Godel"s proofs can be seen to implicitly make use of
formalized versions of several key skeptical "tropes," as they are
called, and which you summarized. That is again just a description of
the logic of Godel"s theorems; I think it might qualify as an example
of what Kreisel called "informal rigor," the translation of informal
philosophical arguments into precise mathematical problems. Vis a vis
Pyrrhonism, it should be recognized that this it is one of the major,
major influences in the development of modern science, as discussed in
Richard Popkin"s classic The History of Scepticism; this is standard
history of science and ideas, not a minor offshoot. In the 20th
century Pierre Duhem was very influenced by Pyrrhonism; so were Paul
Feyerabend and Imre Lakatos. By the way, this simultaneous
characterization of postmodern "chaos," Godelian method, and
skepticism shows that the epistemic structure of pomo is mostly just a
version of skepticism; no big new ideas, and its "effects" are
predicable; the conceptual analogy to Godel is made precise and
completely explained.
3. Now, assume for the sake of argument that Godel"s proofs do have
the skeptical heuristic structure I describe. Once you see how the
different skeptical tropes are applied (crudely: to move in and out of
various "foundations," seeing where they lead, but then criticizing
their "dogmatic" status; the details go much further), it is easy to
see the fragmentation of mathematical foundational studies as a kind
of skeptical practice, and regardless of the intentions of the
participants, of course. Again, this is just descriptive. Godel"s
"legacy" in my title, therefore, is this broad skeptical practice of
creating foundations and taking them apart, moving from one to
another, and never settling down. "Skeptic" meant "searcher" in
Greek, and this describes that postmodern conditionof knowledge. We
can argue about how good a description that is for fom, how far it
goes, and so on, but let"s just accept it for the moment. I think it
is true enough.
4. What"s the problem with pomo as a normative perspective? Well,
you just get this mess in fom. What"s the point of it all? It"s
nihilistic, it appears to have no meaning or purpose, it"s not making
progress, etc. It"s up to the individual to decide how a bad a
problem this really is in fom, and that"s a good side-debate on its
own, related to the more general question of whether much of
contemporary mathematics has just become hugely irrelevant to
outsiders, even other mathematicians, but also many scientists.
Again, I want to provoke people into thinking about this situation via
the "pomo" epithet.
5. Then what is a response for fom in particular? In my paper, I
wanted to show that Godel"s theorems, when you look at them closely
and non-superficially, don"t just "give us" incompleteness and the
unprovability of inconsistency, and skeptical method: there are
important choices which have to be made to get the theorems to "work"
as we want, namely the correct formulation of the formalized
consistency statement,and identification of the HBLob conditions. The
little history I provide, from Godel to Rosser to Lob to Kreisel and
Feferman and then to Kripke and Solovay is just to show that one of
the greatest pieces of formal logicis itself a piece of informal
mathematics which had to go through its own historical development.
The key word is "historical." Postmodernism, as many marxists argued,
is normatively unattractive because it picks and chooses from the
past, like much modern popular culture, without any principled view of
where that is taking us into the future. The normatively unattractive
aspect of postmodern is its ahistoricism, and Godel"s theorems almost
seem to "provide" it; my little history is an argument against that.
6. Hence the dilemma I try to force on the reader, and those
interested in fom is: Take your pick, either the "chaos" of
postmodernism inducedby skeptical practices implicit in the Godelian
metamathematical paradigm,OR mathematical historicism (a la Lakatos in
Proofs and Refutations). I opt for the latter, myself. There is no
"foundation" in the classical dogmatic sense, but a historical view of
the problem of foundations in the history of mathematics, just like we
have for algebra, geometry, probability, etc.
7. Now, you say that " In my opinion, Kadvany is barking up the wrong
tree here, because there is no serious challenge to the naturalness of
the standard proof predicate, and if any doubt remains, the
Hilbert-Bernays derivability conditions take care of it." In my paper
I don"t claim the HBLob conditions are wrong; the idea is that they
are the result of the same kind of trial and error learning as found,
say, in the developoment of theories of the integral, or trigonometic
series, or zillions of other mathematical concepts and theorems, again
in the spirit of Lakatos" Proofs and Refutations. I think that"s an
accurate description of the history, even if it is not such a big deal
mathematically; I refer to Feferman"s work, and Kreisel"s comments as
justification that at least some people thought it was worth being
careful about. My personal belief is that the intensional structure
represented by the HBLob conditions has mathematical content yet to be
discovered and exploited. I critcize Boolos" exceptionally fine
Unprovability of Consistency for just his way of imposing on the
reader the "naturalness" of the canonical proof predicate, when he
should do a better job on the historical evolution. In a small way,
this is a kind of antihistoricism which I believe is bad for
mathematical practice. This kind of anipathy toward getting the
history of theorems and proofs right is a big point of Lakatos" Proofs
and Refutations; it is interesting to see it happening in mathematical
logic itself, which just reinforces Lakatos" claim that logic is just
more informal mathematics.
8. You can seek for a foundation for math outside of mathematics, as
you suggest, and I can"t prove that"s impossible to find, but there
sure are a heck of a lot of arguments about why about a zilllion
approaches will not work.
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