FOM: unexplained provocative statements
john.kadvany@us.pwcglobal.com
john.kadvany at us.pwcglobal.com
Fri Mar 5 19:13:50 EST 1999
Following are my (John Kadvany?s) replies to Harvey Friedman?s posting FOM:
unexplained provocative statements 03/04/99 06:58:39. The text is
structured as: Harvey?s selection of something I wrote, then followed by
Harvey?s question marked HF, followed by my current response marked JK.
Harvey is rightly concerned that my earlier responses are not
self-contained, but I was responding to a reading of my paper, and I cannot
repeat the paper. I?m doing my best here to balance effort and accuracy in
the world of email sound bites.
>1. My take on pomo [postmodernism] is that the debates thoroughly confuse
>normative >and descriptive ideas.
HF: Give an example of such confusions.
JK: Read just about any heated discussion on the Sokal hoax, or accounts
in the New York Times. We can distinguish between describing certain areas
of knowledge as having a postmodern character from the normative
endorsement of that condition as a desirable end. The Marxist Frederic
Jameson developed lots of the ideas about what postmodernism is, as a
description of contemporary consumer culture, but is normatively opposed to
much of it, including its ahistoricism, meaning the absence of a critical
and postive relationship to the past and how we see ourselves as moving
forward as a society; I agree with that basic division, and I think much of
descriptions of contemporary culture are valid. His book is Postmodernism:
The Cultural Logic of Late Capitalism. But note that it is not even so
obvious that disunified science is a bad thing; see Peter Galison?s The
Disunity of Science for a substantive discussion.
>There is also a lot of bad writing and hype
>associated with the subject, but that's true of lots of intellectual
>topics.
HF: Give examples of bad writing and hype.
JK: For bad writing, see writings of Judith Butler, recently discussed by
Martha Nussbaum (U Chicago) in The National Review.
For hype see Sokal after his hoax and others who fan the reactionary flames
against the perceived threats of postmodernism.
>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.
HF: High level f.o.m. *is* so unified and has evident criteria of progress
by
traditional standards.
JK: Please then identify the criteria by which people will abandon work on
any number of incompatible set theories and settle down with just one, for
example. Maybe that is an out-moded view of fom, but I think it is common
enough to justify as a starting point. I apologize for not knowing much
about recent developments and a more nuanced view.
>This should be an uncontroversial observation/definition.
HF: Is my previous statement noncontroversial?
JK: It probably is controversial; I admit I don?t know enough to pursue
it. This is a good issue for future discussion, also related to whether
Godel?s program of looking for low-level consequences of set-theoretic
axioms needs to be revisitied and improved upon.
>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.
HF: By many foundations, what do you mean? What are these many foundations?
You
could be referring to any number of kinds of things here. The readers of
FOM cannot be expected to tell which. There are many ways to look at
"foundations" so that it is very much a philosophical problem of great
weight. What do you mean by the "problem of foundations?"
JK: I mean the basic idea of multiple incompatible set theories for
reconstructing most of mathematics. This perhaps is a bad start but it is
also fairly common. It could be more nuanced.
>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 naive 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.
HF: This paragraph makes absolutely no sense to me, and I suspect, to
hardly
anyone on the FOM. Please explain.
JK: I?m trying to make a lengthy discussion accessible through
shorthands; this is necessary for this type of discussion. Beside, what you
quote is an introductory paragraph, which was then followed by the promised
explanation, and implicit in what follows below.
>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.
HF: What multiple competing foundations are you referring to? Who calls
what
isostheneia?
JK: Again, just the old idea of incompatible, typically incomplete
theories, like incompatable extensions of ZF. Isostheneia is the old Greek
term, used by the skeptics, for what is an undecidable statement relative
to some theory in modern times. Undecidablility relative to a theory is a
great conceptual discovery of the ancient skeptics.
>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.
HF: What are these "skeptical "tropes""?
JK: Steve summarized some of these. Some is explalined below.
>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.
HF: What "translation of informal philosophical arguments into precise
mathematical problems" are you referring to?
JK: Primarily skeptical arguments for isostheneia/undecidability, but also
including other informal devices which get formal treatment in Godel?s
proofs. Again, this is getting detailed and is the topic of the paper.
>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.
HF: Explain Pyrrhonism.
JK: Pyrrhonism is a set of general methodological ideas for dealing with
undecidability and the criticism of so-called dogmatically asserted claims,
like certain foundations in the mind, physical world, senses, or logic, and
coherently defusing such claims. Godel?s theorems provide a paradigm case;
that?s what I try to explain in the paper. This is a bit like trying to
explain marxism or Aristotle or a difficult proof or whatever; it?s just
too much for an email. I suggest reading Popkin or Burnyeat. If you don?t
like these books I will buy them from you second-hand.
>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.
HF: What conceptual analogy? Also, I rarely see anything philosophical
"completely explained."
JK: The idea here is that Godel?s proofs have long been taken up to show
all kinds of speculative ideas about knowledge, with some justification but
lots of confusion. Once you see the skeptical structure underlying the
heuristics of the proofs, then it is clear where people get some of these
ideas and how to clarify them. I still think I?ve pretty much explained
what is going on here. There's also no basis for popular ideas about
Godel's proofs and the limitations or capabilities of the human mind. You
can argue for that kind of thing, but then your argument would probably
apply just as well to ancient skepticism, which seems odd.
>3. Now, assume for the sake of argument that Godel"s proofs do have
>the skeptical heuristic structure I describe.
HF: What skeptical heuristic structure?
JK: Key heuristics of Godel?s proofs (construction of undecidable
sentences, development of undecidability within a number system like PM or
PA without appeal to an external standard of truth, proof of the second
incompleteness theorem by reflecting on construction of the first theorem,
etc.) turn out to be precise formal equivalents of several of the most
important Pyrrhonian tropes, or methodological techniques for destroying
claims to dogmatic truth, with that position being filled in by Principia
Mathematica, ZF, or a Hilbert-like stand-in or whatever. I really would
like people to refute this analysis if it is incorrent; it seems almost
obvious to me once understood.
>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.
HF: What "fragmentation of mathematical foundational studies?" What
"skeptical
practice?"
JK: Here you can talk to Steve Simpson on fragmentation; he seemed to
recognize something. The skeptical practice is the ability we know have to
work among a variety of incompatible foundations, with foundation being
taken in the old Principia sense, but without having to really decide among
them. This is the so-called postmodern condition. There is lots of work
being done on constructible sets, versions of the axiom of determinacy,
large cardinals and so on: What would eventually decide among all these in
some philosophical-foundational sense, and what would that mean?
Mathematically there may be much interest in pursuing all or various
subsets, but that?s not the issue.
>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.
HF: Who is "creating foundations and taking them apart, moving from one to
another, and never settling down" and where?
JK: See previous. There is also a skeptical idea of ataraxia, which
refers to the goal of people calming down and not getting so worked up
about foundational issues. The Pyrrhonists were actually therapist-types,
sort of consultants, who worked in the medical community in Alexandria
where there were heated debates about theoretical versus empirical
medicine. FOM could use at bit of that, in my humble opinion.
"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.
HF: Do you mean "skeptic" is a description of fom? Explain.
JK: Yes, fom practice is skeptical in this way. It?s not the practice of
creating a single theorem, but the pattern of mathematical interest and
fragmented progress across the board, and across time, as reflected in
publications. Godel?s methodological legacy included techniques for
generating a kind of implicit skeptical practice in the development of
mathematical theories, and this has generated a kind of postmodern
condition of knowledge. Part of this is to take the perspective of
stepping back and looking at mathematics as a longer-term process, not a
bunch of static results, though the latter form an essential part of
mathematical content also.
>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.
HF: Why don't you explain what pomo is in simple clear terms?
JK: I did in an earlier email. Ask Steve Simpson, who seems to know, or
ask Sokal, who is so worked up over it. It?s not that deep an idea and is
expressed in several of the above answers.
>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.
HF: What important choices? What substantive issues do you see here?
JK: Choices are the selection of the Hilbert-Bernays-Lob conditions as
characterizing the desirable properties of a proof predicate. This to me is
one of the most interesting and important developments in all 20th century
logic.
>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.
HF: To you mean "informal" in the trivial sense that any really interesting
theorem is?
JK: No, it means that the theorem-proving process, and the theorem?s
conceptual content, contains important features which are not at all
characterized by a formalism, or at least a single formalism. This is
getting detailed and I refer you to Lakatos? Proofs and Refutations. The
point is when such informal features are missed and divert attention from
new ideas which may be outside the bounds of contemporary mathematics, but
which could be usefully incorporated.
>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.
HF: This paragraph makes no sense to me.
JK: Many paragraphs in mathematical papers make no sense to me until I
read them several times, discuss them with my friends, or follow up on the
references indicated in the paper. Also, you can say instead 'Please
explain this paragraph with concrete examples.'
>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.
HF: What is "the Godelian metamathematical paradigm" and what is
"mathematical
historicism" and what is "foundation" in the classical dogmatic sense and
want is "the problem of foundations in the history of mathematics" and what
is "the problem of foundations for algebra, geometry, probability, etc.?"
HF: "the Godelian metamathematical paradigm": JK: Lots of techniques and
appraoches for coding up large chunks of math and specific problems as
(meta)mathematical problems of various types. Most of this depends on
routine applications of metamathematical ideas and limits due to
unprovability or undecidability created by Godel, at least to a first
approximation.
HF: What is "mathematical historicism": JK: See Proofs and Refutations.
Mathematics is a thoroughly historical subject, and requires philosophiical
approaches which are historically oriented, all in contrast to most ideas
of a priori knowledge, and most philosophy of mathematics.
HF: what is "foundation" in the classical dogmatic sense: JK: You expect
Principia Mathematica or whatever to provide some kind certain basis or
framework which justifies mathematical knowledge in a very strong way.
HF: what is "the problem of foundations in the history of mathematics": JK:
The emergence of foundational issues through the 19th century and mostly
after 1900, culminating perhaps in the Hilbert programme, but not
exclusively.
HF: what is "the problem of foundations for algebra, geometry, probability,
etc.?": JK: Geometry and probability used to be much more informal, were
the topics of great philosophical debate and use, such as Kant?s use of
geometry, but have now become largely normalized mathematical subjects.
There are still important philosophical questions, but not so much the old
ones, and the concepts have changed quite a bit through this process of
mathematicization. I don?t this is sufficiently recognized vis a vis logic
and foundations and this perspective is hindering progress.
>7. Now, you say that [following internal quote is Steve Simpson:]" 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.
HF: What is the import of your invoking Lakatos in this?
JK: Lakatos is all about the heuristic-historical development of
mathematical proofs. See Hersh and Davis, The Mathematical Experience for
an introduction.
>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.
HF: What work of Feferman are you referring to? He has written many papers.
JK: Sorry. 'Arithmeticization of metamathematics in a general setting,'
Fundamenta Math. 195?.
>My personal belief is that the intensional structure
>represented by the HBLob conditions has mathematical content yet to be
>discovered and exploited.
HF: What do you mean by intensional structure?
JK: The HBLob conditions are characterized by modal-logical conditions
formalized using Kripke semantics, and modal logic is all about formalized
intensional relations. Intensional (Feferman noted this in his Fund.
Paper, but it is general usage) means 'depends on the description used to
define the set or predicate,' versus extensional, meaning does not so
depend. So we have to get the description of the proof-predicate correct,
not just its extension, to get the incompleteness theorems to work as we
want. This is very unusual for mathemtical practice, where nearly all
reasoning is extensional.
> 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.
HF: Was he writing history?
JK: He should have been, a bit. The book suffers in this respect. I?m not
saying write 20 pages per chapter. I?m saying probably 10 pages for the
whole book, and more importantly, to provide the right heuristic discussion
motivating the defintions and proofs. "Naturalness" is a sign that a
difficult and important series of steps has been omitted. The book suffers
in this regard. Also, does anybody know why this book is out of print?
What can be done about that?
>In a small way,
>this is a kind of antihistoricism which I believe is bad for
>mathematical practice.
HF: Why?
JK: It buries the heuristic structure of the theory, and prevents progress
and hinders learning. See Proofs and Refutations, appendix 2. This also
has a long history in the Greek tradition of analysis-synthesis, with
Descartes, John Walllis and others complaining about discovery methods of
the ancients being covered up.
>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.
HF: Why should Boolos - or anyone else - write about all aspects of things
when
they are writing about some aspects of things? What do you think of the
patently absurd and/or meaningless claim "that logic is just more informal
mathematics?" In fact, what does this mean?
JK: Just because his book was imperfect in important ways, that does not
mean writing a whole other book. Boolos and all mathematicians have the
responsiblitiy to make their public work of maximum social and pedagogical
value, even if only to the mathematical community, and to be conscious
about the implications of their approach. I think lots of mathematicians
today recognize the need to revise the way math books are written, at least
compared to say twenty years ago.
On logic being just more informal math: this is mentioned above.
Informality of logic can?t be that obviously absurd, since the default
assumption of many philosophers is to characterize mathematics and even all
of human language only through some type of formal system. You miss much
of great importance by that sometimes scientistic assumption. See Proofs
and Refutations. To me the interesting issue here is the relationship
between informal mathematics and its formal representations; I think the
intensionality of Godel's second theorem is at just this boundary, and
bears closer attention for that reason to understand better the
relationship between extensional and intensional reasoning.
>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.
HF: What do you mean by "outside of mathematics?" Depending on what you
mean,
we obviously have it, or it's impossible to have.
JK: Like in the physical world, or the human mind, or the senses, or some
metaphysical doctrine. We do not obviously have or not have such
foundations, and they are not logically impossible, if that is what you
mean.
John Kadvany
Applied Decision Analysis, Inc.
Menlo Park, CA 94025.
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