FOM: unexplained provocative statements

[Harvey Friedman]

This is a reply to kadvany 3:38PM 3/2/99. I will follow this by replies to
the net two postings of kadvany.

In general, these postings contain provacative statments whose meanings are
left unexplained. The only impression that an f.o.m. expert is likely to
get is that aspects of f.o.m. are being criticized in some way. Readers of
FOM cannot be expected to have read Kadvany's writings or the writings he
refers to. This is a typical situation for the FOM, and so many FOM posters
have appropriately strived to keep their comments fairly self contained and
self explanatory. The point of these replies to Kadvany is to get Kadvany
to make his points sufficiently precise and self contained so as to be
useful for the FOM list.

>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 [postmodernism] is that the debates thoroughly confuse
>normative
>and descriptive ideas.

Give an example of such confusions.

>There is also a lot of bad writing and hype
>associated with the subject, but that's true of lots of intellectual
>topics.

Give examples of bad writing and hype.

>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.

High level f.o.m. *is* so unified and has evident criteria of progress by
traditional standards.

>This should be an uncontroversial observation/definition.

Is my previous statement noncontroversial?

>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.

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?"

>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.

This paragraph makes absolutely no sense to me, and I suspect, to hardly
anyone on the FOM. Please explain.

>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.

What multiple competing foundations are you referring to? Who calls what
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.

What are these "skeptical "tropes""?

>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.

What "translation of informal philosophical arguments into precise
mathematical problems" are you referring to?

>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.

Explain Pyrrhonism.

>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.

What conceptual analogy? Also, I rarely see anything philosophical
"completely explained."

>3.  Now, assume for the sake of argument that Godel"s proofs do have
>the skeptical heuristic structure I describe.

What skeptical heuristic structure?

>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.

What "fragmentation of mathematical foundational studies?" What "skeptical
practice?"

>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.

Who is "creating foundations and taking them apart, moving from one to
another, and never settling down" and where?

"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.

Do you mean "skeptic" is a description of fom? Explain.

>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.

Why don't you explain what pomo is in simple clear terms?

>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.

What important choices? What substantive issues to you see here?

>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.

To you mean "informal" in the trivial sense that any really interesting
theorem is?

>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.

This paragraph makes no sense to me.
>
>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.

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.?"

>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.

What is the import of your invoking Lakatos in this?

>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.

What work of Feferman are you referring to? He has written many papers.

>My personal belief is that the intensional structure
>represented by the HBLob conditions has mathematical content yet to be
>discovered and exploited.

What do you mean by intensional structure?

> 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.

Was he writing history?

>In a small way,
>this is a kind of antihistoricism which I believe is bad for
>mathematical practice.

Why?

>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.

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?

>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.

What do you mean by "outside of mathematics?" Depending on what you mean,
we obviously have it, or it's impossible to have.