[FOM] The irrelevance of Friedman's polemics and results

Joseph Vidal-Rosset joseph.vidal-rosset at univ-nancy2.fr
Sat Feb 4 04:30:45 EST 2006

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On Thu, 2 Feb 2006 13:39:54 -0800 (PST)
John Steel <steel at Math.Berkeley.EDU> wrote:
> As I understand it, predicative mathematics is not such a "good
> point", as the existence of least upper bounds for arbitrary sets of
> reals is not predicatively justified. Not many would want to revamp
> the way we teach undergraduate Analysis in a way that paid attention
> to predicativity. Certainly not our revered "core mathematicians".
> So "core mathematicians" (whoever those people are-- the editors of
> Annals of Math?), who have learned the naive lub principle, are
> forever in danger of using it in a necessary way.
> Is this a problem, an accident waiting to happen, so to speak?
> Should we fight to reform our curriculuum, so as to encourage
> only "safe" predicative mathematics?
> Pure mathematicians get out ahead of applications. They find natural
> structure, and later the less pure use it. They of course look for
> applications, so as to see the structure they have found is important.
> Mostly, these applications are to other parts of pure math. Much of
> what Avron writes seems to be a criticism of this procedure in itself.

Yes, because that's precisely the point of predicativism: its goal is
to work into secure axiomatics i.e. without impredicative definitions.
Poincaré had already pointed out that Zermelo's set theory was not
secure enough dealing with sets which are both infinite and
impredicatively defined. (Poincaré, "La logique de l'infini", Revue de
Métaphysique et de Morale, t. XVII, juillet 1909). The story of the
term "impredicativity" is complicated. 

It seems to me that, predicativism shares with intuitionism the view
that there are procedures in classical mathematics (as ZFC) which are
not absolutely true, even if they can disagree on the diagnosis. 

I am interested in the question of the possibility of drawing a
distinction between philosophical points and mathematical points in
this sort of debates. 

A Tolerance Principle à la Carnap could maybe solve the problem from a
strict scientific point of view: "truth" in mathematic or mathematical
logic depends on the axiomatic system (there is no Moral in Logic). 

My question is both to  Arnon Avron and Harvey Friedman: is Carnap's
Tolerance Principle wrong (and then intolerable) ? 

- --
Joseph Vidal-Rosset
Université de Nancy 2
Département de Philosophie
Nancy - France

About impredicativity http://jvrosset.free.fr/philmath.pdf
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