Re: [HM] Mengenlehre, a disease from which one has recovered

Graham White (graham@dcs.qmw.ac.uk)
Sat, 12 Sep 1998 13:24:48 +0100 (BST)

> J.G. Cabillon transcripted the following quote of Poincare'
>
> > Un des traits caracteristiques du cantorisme, c'est qu'au lieu de
> > s'elever au general en batissant des constructions de plus en plus
> > compliquees et de definir par construction, il part du _genus
> > supremum_ et ne definit, comme auraient dit les scholastiques, que
> > _per genus proximum et differentiam specificam_.
> (ie One of the main features of cantorism is that he begins with the "genus
> supremum" and defines only "per genus proximum et differentiam specificam",
> as would have said the Scholasticians, instead of rising to generality by
> building more and more complicated constructions and making constructive
> definitions.)
>
> This quote is very interesting since it establishes a clear analogy between
> set theory and the aristotelian philosophy (and its scolastic formulation).
> For Aristotle, being "to on", is the more general "concept" (to on esti
> katholou malista panton, Met.B4,1001a21), but its generality is not that of
> a genre (oute to on genos, Met.B3,998b22). It recalls the paradoxes of set
> theory, with "the totality of all sets"="a proper class"=not a set.
> Aristotle showed that "being" is not a genre by exhibiting a paradox
> somewhat similar to that of Russell, but he could overcome the difficulty by
> building a whole ontology.

I don't think the analogy is at all clear. Poincare's description of
Cantor is that he first starts with a highest genus, which is presumably
that of sets. Now Aristotle (for precisely the reasons which Olivier
gives) would not have talked of a highest genus per se, but only a genus
which was highest within a particular category; and there is, for Aristotle,
a fixed list of categories (none of which is "mathematical objects").
Aristotle's own philosophy of mathematics (which is surprisingly little
investigated) would probably start with his account of abstraction
(beginning of the Physics, if memory serves). And what you would get would
probably be any theory of mathematical objects.

The quote about "per genus proximum et differentiam specificam" is,
of course, "scholastic" in some sense (Boethius, again if memory serves).
However, what that means for Poincare is anybody's guess. In the period
when he was writing, there had been very little historical investigation
of the scholastics (with the exception of Duhem's work), so what
"scholastic" normally meant was neo-scholasticism: that is, it was built
round doctrinaire Thomism and had its logical content reduced to a series
of slogans like the one Poincare quotes. So unless there is more evidence,
what we have is something like Hilbert's quotation of Kant that was
discussed on this list a while back: that is, it is something that would
be part of an educated person's repertoire at the time, but needn't betray
any deeper acquaintance.

> I'd like to rise two questions:
>
> 1) Are there another quotations, from Poincare' or others, which also
> describes this analogy between set theory and aristotelian philosophy?
> (beside Dauben's works)

I have a few questions about Poincare.
i) Are there any citations (of Aristotle or the scholastics) in Poincare
that go beyond the usual neo-Thomist repertoire? In particular, does he ever
say anything other than "the scholastics" in this undifferentiated way?
ii) Is there any interesting relationship between Poincare and Duhem?

>
> 2) That recalls me some tentatives to make a theory of concepts out of the
> theory of sets that have been made in the '70s around W. Reinhardt and Gaisi
> Taikeuti. Does anybody know more about it?
>
> ---
> Olivier Souan
> (Paris IV)
>

Graham White