> 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'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)
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)