[FOM] Re: Papers of Poincare (Bill Everdell)

[Everdell@aol.com]

In a message dated 4/11/03, Colin McLarty writes on Poincaré:

<< Many logicians since have claimed he implicitly posed limitations - or 
that he should have posed limitations. But the limitations are not to be 
found in his math or his philosophy. He even wrote one philosophic paper, a 
month before he died, saying that some mathematicians reject the 
well-ordering theorem as non-constructive. But he explicitly claimed to 
understand the issue better than those mathematicians do. He says the 
constructivists (or, in his words, pragmatists) and the Cantorians both only 
understand one side of the argument, while he understands both sides. (This 
is the 1912 "logique d l'infini" translated in MATHEMATICS AND SCIENCE under 
the title "Mathematics and logic".) >>

I'm trying to get these references straighter.  In 1909 (not 1912) Poincaré 
published "La logique de l’infini" in _Revue de métaphysique et de morale_ 
(1909).  This I have read as it was reprinted in Poincaré's posthumous 
_Dernières Pensées_, and always thought had appeared as "The Logic of 
Infinity" in the translated book, _Mathematics and Science, Last Essays_ (NY: 
Dover, 1963).  This is the one to which Russell replied in 1910, and contains 
the quotation given by Lucas Wiman:
"Knowledge of the genus does not result in your knowing all its members; it 
merely provides you with the possibility of constructing them all, or rather 
constructing as many of them as you may wish.  They will exist only after 
they have been constructed; that is, after they have been defined; X exists 
only by virtue of its definition." (Mathematics and Science: Last Essays; 
quoted in Chihara, Ontology and the vicious-circle principle, quoted by Lucas 
Wiman, [FOM], 9Apr03).

In the same year of 1909, Poincaré published "Über Transfinite Zahlen" in 
Acta Mathematica (1909) later reprinted in his Oeuvres, Paris: 
Gauthier-Villars, v11, pp120-124.  In this article he viewed transfinites as 
an aberration.

None of these, I suppose, is the 1912 "logique de l'infini" cited above by 
McLarty.  Or have I missed something?

Peter Galison has a book on Poincaré in the offing, which should be as fresh 
as Lucas Wiman might want, but I gather it does not say much about 
mathematical foundations.

-William R. Everdell, St. Ann's School, Brooklyn, NY, USA