[FOM] Re: Papers of Poincare (Akihiro Kanamori)

[Lucas Wiman]

Colin,

 >The Dover translation of 1952 has very substantial deletions. It omits 
the
 >most mathematical sections, and the most polemical arguments with the
 >logicists - the parts deemed least interesting to the general public. 
These
 >arguments are the funniest and meanest parts of the paper. I do not 
know if
 >Ewald has a new translation or used the same abridged one.

Unfortunately, this version seems to have been reprinted in the modern 
library's "The Value of Science."  That's a shame, as it would have made 
the book much more useful to most audiences (especially me!) to include 
it.

 >Poincare believed
 >that no axiomatization of arithmetic could ever be complete. [...]
 >This was long before Godel, and Poincare had no kind of
 >proof of his claim. So you could take Poincare's view as a mere 
prejudice.

I would say that this shows a very good intuition on Poincare's part.  
By the same argument, it was only Hilbert's "prejudice" that such an 
axiomatization could be achieved.  (Quite explicitly:  "There is no 
ignoramibus in mathematics!")

 >Poincare enthusiastically admired Hilbert's FOUNDATIONS OF GEOMETRY. He
 >championed Cantor's set theory and was among the first mathematicians to
 >actually use it. (He used it to study limit points of orbits in 
topological
 >dynamics.)

Hilbert's book was a mathematically significant work which showed the 
legitimacy of non-euclidean geometries, and also presented Euclidean 
geometry beautifully.  Still, I wouldn't say that someone who said that 
he does "not blame Hilbert for this formal character of his geometry.  
[Hilbert] was bound to tend in this direction, given the problem he set 
for himself" enthusiastically admired this work.  That's not exactly 
glowing praise--not blaming someone for something they did.

I don't think he was very fond of Cantor:  "{\it There is no actual 
infinity} The Cantorians forgot this, and so fell into contradiction.  
It is true that Cantorism has been useful, but that was when it was 
applied to a real problem, whose terms were clearly defined, and then it 
was possible to advance without danger."  You're quite right that 
Poincare didn't seem to mind Cantor's set theory per se (he said that 
"the services [Cantorism] has rendered to the science [of mathematics] 
are well known."), but he did oppose its use of completed infinities and 
impredicative definitions (which did lead it to contradiction for some 
time).  At one point, he compared Cantorism to scholastics in the 
arbitrariness of its definitions.  As we now know, applying Poincare's 
restrictive principles to set theory would emasculate it, leaving a 
bland, uninteresting theory.  Explicitly or implicitly, he was 
fundamentally opposed to Cantor's set  theory.

 >So you can read him as much friendlier to logic than Russell
 >and Goldfarb do. That way you can relate his philosophy of math more
 >directly to his mathematical work.

I could, but I think it would probably be revisionist to do so.  He 
certainly criticized Russell's logic at great length, and seemed to feel 
that Russell's and others' work (Couturat, Peano and Hilbert) was a 
pointless waste of time.  He was ambivalent to logic in much the same 
way as his heirs, the intuitionists: logic is a branch of 
mathematics--not a foundation for it.  At the core of mathematics lies 
reasoning principles and creation, not meaningless symbols on paper.

 >I have written about this in "Poincaré: Mathematics & Logic & 
Intuition",
 >Philosophia Mathematica (3) 5 (1997) 97-115.

I will read your paper; it sounds quite interesting.  I would love to 
see what sounds like a fresh and innovative reading of Poincare.

Thanks,
Lucas Wiman