[FOM] Re: Papers of Poincare (Lucas Wiman)

[Colin McLarty]

Lucas Wiman <lrwiman at ilstu.edu> raises arguments on Poincare that are 
widely supported in the literature, but really not by Poincare's own writing.

As to Hilbert's FOUNDATIONS OF GEOMETRY, Poincare published a long admiring 
review, which he later extended as an essay recommending Hilbert for the 
Lobachevsky Prize. The first version is reprinted in his OEUVRES volume 11. 
I believe the other is translated in REAL NUMBERS, GENERALIZATIONS OF THE 
REALS AND THEORIES OF CONTINUA, edited by Philip Ehrlich.

Poincare was impressed by the wide range of geometries Hilbert created in 
the FOUNDATIONS OF GEOMETRY, going beyond the non-Euclidean. He certainly 
did say the book was only on the formal aspect of geometry, but, he wrote: 
"Incomplete, we must all resign ourselves to be. It is enough that through 
this book the philosophy of mathematics has made a considerable advance, 
comparable to those we owe to Lobachevski, to Riemann, to Helmholtz, and to 
Lie" (Poincare's OEUVRES vol.11, page 112).

Poincare stressed a point that Hilbert himself had probably not appreciated 
so well in 1899: that the FOUNDATIONS must use an entirely mechanical, 
formal "logic" to succeed at its project. He compares it to Jevons's 
"logical piano", a machine for calculating syllogistic conclusions. This 
formality, he says, is why Hilbert could discover so many new geometries. 
Yet Poincare also says this mechanical "logic" captures only a part of 
actual logic.

Wiman also wrote:

 >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).

Poincare consistently distinguished Cantor's set theory from "Cantorism". 
The theory was useful and Poincare was among the first to use it. The 
"ism", which he associated with logicists, was to him an odd confusion.

It is important to remember that each of the phrases "actual infinity", 
"completed infinity", and "impredicative definition" has meant very 
different things to different people at different times. Poincare's 
rejection of the "actual infinite" did not prevent him from using Cantor's 
diagonal argument as a philosophic example of the right way to handle 
infinite sets.

 >As we now know, applying Poincare's restrictive principles to set theory
 >would emasculate it, leaving a bland, uninteresting theory.

These "restrictive principles" are not given by Poincare. Poincare has been 
"explicated" to death by later logicians.

He indeed criticised Russell's logicism and Hilbert's logic, up to 
Poincare's death in 1912. Most of his criticisms were explicitly quoted 
from Russell's own critiques, in that period before PRINCIPIA MATHEMATICA. 
His chief difference from Russell was that he, Poincare, believed any 
system of formal logic would require a substantial intuitive explanation in 
ordinary language - what we now call an informal metatheory. Russell sought 
a self-contained, formalized language, usable with no metatheory.

For Poincare logic was emphatically not a part of mathematics. It was the 
laws of thought and was presupposed by mathematics the same as by any 
thought. The first page of "The logic of infinity" insists that the 
ordinary rules of logic not only can be applied to infinite sets, but must. 
Logicists go wrong, he says, when they try to replace this ordinary logic 
with formal systems. (This essay is reprinted in Dover's 1963 MATHEMATICS 
AND SCIENCE, LAST ESSAYS, among other places.)

He clearly regarded his mathematical work as cohering to his philosophy. He 
often cites the work in his philosophic discussions. And the work is indeed 
not Russell's or Couturat's idea of "logistic" (Poincare despised Couturat 
and the feeling was mutual). But it freely uses classical logic on infinite 
sets.

best, Colin