[FOM] Papers of Poincare (Friedman and Everdell)

[Colin McLarty]

William R. Everdell notes, correctly, that

 >In 1909 (not 1912) Poincare published "La logique de l'infini" in _Revue
 >de metaphysique et de morale_

On the other hand, in 1912  Poincare published a different "La logique de 
l'infini" in Scientia 12, 1-11. It is translated as "Mathematics and logic" 
65-74. in Poincaré 1963 Mathematics and science: last essays. I have 
mentioned both. I distinguish them as the 1909 "La logique de l'infini" and 
the 1912 "La logique de l'infini".

I can send a fairly full annotated bibliography of Poincare on foundations, 
as a Word doc,  to anyone who e-mails me for it. It compares the original 
publications to the English collections. I have not done what Alasdair 
Urquhart did, by also comparing the first French collections.


Harvey Friedman makes characteristically vigorous points. When I said 
Poincare never suggested any limitation on classical mathematics, he wrote:

>This is in direct contradiction to what Feferman writes in his book
>"In the Light of Logic". Here are some excerpts from that book.

I try to be clear that I disagree with many logicians who interpret 
Poincare along Russell's lines. But the excerpts include no quotes of 
Poincare, only attributions to him, and then never with specific sources 
cited. So I have no new response.

>Do you know of any example of interesting work on "patterns of
>correct actual reasoning" that does not use, or at least cannot be
>made much more interesting, through the use of formal systems?


Poincare did say, the only reason Hilbert's FOUNDATIONS OF GEOMETRY could 
be so creative, and so rigorous,  was that it used purely formal logic. Of 
course today we criticize FOG for not formalizing its logic, but Poincare 
regarded it as purely formal and stresses that this means a machine could 
perform it. (This is in Poincare's review of FOG which is fully cited in my 
article and in earlier e-mails. The most accessible English version is in 
REAL NUMBERS, GENERALIZATIONS OF THE REALS AND THEORIES OF CONTINUA, edited 
by Philip Ehrlich).

As to logistic itself, Poincare said "Russell and Hilbert have each made a 
vigorous effort; they have each written a work full of original views, 
profound and well warranted.... Among their results, some, many even, are 
solid and destined to live" (In English, this is most accessible in the 
book Poincare, FOUNDATIONS OF SCIENCE, p. 470). We know he was sincere 
about this since he tried to give his own definition of "predicativity" as 
well - adopting the terminology from Russell. But I believe he was never so 
specific about it as later logicians have tried to make him.

For Poincare, though, however valuable formalization is, it always 
presupposes informal reasoning. So no formalization can be the foundation 
of mathematics itself.

>If Poincare had sufficient logical insights, he would have seen the
>great importance of formalizing set theory. Of course, most people,
>including Poincare, have their limitations.

His praise of Russell and Hilbert's efforts suggests he did think it worth 
trying. He felt it was being done very badly. He despised Couturat's and 
Burali-Forti's efforts. And he criticized Zermelo's separation axiom as 
vague and arbitrary. Recall that Zermelo did not formalize the axiom and 
rejected Skolem's later idea of doing that. And, while Poincare agreed some 
limitation was needed to avoid paradox, he wanted a more rational account 
of it than Zermelo gave. So did Russell. Citations are in my paper.

Harvey makes a challenging claim about logic, saying that perhaps "reals 
describable in finitely many words" does have some sense not merely 
relative to fixed means of definition. As to the paradox of definability he 
writes:

>Another possible way out of the Paradox might be to deny induction in
>a spectacularly strong way - if it involves arbitrarily abstract
>notions like "define".


This is the heart of the issue, I think. It would be valuable to work this 
out. But for Poincare induction is the basic intuition of  number and he 
would never deny any part of it. He relies entirely on the paradox of 
definability because he believes that all concepts must be treated alike, 
whether they apply to finite sets or infinite. This is explicit in the 1909 
"Logique de l'infini" translated in his book MATHEMATICS AND SCIENCE esp 
pages 45 and 46. And he never dreamed of distinguishing some concepts as 
properly mathematical from others which are semantic or something.

Again, many logicians since Poincare interpret his remarks as implying or 
requiring limitations of classical logic, or on induction, or the least 
upper bound principle. Brouwer certainly did. He thought Poincare *should* 
pose such limitations. Yet Brouwer knew well that Poincare did not agree. 
He knew Poincare never accepted any limitations on classical mathematics.

It raises a very good question, and I think it would help philosophy of 
math if we spent more time on things like this, when Harvey says:

>I am no historian, but I am struck as to how people can read Poincare
>so differently. I would like to understand how this can happen.

To make a start, as I see it: most of us read Russell on Poincare long 
before we read Poincare. I did. I first began reading Poincare's philosophy 
of math, to see if I could relate his "constructivism" to his practice in 
topology. After some long reading I found there is no constructivism in his 
philosophy. Certainly there is none in his practice.

We forget how far most of us are today from agreeing with Russell's 
detailed views on logic. Godel offers a sharp reminder in his essay 
"Russell's mathematical logic", in P. Schilpp ed. The Philosophy of 
Bertrand Russell, Open Court. 123-154. But we still tend to see Russell as 
being on our side, and so his opponent Poincare must be on the other side.

More than that, the constructivist reading of Poincare raises interesting 
issues of logic. Many variant constructivisms are suggested for him and it 
is interesting to explore various senses of "predicativity". This is 
Feferman's strong point. But then, the many variants are possible because 
Poincare himself never adopts one. Valuable logic is not necessarily 
accurate history of Poincare's ideas.

best, Colin
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