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

[Colin McLarty]

Thanks to Lucas Wiman for some interesting questions here.

I think the main point to make is: Poincare never suggests any limitation 
on classical mathematics. Indeed he faults logicism for offering to limit 
it in some way. When Russell conjectured a few solutions to the paradoxes 
of set theory, and said it would take time to see how much of mathematics 
could be preserved by the best solution, Poincare noted that real math was 
not at stake. Rather logicism was at stake. By real math he meant the kind 
of thing he did, in his research publications, which was in no way 
constructive or limited by "predicativity" (these passages are abbreviated 
on p. 480 of THE FOUNDATIONS OF SCIENCE, cited with the originals in my paper).

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".)


As to whether logic for Poincare was a part of mathematics

>Yes--that is precisely the point I tried to make that logic (as it is 
>commonly understood *now*--in Poincare's time, the correct term seemed to 
>be logistic) is a part of mathematics--not a foundation for it.
>Poincare repeatedly (and correctly) stated that any logic claiming to give 
>a foundation for mathematics would have to already include fundamental 
>mathematical concepts of arithmetic.

Probably we do not disagree in substance. But I would say that logic then 
and now is commonly understood to be the patterns of correct actual 
reasoning. Anyway that is what Poincare meant by logic. It is not the study 
of formal deductive systems. Formal deduction systems take their value from 
their depiction of the correct informal patterns.

> >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.
>
>This is not the case.  Perhaps he did distinguish between them, but if so, 
>he did not distinguish consistently.


I did overstate the consistency. Poincare's terminology changes. But he 
consistently supported informal set theory and derided logicist efforts to 
formalize it. Of course Russell also rejected every formalization that 
Poincare lived to see and Poincare most often based his criticisms on 
Russell's.  As I say in my paper, Poincare's objections are also close to 
what Godel would say much later about PRINCIPIA MATHEMATICA.

>Even after reading your paper, it is difficult for me to see exactly how 
>it is that philosophers have misinterpreted him.  There are real numbers 
>(in the arithmetic account of the continuum of which Poincare seemed fond) 
>that cannot be described in a finite number of words--these numbers, it 
>seems to me, would have bothered Poincare (and very well might have).


Your statement on describable reals is too hasty, and Poincare knew it. 
Asking whether a real number is "described in a finite number of words" 
only makes sense relative to fixed means of definition. By diagonalizing on 
those means, we get new explicit means of definition that let us define new 
reals in finitely many words. Poincare liked this familiar paradox a great 
deal and made it an argument against taking any formalization as 
foundation, in his 1909 "logique de l' infini" (translated in MATHEMATICS 
AND SCIENCE under the title "logic of infinity"), and elsewhere.

Russell had thought such things bothered Poincare. Specifically he tried to 
confound Poincare by the paradox of "the smallest integer which cannot be 
defined by fewer than one hundred English words". Poincare happily said, 
look, this only shows that the word "defined" is ambiguous. You have to say 
"defined by such-and-so means". And the paradox itself shows that no such 
means are complete. So there is no hope of formalizing all mathematics.


>Given a sequence of real numbers (which, as I understand Poincare, would 
>be a rule-based sequence), it is possible to construct a real number to 
>which it converges, so he certainly wouldn't have had a problem with 
>that.  However, he probably would have disliked the general least upper 
>bound principle.


Poincare considered the least upper bound principle the key intuition of 
continuity and he accepted it in full generality. That is why no one can 
quote him offering any restrictions. For him every bounded sequence of 
reals has an l.u.b. and not only the sequences defined in some 
formalization or given by some rule. Of course everyone agrees that to 
actually "give" a sequence you have to give some rule for it. Intuition, 
for Poincare, always goes beyond any formalization and this is his basic 
argument against logicism.

Poincare certainly says

>"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." (From Mathematics and Science: 
>Last Essays; quoted in Chihara: Ontology and the vicious-circle principle).

This is explicit: you can know a genus without knowing all its members. We 
can even speak about those of its members that do not "exist" yet because 
we have not defined them individually. As a specific example, Poincare uses 
the full power set of the reals, in practice and in his philosophy, while 
not all of its members are "given" at any one time. In terms of the quote 
above: For Poincare, the whole genus of bounded sequences of reals has 
least upper bounds, not only the the ones we have "given" so far.


>It's not entirely clear to me how Poincare would have responded to the 
>idea that the reals are defined impredicatively in the arithmetic 
>formulation that he so loved.  Perhaps he would have rejected the 
>arithmetic formulation of the continuum, or perhaps his injunction against 
>impredicative arguments (however one interprets that).  Do you know of 
>anywhere where Poincare directly dealt with this?


He talks specifically about people who object to the standard arithmetic 
continuum as non-constructive. He says these people "forget that possible, 
in the language of geometers, simply means free from contradiction". (p.44 
of his book FOUNDATIONS OF SCIENCE. Fuller citation is in my paper.) He 
defends the geometers on this and insists they have the true definition of 
the continuum.

And that is what he says more abstractly in the passage you quote from 
Chihara's book. We know the arithmetic continuum even though not all its 
members are given at any one time. Each impredicatively defined real "will 
exist after it has been constructed" and we can "construct as many of them 
as we wish".

Poincare took that to mean, precisely, that we can freely do analysis by 
all classical means, since the reals we need will exist as soon as we need 
them. They "exist by virtue of their definitions". Throughout his career 
Poincare insisted "What does the word exist mean in mathematics? It means, 
I say, to be free from contradiction."

This principle was central to his work in non-Euclidean geometry, and 
especially to his philosophy of geometry. In fact, the first clue that 
logicians have misinterpreted Poincare is that they so rarely mention his 
views on geometry, while nearly all his mathematics and far the greatest 
part of his philosophy is on geometry. The arithmetic views they attribute 
to him, are nowhere in his writings.

Brouwer himself knew that Poincare was no kind of constructivist, and 
complained about it. He thought, based on his reading of Poincare on 
intuition, that Poincare *should* be a constructivist. Chihara agrees with 
this, as do many logicians today. But Brouwer also sharply faulted Poincare 
for sounding just like Russell on the question of mathematical existence 
(as quoted in my paper) while logicians today generally do not know what 
Poincare actually said. They show that he should be a constructivist by 
their standards, and conclude that he was. Goldfarb's paper refers 
extensively to Poincare's works. But unfortunately he gives few page 
citations or direct quotes.