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

[Harvey Friedman]

Response to McLarty 4/10/03 1:36PM.

>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.

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

1. page ix. "...This is a semiconstructive philosophy, going back to
ideas of Henri Poincare in the early part of the twentieth century,
whose point and programmatic development beginning with that of Weyl
(mentioned above) are not nearly so well known as strictly
constructive programs such as Brouwer's intuitionism and those of
more modern schools of that character...."

2. page 52. "The vicious circle principle, first enunciated by
Poincare, was designed to block certain purported definitions, in
which the objects introduced  somehow defined in terms of itself.
According to Poincare, all mathematical objects beyond the natural
numbers are to be introduced by explicit definitions. But a
definition which refers to a presumed totality - of which the object
being defined is itself to be a member - involves one in an apparent
circle, since the object then itself ultimately a constituent of its
own definition. Such "definitions" are called impredicative, while
proper definitions are called predicative; put in more positive
terms, in the latter one only refers to totalities which are
established prior to the object being defined. ...Poincare raised his
ban on impredicative definition thinking doing so would exclude the
paradoxes....However, that succeeds only by taking a very broad
reading of what it means for a definition to refer to a
totality....On the other hand, Poincare's principle as it stands
would certainly exclude impredicative definitions in analysis...; the
objection to such definitions is not that they are paradoxical, but
rather that they are implicitly circular and hence not
proper....Russell was one of the first to accept Poincare's ban on
impredicative definitions as applied to sets,..."

3. page 254. "In Das Kontinuum, Weyl to a large extent accepted Henri
Poincare's definitionist philosophy of mathematics, which featured
the following general ideas: (i) the conception of the natural
numbers sequence is an irreducible minimum for (abstract)
mathematical thought, and the associated principles of proof and
deifnition by induction are to be accepted as basic; ii) all other
mathematical concepts, in particular those of sets and functions, are
to be introduced by explicit definition; iii) there are no completed
infinite totalities; and iv) great care must ge taken to exclude
apparent definitions which seem to single out an object from a
presumed completed totality of objects, by essential reference
(either explicit or implicit) to that totality. Poincare did not
elaborate these ideas in detail; in particular. he was silent about
the choice of underlying symbolism for definitions and of underlying
logic for proofs. ...Improper (apparent) definitions of an object A
which essentially assume the existence of a competed infinite
totality S containing A as member are said to be impredicative, while
proper ones are said to be predicative. Thus Poincare's definitionist
philosophy of mathematics is also called that of predicativity. The
idea iv) was embodied in Poincare's viscous circle principle, which
banned the use of impredicative definitions, and thence of the
assumption of totalities S underlying such definitions. Poincare was
led to this principle by his analysis of the paradoxes, which he
viewed as implicitly admitting such totalities as the set S of all
sets;..."

3. page 266. "Here we find him (Weyl) reiterating his rejection of
the Dedekind-Cantor-Zermelo (philosophically platonistic) program of
set-theoretic foundations, and his agreement with Poincare on the
opposed need for definitionist (predicative) foundations, with the
vicious circle principle as a crucial test for acceptability of
proper definition principles..."

There are other relevant quotes, but this is enough.

>... 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.

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?

>...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.

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.

>
>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.

That's the party line, but perhaps that's not true. Perhaps one can
make sense of it.

>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.

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".

In any case, using this to dismiss the great importance of
formalization is a great mistake. One could still have a
formalization that is complete for "normal mathematical purposes".

>
>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.

Again, that conclusion is misleading. E.g., you can get out of it by
denying that discussions of the meaning of arbitrary one hundred
length sentences of English words is mathematics.

>
>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.


Feferman obviously takes a different view: that Poincare rejected the
least upper bound principle.

>
>
>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 view attributed to Poincare is on the face of it incoherent. To
put coherence into this would require careful use of formalization!

>
>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.

Feferman cites

Poincare, 1913, Dernieres pensees (Flammarion, Paris); English
translation in Poincare (1963).

Poincare, 1963, Mathematics and Science: Last essays. English
translation of Poincare (1913),Dover press, New York.

>
>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.


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.