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

[Lucas Wiman]

Colin,

 >Yet Poincare also says this mechanical "logic" captures only a part of
 >actual logic.

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.

 >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.  In his essay "The Future of 
Mathematics" in Science and Method, Poincare has a section on 
"Cantorism."  He states that its services to mathematics are well 
known.  Not Cantor's work, but Cantorism itself.  (This may be an 
artifact of bad translation; I don't have access to the original).  If 
what you are saying is true, then it seems to me that this makes no 
sense.  Cantorism, according to your definition, had not served 
science--it was a perversion of something which had.  Poincare seemed to 
feel that Cantor's set theory was only meaningful when applied to 
clearly defined circumstances.  The examples you cite in your paper are 
sets of points and curves, which seem to accord to this thesis.  Cantor 
had a much more general notion of set in mind.  (Also, why name an "ism" 
after someone whose ideas were perverted into the ism.  That seems to 
make no sense.)

 >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.

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).  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.  The quote you give in your paper on pp. 
106 seems to quite clearly show this:

"The word all has a clear meaning when applied to a finite number of 
objects; for it still to have one when the objects are infinite in 
number would require that there be an actual infinity.  Otherwise all 
the objects cannot be conceived as given prior to their definition and 
if the definition of a notion N depends on all the objects A, it might 
be spoiled by a vicious circle, if some of the objects A cannot be 
defined without using the notion N itself."

And also:

"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 priniple).

If we try to quantify over all sets of reals, certainly some of them 
will not even be definable without reference to the power set of the 
reals itself.  Thus we would be stuck in a circle (though I wouldn't 
consider it vicious).  Thus the least upper bound principle would only 
make sense to Poincare if it were restated in terms of definable sets of 
reals.

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?

 >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.

Cantor's diagonal argument is a perfectly constructive result; I don't 
see why its acceptance has anything to do with the admission of an 
actual infinite into mathematics.

 >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.

Yes, this much is clear.  When I mentioned logic before, I was referring 
to its modern usage, which Poincare would almost certainly consider a 
part of mathematics.  He made it quite clear that in order to study 
these formal systems, one needs to assume a background mathematical 
reasoning ability (what you call "informal metatheory"), and that any 
attempts to justify mathematics based upon formal systems would fail.

- Lucas Wiman