[FOM] : Question of an old paper in Set Theory
Irving
ianellis at iupui.edu
Tue Oct 26 04:36:59 EDT 2010
In regard to A. Mani's question of an old paper in set theory
The reference is to a paper by Philip Edward Bertrand Jourdain
(1879-1919). From the historical (rather than technical point at issue,
the paper by Jourdain to which Mani refers, and which was a point of
acrimonious contentious disagreement between Jourdain and Russell on
the Axiom of Choice, was received only in draft manuscript notes by
Gosta [Magnus Gustav] Mittag-Leffler (1846-1927), and published in
Mittag-Leffler's Acta Mathematica as "A Proof that Every Aggregate Can
Be Well-Ordered" (in Acta Mathematica 43 (1922), 239-261). It was
published by Mittag-Leffler as a courtesy to a recently deceased
colleague who had been seriously ill.
The proof that Jourdain presented in "A Proof that Every Aggregate Can
Be Well-Ordered" was indeed based upon an error, one which Cantor
himself had made in working out a similar proof several decades
previous, and which he had since recognized as flawed and had,
consequently, discarded (see Gregory H. Moore, Zermelo's Axiom of
Choice: Its Origins, Development, and Influence (New
York/Heidelberg/Berlin: Springer-Verlag 1982, pp. 188–192) for an
account of the history of Jourdain's "proof"). Mittag-Leffler was
convinced that Jourdain's paper was seriously flawed, but he published
it nonetheless, as a goodwill gesture, evidently, to Jourdain's memory.
Hence Mittag-Leffler's editorial footnote (p. 239n.): "The undersigned
does not accept the principal wiew [sic] on which is based the above
paper of the regretted, highly esteemed mathematician Philip B.
Jourdain, which paper seems to be the last one written by him. But it
contains so many points of wiew [sic] that I have thought I would do
the mathematical Public a service by publishing it."
Jourdain had written the paper in question purporting to contain a new
proof of the well-ordering principle (specifically, Zermelo's axiom or
the multiplicative axiom, i.e. the axiom of choice) that Jourdain sent
to Russell for proofreading and comment, Russell evidently refused to
check Jourdain's proof, making it conditional upon a payment from The
Monist chief editor Paul Calvin Carus (1852–1919) for "The Philosophy
of Logical Atomism". This led Jourdain to tell Russell, in a letter of
24 May 1919 (as quoted by Grattan-Guinness at pp. 149-150 of his
“Russell and Philip Jourdain: A Study of Their Relationship”, Russell
(o.s.) no. 8 (Winter 1972-73), pp. 7–12) that:
<<Your suggestion that you will examine my proof when the Monist pays
you strikes me as unworthy. It is practically asking me to bother Mrs.
Carus specially for you in order that I may get some approval from you
of what I have written. But you seem to have overlooked another aspect
of the case: by refusing to look at my proof you are deliberately
shutting your eyes to truth.
...It is a pity to treat this principle as if it would be doing me a
personal favour to examine it. Personal questions are surely out of
place in such a matter. But since you have thought fit to plead poverty
as a reason for not doing what you seem to think is a personal favour I
may perhaps mention that rather more than five years ago when you
wanted some money badly I had to take the advancing of £100 to you as a
responsibility on myself, since at first Carus would not contemplate
paying £100 for the Lowell lectures [Our Knowledge of the External
World. I did this out of a sense of duty or friendship although I could
not afford it: my total earnings were £200 a year and my total private
income £12.>>
Russell apparently replied hastily to this accusatory missive from
Jourdain; he must in the final analysis have told Jourdain that he had
sought to “spare” Jourdain as long as possible from the news that his
proof was flawed, since Jourdain next wrote to Russell on 27 May (as
quoted by Ivor Grattan-Guinness, Dear Russell - Dear Jourdain: A
Commentary on Russell's Logic, based on His Correspondence with Philip
Jourdain (New York: Columbia University Press and London: Duckworth,
1977), p. 150) apologizing for the accusation that Russell had refused
to read the proof until being paid by Carus, and adding that he did not
object to anyone thinking the proof invalid. John Edensor Littlewood
(1885-1977) also found Jourdain’s proof to be invalid (see p. 129 of
(Béla Bollobás, editor), Littlewood's Miscellany (Cambridge/London/New
York: Cambridge University Press, 1986; revised ed., first published in
1953 as A Mathematician’s Miscellany), as evidently, did Whitehead,
for, in a letter of 11 March 1919 to Russell, Jourdain complained that
neither Russell nor Whitehead had gotten the main point of his proof;
nevertheless, Russell's shabby manner of treating Jourdain and
Jourdain's proof culminated with Jourdain's distrust and hostility
towards Russell and embittered his final days when Russell ignored his
[Jourdain’s] desperate plea for Russell to visit him on his deathbed to
discuss the proof (see Grattan-Guinness, Dear Russell - Dear Jourdain,
pp. 151-153). According to Dora Russell (Bertrand Russell's wife from
1921 to 1935), Littlewood and Russell debated whether to go to Jourdain
and whether, to spare a dying man from being told his proof was correct
when they held that it was incorrect, to tell him whether they thought
it was or was not correct, that, in the event, Littlewood went as a
joint representative of himself and Russell, to "carry greetings and
what comfort could be given from both of them" (as quoted at p. 20 in
Grattan-Guinness, "Russell and Jourdain: An Exchange", Russell (o.s.)
no. 9 (Spring 1973), pp. 20-21).
Regarding this episode, Littlewood wrote (p. 129):
<<I had to go away for a painful two days at P. E. B. Jourdain's
death-bed. Jourdain had for some time thought he had a proof of the
Multiplicative Axiom (or Axiom of Choice), and would have died happy if
it were accepted. When discussing with Russell the difficulty of
dealing with this, I rashly opened my mouth with a suggestion, and in
the end it was I (and I think Dorothy Wrinch) who paid the visit.
Jourdain expounded his latest version verbally, and I took the line
that there was a new point involved which I should have to think over
carefully, for say a day. Jourdain instantly said: "My dear man, you
know perfectly well you can tell in 10 minutes whether a proof is right
or wrong." [Here Littlewood adds in a footnote that Jourdain “became
rhetorical and emotional at this fallacious point."] I brazened it out
-- only thing to do -- and I suppose all went as well as one could
hope. It was no good just lying (and finding a mistake if after all he
did recover) because of his wife. It turned out she didn't care a damn
about his intellectual life.>>
(Excerpted from pp. 199-200 of my Evaluating Russell (2006).)
Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info
More information about the FOM
mailing list