[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
URL: http://www.irvinganellis.info

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