[FOM] Did G?del's result come as a surprise to Bertrand Russell?

Irving ianellis at iupui.edu
Tue Mar 30 13:47:00 EDT 2010

Leon Henkin recorded that in response (letter to Bertrand Russell, 
March 26, 1963) to his [Henkin's] request for comments on his 
award-winning article “Are Logic and Mathematics Identical?", Science 
38 (November 1962), pp. 788–794, Russell expressed the thought that 
Gödel's incompleteness theorems mean that 2 + 2 = 4.001 is permissible 
in "school-boy arithmetic". Russell's reply (letter to Leon Henkin, 1 
April, 1963; transcribed in Ivor Grattan-Guinness, The Search for 
Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations 
of Mathematics from Cantor through Russell to Gödel (Princeton/London: 
Princeton University Press, 2000), pp. 592-593) was that Gödel's 
theorem showed, not that (primitive recursive) arithmetic is 
incomplete, but that it is inconsistent, that it permitted "school-boy" 
arithmetic to allow that "2 + 2 = 4.001." This reply (and its "April 
Fool's" date) prompted Henkin to ask me (January 6, 1983, during the 
Special Session on Proof Theory, American Mathematical Society Annual 
Meeting, January 5-9, 1983, Denver Colorado) whether Russell was 
joking; but the entire tenor of the letter, together with the 
philosophical background on which Russell drew to conclude that Gödel’s 
results allowed school-boy arithmeticians to have 2 + 2 = 4.001, shows 
that Russell really was in earnest. In his reply to Russell (letter to 
Bertrand Russell, July 17, 1963), Henkin therefore actually found it 
necessary to explain to Russell that Gödel's results did not say that 
arithmetic is inconsistent, but that it is incomplete.

Logician and Gödel scholar John Dawson, on p. 96 in "The Reception of 
Gödel's Incompleteness Theorems", in Thomas Drucker (editor), 
Perspectives on the History of Mathematical Logic (Boston/Basel/Berlin: 
Birkhäuser, 1991), 84–100, upon examining this episode, wondered 
whether Russell's response reflected Russell's momentary bewilderment 
upon learning of Gödel's theorems, or a continuing "puzzlement". Dawson 
(p. 96) asked whether Russell was saying that “intuitively, he had 
recognized the futility of Hilbert's scheme of proving the consistency 
of arithmetic but had failed to consider the possibility of rigorously 
proving that futility", or if he actually was "revealing a belief that 
Gödel in fact had shown arithmetic to be inconsistent," and he notes 
that Henkin, for one, assumed that Russell supposed Gödel to have 
proven the inconsistency of arithmetic. Dawson (pp. 96-97) adds [that, 
in either case, Gödel eventually received a copy of Russell's letter 
and consequently remarked to Abraham Robinson on 2 July 1973 that 
"Russell evidently misinterprets my result; however he does so in a 
very interesting manner
." Another interpretation of Russell’s reaction 
to Henkin could arise from consideration of the fact that of course 
Gödel’s results do not assert merely the impracticality of obtaining a 
proof of the decidability of a theorem, but that it is theoretically 
impossible to find such a proof, so that the effect of Gödel's work was 
to deflate the sails of Russell's claims for logicism. Grattan-Guinness 
(op. cit., p. 593) renders an even more austere judgment, declaring not 
only that Russell "misunderstood Gödel's [incompleteness] theorem," but 
that "Russell was still struggling with the theorem at the end of his 
life when he wrote an addendum (pp. xviii-xix) to his replies for a new 
edition (1971) for B. Russell, "My Mental Development", in Paul Arthur 
Schilpp (editor), The Philosophy of Bertrand Russell (Evanston/Chicago: 
Northwestern University Press, 1944; 1971), pp. 3–20. A contrary 
account is given by Francisco A. Rodríguez-Consuegra, "Russell, Gödel 
and Logicism", in Philosophy of Mathematics (Kirchberg am Wechsel, 
1992), Schriftenriehe Wittgenstein-Gesellschaft 20, nr. I (Vienna: 
Hölder-Pichler-Tempsky, 1993), pp. 233–242, who asserts that Russell 
did fully and accurately understand Gödel incompleteness.

This of course does not, however, directly or explicitly answer the 
specific question of whether Gödel's result come as a surprise to 
Bertrand Russell, but in the absence of additional evidence and in 
light of his self-professed report that he did not pay attention to 
work in logic after writing the Introduction to Mathematical Philosophy 
(1919), we are perforce led to presume that he either was unaware of 
these results until reading Henkin's article, or at least professed to 
be. The main thrust of Dawson's paper on "The Reception of Gödel's 
Incompleteness Theorems" is that logicians who were cognizant of 
publications in logic during the period were not particularly surprised 
by Gödel's results. Dawson also noted that Gödel in fact briefly 
announced his results at a philosophical conference in Königsberg in 
September 1930 at which Carnap, Von Neumann, and Heyting presented 
their foundational philosophies of mathematics, prior to their 
publication in 1931 in the Monatshefte für Mathematik und Physik, and 
did a preliminary publication of his results in "Einige 
metamathematische Resultate über Entscheidungsdefinitheit und 
Widerspruchsfreiheit", Anzeiger der Akademie der Wissenschaften in Wien 
67 (1930), 214–215; English translation by Stefan Bauer-Mengelberg as 
"Some Metamathematical Results on Completeness and Consistency", in 
Jean van Heijenoort (ed.), From Frege to Gödel: A Source Book in 
Mathematical Logic, 1879–1931 (Cambridge, Mass.: Harvard University 
Press, 1967), 595-596; reprinted, German and English on facing pages: 
K. Gödel, Collected Works, vol. I: Publications 1929–1936 (Oxford: 
Clarendon Press/New York: Oxford University Press, 1986), 140–143.

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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