Did Russell understand Gödel's incompleteness theorems?

Question

Russell was active in philosophy (although no longer in math) for many years after the Gödel's 1931 publication. Gödel's paper were not obscure, and Russell would have been aware of their effect on the Principia and his logicism (and Hilbert's formalism). Logicomix (a partially fictionalized accounts of Russell's life) and common sense suggest that Russell would have grasped Gödel's theorem and its drastic effect to his philosophy. On the other hand writers such as Hofstadter (in "I am a Strange Loop") suggest that Russell never understood Gödel's theorem: going as far as to rudely compare Russell to a dog staring blankly at a TV screen.

Is there any writing of Russell's thoughts on Gödel's incompleteness theorem? Is there any reliable historic/biographic source on Russell's understanding of Gödel? Did Russell understand Gödel's incompleteness theorems?

Answer 1

After a bit of searching, I found some promising leads (and quite a few consistent descriptions) which suggest that Russell thought Gödel's results were of cardinal importance, but misunderstood their implications. In particular, he thought that Gödel's result essentially entailed that Peano Arithmetic was inconsistent rather than incomplete; but also realized that this is not something which Gödel was likely to be claiming.

I realized, of course, that Godel’s work is of fundamental importance, but I was puzzled by it. It made me glad that I was no longer working at mathematical logic. If a given set of axioms leads to a contradiction, it is clear that at least one of the axioms must be false. Does this apply to school-boy's arithmetic, and, if so, can we believe anything that we were taught in our youth? Are we to think that 2 + 2 is not 4, but 4.001? Obviously, this is not what is intended.

(From Russel, Gödel, and Logicism.) It would be interesting to have a more complete record of how Russell came to this misconception: was it that something was being established as a Theorem, whose content was to establish as true a statement which was provably not a theorem (of another formal system)? Of course, Russell may not have exponded at much length why he interpreted the Incompleteness Theorem as he did; he had, after all, stopped working in mathematical logic. As a prodigious writer and an obvious person to ask about Gödel's results, it does seem plausible that my cursory search has revealed only the top tenth of the iceberg. This hypothesis is supported by the record of Gödel's reaction to Russell's reaction to his Incompleteness Theorem:

Russell evidently misinterprets my result; however he does so in a very interesting manner.

(From Information and Randomness: An Algorithmic Perspective.) Perhaps Gödel simply found Russell's confused concern a refreshing change of reaction from that of others'; perhaps he said this to heighten the contrast against Wittgenstein's reaction to the Incompleteness Theorem (which was trivializing, but what more should one expect from someone who considers set theory akin to a childhood disease?); or perhaps Gödel was simply being polite to an elder statesman. But if he genuinely found Russell's reaction interesting, this would suggest a meatier misinterpretation.

Answer 2

The following is from a late paper of Russell's titled "Logical Positivism". It can be found in "Logic and Knowledge"

It appeared that, given any language, it must have a certain incompleteness, in the sense that there are things to be said about the language which cannot be said in the language. This is connected with the paradoxes - the liar, the class of classes that are not members of themselves, etc. These paradoxes had appeared to me to demand a hierarchy of 'logical types' for their solution, and the doctrine of hierarchy of languages belongs to the same order of ideas. For example, if I say 'all sentences in the language L are either true or false', this is not a sentence in the language L. It is possible, as Carnap has shown, to construct a language in which many of the things about the language can be said, but never all the things that might be said: some of them will always belong to the 'metalanguage'. For example, there is mathematics, but however mathematics may be defined, there will be statements about mathematics which will belong to 'metamathematics', and must be excluded from mathematics on pain of contradiction.
There has been a vast technical development of logic, logical syntax, and semantics. In this subject, Carnap has done the most work. Tarski's "Der Begriff der Wahrheit in den formalisierten Sprachen" is a very important book, and if it compared with the attempts of philosophers in the past to define 'truth' it shows the increase of power derived from a wholly modern technique. Not that difficulties are at an end. A new set of puzzles has resulted from the work of Godel, especially his article "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (1931), in which he proved that in any formal system it is possible to construct sentences of which the truth or falsehood cannot be decided within the system. Here again we are faced with the essential necessity of a hierarchy, extending upwards ad infinitum, and logically incapable of completion.

It looks like here he came a little closer to understanding it, but still pretty far off. He seems to be confusing Turing's decidability, the Tarski definability theorem, and incompleteness into one homogeneous lump. His statement of Gödel's theorem is either trivially false or interestingly true depending on what he means by "decidable in a formal system": the man does have a knack for statements which skirt the line between the two.

Answer 3

Russell's comments on Gödel were scanty, but it was very unlikely that Russell did not understand what Gödel was talking about. The paradox presented by Gödel sentence was nothing new; it was the same old vicious circle paradox, which had been abundantly dispelled by Russell's Theory of Types[source 1]. Russell discovered the Theory of Types in 1906. The Theory of Types provided no shelter for vicious circles.[source 3] On the other hand, Gödel's raising this paradox anew in 1931 indicated that Gödel probably never understood Russell's Theory of Types.

In Russell's Theory of Types, meaning is fundamental. A self-referential sentence G's meaning cannot be determined until each of its constituent's meaning is determined; one of G's constituent is G itself, thus G's meaning cannot be determined because G contains a vicious circle.

No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the "whole theory of types").

Wittegenstein, Tractatus 3.332

Gödel, from formalist's point of view, regarded symbols in PM as meaningless empty signs[source 9], and, by means of numbering, Gödel managed to show that G belonged to the body of propositions for which PM was supposed to be a foundation - this was the point of contention: by Russell's type theory, G had no meanings and thus did not belong to the body of propositions: all self-referential sentences were specifically weeded out by theory of types as nonsensical. Gödel's attack on Principia was similar to planting a bag of weed in one's roommate’s car then accusing him of illegal possession of drugs, or to accusing one's room-mate, who was actually speaking another language on the phone, of making threats about the President of the United States

The root of the problem was formalists' disregard for meanings. In Russell's 1937's "Introduction To The Second Edition" of The Principles of Mathematics, Russell categorically dismissed formalists:

The formalists are like a watchmaker who is so absorbed in making his watches look pretty that he has forgotten their purpose of telling the time, and has therefore omitted to insert any works.

Was Gödel's work below contempt? There are, in this world, lunatics, crooks and dupes. The crooks are below contempt; the lunatics and dupes are not. Russell was very generous; he lavished praises on Wittgenstein and Ramsey who ruthlessly attacked his Principia but at the same time unmistakingly demonstrated their understanding of his theory of types. On the other hand, nothing in Gödel's work indicated that Gödel even had a clue about Russell's theory of types. Although Russell was eager to say something nice, there was really not much he could say; towards fledging philosophers, Russell was tender and caring and was very careful to avoid raising his voices.[source 6] Russell actually came face to face with Gödel in early 1940s'; the following excerpt was Russell's recount of his encounter with Gödel:

While in Princeton, I came to know Einstein fairly well. I used to go to his house once a week to discuss with him and Gödel and Pauli. These discussions were in some ways disappointing, for, although all three of them were Jews and exiles and, in intention, cosmopolitans, I found that they all had a German bias towards metaphysics, and in spite of our utmost endeavours we never arrived at common premises from which to argue. Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal 'not' was laid up in heaven where virtuous logicians might hope to meet it in hereafter.

Russell, Bertrand. "America. 1938-1944." Autobiography. 1967. London and New York: Routledge, 2000. Print. 466.

Gödel's theorems were probably more devastating to formalists because they showed what absurdities were admissible in formalism. Nevertheless, it is possible that PM is incomplete or inconsistent, but proving complete and proving consistent were not PM's concerns. Both completeness and consistency involve all, exact what propositions constitute all is the point of contention.[source 4] PM did not aim at all; it aimed at deducing arithmetics, which was the starting point of ordinary mathematics; PM was on target, and PM made several serendipitous discoveries along the way. Regarding consistency, all that PM could say was probably something like this: "as of today, no contraction has been discovered within PM" - more than that was beyond what inductive reasoning could warrant [source 2]. By deducing arithmetics from logical principles, PM demonstrated that mathematics and logic are identical - this thesis, first proposed in his 1900's Principles, Russell had never seen any reason to modify.[source 5]

If Gödel sentence were true, PM would be inconsistent because, in virtue of PM ✳2.02, which says a true proposition is implied by any proposition, PM implies G. Any body of premises that can deduce G is not a valid foundation because it contains a contradiction; an inconsistent body of premises contains false premises, and a false premise implies any conclusion (PM ✳2.21) - that is why Russell asked "are we to think that 2 + 2 is not 4, but 4.001?" Then again, if principles in PM were asserted, Gödel sentence implicated no one because it was either nonsense or non-contradiction by the Theory of Types.[source 3]

Can you know that a proposition is true before it is proven? Yes, you can. But those propositions are what Wittgenstein called tautologies, none of which is Gödel sentence G. Basically, tautologies are different ways of saying the same thing. Tautology

Like all theories whose justifications are inductive, PM by its nature should be tentative, subject to revisions based new evidence; the second edition of PM demonstrated Russell's attitude more than proved a point: Russell learned a lesson from Newton-Leibniz quarrel, and he had no desire to take the place of Aristotle to establish himself as a towering authority - Russell actually took pains to avoid playing the authority role.7[8]

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Sources: 1. The Vicious Circle Principle

2.The Inductive Nature of Principia Mathematica

3. If Gödel sentence is taken literally, it is nonsense; if it is interpreted in type theory, it is a simultaneous assertion of multiple statements - this is what Russell means by "Here again we are faced with the essential necessity of a hierarchy, extending upwards ad infinitum, and logically incapable of completion. " See Liar's paradox. Note that every time I point to the Liar's paradox, people automatically say I mistook true for provable. Actually, this distinction is irrelevant; what the liar's paradox and G in common is that they are all self-referential. If a sentence can't comment itself, then it is commenting its counterpart one order below itself, thus a hierarchy rises from the second order ad infinitum. There is no first order G, because a proposition about a proposition is at least 2nd order. First order propositions are all about individuals, not propositions. Theory of Logical Types

4. If consistency means there is no contradiction, then one needs to examine all propositions, then again what goes into all? Russell somewhere said Whitehead and himself believed it was impossible to prove that a formal system was consistent.

5.

The foundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify. Russell, Bertrand. Introduction to the second Edition. Second paragraph. Principles of Mathematics, 1937.

6. Russell in several writings mentioned that extremely intelligent people are also emotional unstable; instead of calling for "mental toughness" he advocated separating sensitive children from the crowd. One source I can tell for sure is Education and the good Life. Russell was definitely aware of the fragile mental state in philosophical community. This awareness permeates almost all of his non-technical writings.

7. Somewhere Russell blamed Newton of retarding British Mathematics for 150 years, and he did not know how far British mathematics had fall behind Germany until he visited the US. I can't come up with the source off the top my head, but somewhere he definitely said something like this

8. Russell tried to make do without Axiom of Reducibility in the 2nd. Ramsey accused the AOR of being a matter of brute fact, not a tautology (see Ramey's Foundations of Mathematics); Russell admitted that AOR lacked self-evidence, and was willing to show what it was like without AOR in the 2nd.

9. The following interpretation of PM by formalists is the major misunderstanding of PM by formalists:

The symbols of PM are, however, fully devoid of meanings in the sense that derivation of theorems depends only on following the formal rules of PM.

Source: Nagel & Newman. Gödel's Proof. New York and London: New York University Press, 2001. Print 71.

To have a sense of how the Theory of Types works, take for example the sentence "a set is not a member of itself"; this sentence is neither true nor false; it is meaningless and thus has no membership in the body of propositions for which PM is supposed to be a foundation. Obviously sentences like this are not only permissible to formalists, who do not consider meaning as a guarding criterion that bars the entrance of nonsense, they are even fundamental to ZFC.

Answer 4

As mentioned in a comment, Alasdair Urquhart has written a paper, Russell and Gödel (Bull. Symb. Logic 22 (2016), 504–520), that discusses a number of different topics, including Russell’s view of Gödel’s results. He provides many of the Russell quotes that other respondents here have given, as well as the following quote from an “Addendum” that was written in 1965 but published only posthumously in 1971, in the fourth edition of The Philosophy of Bertrand Russell.

Not long after the appearance of Principia Mathematica, Gödel propounded a new difficulty. He proved that, in any systematic logical language, there are propositions which can be stated, but cannot be either proved or disproved. This has been taken by many (not, I think, by Gödel) as a fatal objection to mathematical logic in the form which I and others had given to it. I have never been able to adopt this view. It is maintained by those who hold this view that no systematic logical theory can be true of everything. Oddly enough, they never apply this opinion to elementary everyday arithmetic. Until they do so, I consider that they may be ignored. I had always supposed that there are propositions in mathematical logic which can be stated, but neither proved nor disproved. Two of these had a fairly prominent place in Principia Mathematica—namely, the axiom of choice and the axiom of infinity. To many mathematical logicians, however, the destructive influence of Gödel’s work appears much greater than it does to me and has been thought to require a great restriction in the scope of mathematical logic. … I adhere to the view that one should make the best set of axioms that one can think of and believe in it unless and until actual contradictions appear.

This quote seems to demonstrate a reasonably good understanding of what Gödel proved. On the other hand, as already noted, there are other remarks by Russell that seem to misunderstand Gödel. Urquhart concludes, “In the end, it is probably impossible to interpret Russell’s comments on the incompleteness theorem in a fully consistent way. His remarks combine correct summaries of Gödel’s work with what appear to be quite confused and muddled ideas.”