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
1. The Vicious Circle Principle
2.The Inductive Nature of Principia Mathematica
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
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.
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.
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.
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
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.
- 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.