Synopsis of Kurt Gödel, ‘La logique mathématique de Russell’
[‘Russell’s Mathematical Logic’]
Like Bertrand Russell, the Austrian mathematical logician Kurt Gödel (1906-78) played a decisive role in twentieth-century debates on the foundations of mathematics.1 Gödel’s article on Russell is ‘notable’, as Charles Parsons observes, as his ‘first and most extended philosophical statement’ and as ‘perhaps the most robust defence of realism about mathematics and its objects’ to be made in the first half of the twentieth century.2. It was originally published in English in a 1944 collection, The Philosophy of Bertrand Russell. It was translated by Jacques-Alain Miller and Jean-Claude Milner for volume 10 of the Cahiers under the title ‘La logique mathématique de Russell’. The English original was later included in Paul Benacerraf and Hilary Putnam’s widely used reader, The Philosophy of Mathematics.
In order to understand ‘Russell’s Mathematical Logic’ it is helpful to have some sense of the history of Russell’s work and its context. The turn of the nineteenth and twentieth centuries witnessed a crisis in the foundations of mathematics. Frege had hoped to place mathematics on an epistemologically secure foundation by demonstrating that the truths of mathematics can be derived, i.e. proved, from purely logical assumptions: a project known as logicism. It is important to be clear that what Frege and his immediate followers meant by ‘logic’ is considerably broader than what contemporary philosophers have in mind when they use that term. In particular, their expanded conception of logic included significant amounts of mathematical set theory; Gödel invokes this conception when he explains, in his opening paragraph, that mathematical logic treats of ‘classes, relations, and combinations of symbols, etc.’ Frege’s logicism failed because it could not avoid a fundamental paradox. For Frege, any concept whatsoever had a collection of objects (i.e. a ‘class’) associated with it, such that any object that ‘fell under’ the concept in question was a member of the class. Russell’s paradox, discovered by Russell in 1901, asks us to consider the class of all classes that are not members of themselves. Is it a member of itself? If it is a member of itself, then it isn’t, and if it isn’t a member of itself then it is. Either way, we have reached a contradiction. As Gödel puts it here, Russell’s paradox served to bring ‘to light the amazing fact that our logical intuitions (i.e., intuitions concerning such notions as: truth, concept, being, class, etc.) are self-contradictory’ (CpA 10.5:89; translation, 452).
Russell’s paradox, along with the discovery of other set theoretic paradoxes, provoked a flurry of activity, as logicians and mathematicians sought to re-secure the foundations of their discipline. Ultimately, this period of crisis gave birth to modern axiomatic set theory. Russell’s own most significant contribution to the debate was the three volume Principia Mathematica (1910-1913)3, co-authored with the English mathematician Alfred North Whitehead; most of Gödel’s attention here is focused on this project.
Gödel begins by explicitly locating Russell’s work within a understanding of logic as foundational for human thought. Logic ‘contains the ideas and principles underlying all other sciences’ (84/ 447). Gödel expresses his regret at the lack of a ‘precise statement of the syntax of the formalism’ (85/448), as developed in Russell’s Principia. The syntax of a formal system consists of the symbolic vocabulary, rules for using that vocabulary to construct formulae, and rules for deriving formulae from other formulae. Gödel is especially concerned here about the status of ‘incomplete symbols’. An incomplete symbol is, for Russell, one which does not have significance in and of itself, but acquires it only in combination with other symbols. An example from natural language, which we will discuss in a moment, is a definite description of the type, ‘the present King of France’.
After criticising Russell’s treatment of syntax, Gödel moves on to a more positive philosophical engagement with Russell’s mathematical logic. The ‘realistic attitude’ which Gödel recognises in Russell is one which sees logic as being concerned with the world, rather than simply with human knowledge or with formal systems. Logic is a science like any other, distinguished from the likes of zoology only by its generality. This much is uncontroversial exegesis of Russell’s earlier work. However, Gödel’s goes on to identify a further similarity between logic and mathematics on the one hand and the natural sciences on the other. Drawing on an early piece of Russell’s, Gödel refers to what he calls logical ‘evidence’, i.e. evidence that plays a justificatory role with respect to logical and mathematical axioms similar to that played by sensory data in justifying our beliefs in scientific laws. Thus ‘axioms need not be evident in themselves, but rather their justification lies (exactly as in physics) in the fact that they make it possible for these “sense perceptions” to be deduced; which of course would not exclude that they have a kind of intrinsic plausibility similar to that in physics’ (86/449). Gödel does not expand here on the nature of such ‘evidence’, but his emphasis on the continuity between logic/mathematics and the natural sciences has proved influential in subsequent philosophy of mathematics (for instance in Penelope Maddy’s work on the sorts of justification at issue in acceptance of set theoretic axioms).
Perhaps Russell’s most well-known logical enterprise is his analysis of ‘definite descriptions’ – typically, noun phrases beginning with the word ‘the’. This first found expression in his 1905 essay ‘On Denoting’, and was part of the work leading up to Principia; a more accessible exposition is to be found in Russell’s Introduction to Mathematical Philosophy (1920). The theory of descriptions allowed Russell some metaphysical leeway, in that it opened the door to making sense of the suggestion that there are meaningful parts of language which fail to latch on to any existing object (‘the present king of France’, ‘the largest prime number’). Gödel’s engagement with the theory here is mainly expository in nature. Noting that Frege had thought the sole function of phrases such as ‘the author of Waverley’ was to signify objects (in this case, Walter Scott), Gödel outlines Russell’s motivation for his theory by showing that Frege’s doctrine has the unhappy conclusion that the proposition ‘Scott is the author of Waverley’ is identical in content to ‘Scott is Scott’. This conclusion sits comfortably with Frege’s curious doctrine that all true sentences refer to an object known as ‘the True’. Russell responded to this worry by proposing that definite descriptions are incomplete symbols, that sentences of the form ‘the F is G’ mean the following three things in combination:
(1) There is at least one thing which is F.
AND (2) There is at most one thing which is F.
AND (3) Anything which is F is also G.
Clauses (1) and (2) together imply that there is exactly one thing which is F. On Russell’s theory, for example, ‘the present king of France is bald’ turns out to be false, since there is no present king of France, falsifying (1). A further implication of the theory is that negative existential statements, such as ‘the present king of France does not exist’ can be given a satisfactory analysis: the sentence means ‘there is no thing x, such that x is a present king of France’.
Gödel then moves on to consider Russell’s own response to the paradox that bears his name, and to other set theoretic paradoxes. Two families of responses to the paradoxes can be identified in the history of mathematical logic. What Gödel terms the ‘zig-zag’ response finds its most well-known expression in Quine’s ‘New Foundations’ (1937).4 The alternative and more widely accepted approach is based on the ‘limitation of size’ principle: it diagnoses the problem with Frege’s account as being that it admits collections which are somehow too big (90/452-453).5 In the Principia, as Gödel observes, Russell does not pursue either response, but rather embraces a ‘no class theory of classes’. There are no classes in reality, but rather our talk of classes in mathematics is just a convenient ‘logical fiction’, a ‘façon de parler’.
Rather than explore the details of Russell’s no class theory, Gödel opts to examine the ‘general logical principles’ underlying the Principia’s response to the paradoxes, in particular the so-called ‘vicious circle principle’. This
forbids a certain kind of ‘circularity’ which is made responsible for the paradoxes. The fallacy in these, so it is contended, consists in the circumstance that one defines (or tacitly assumes) totalities, whose existence would entail the existence of certain new elements of the same totality, namely elements definable only in terms of the whole totality. This led to the formulation of a principle which says that no totality can contain members definable only in terms of this totality, or members involving or presupposing this totality (91/454).
What Gödel has to say about the vicious circle principle (VCP) is the most enduringly important and influential component of his paper. He first makes a subtle distinction. What is at stake in the idea of a member ‘involving’ or ‘presupposing’ a totality?
Corresponding to the phrases ‘definable only in terms of,’ ‘involving,’ and ‘presupposing,’ we have really three different principles, the second and third being much more plausible than the first. It is the first form which is of particular interest, because only this one makes impredicative definitions impossible and thereby destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of modern mathematics itself (92/455).
An impredicative definition is one in which an object is defined with reference to a totality of which that object is itself a member; an example would be the definition of the least upper bound of a set of real numbers.6 Classical mathematics makes frequent use of impredicative definitions. These definitions, however, had been the object of attack by mathematical constructivists (notably Henri Poincaré, who believed that mathematical objects do not exist independently of the constructive activity of the mathematician. Gödel links the objection to impredicative definitions to the first form of VCP and goes on to defend impredicative definitions in terms of mathematical realism.
A mathematical realist in Gödel’s sense will assume that duly accepted theorems of mathematics are true, and are true in virtue of the existence of mathematical objects independent of human thought or language. If we are realists – and Gödel, on the basis of his insistence on the continuity between mathematics and natural science, believes that we have every reason to be as much realists about the posits of mathematics as about those of physics – then impredicative definitions present no special problem. The first form of the VCP applies only if the entities in question are ‘constructed by ourselves’. Gödel continues, in an often-cited passage,
In this case there must clearly exist a definition (namely the description of the construction) which does not refer to a totality to which the object defined belongs, because the construction of a thing can certainly not be based on a totality of things to which the thing to be constructed itself belongs. If, however, it is a question of objects that exist independently of our constructions, there is nothing in the least absurd in the existence of totalities containing members which can be described (i.e. uniquely characterized) only by reference to this totality (93/456).
Gödel contrasts the kind of realism which rejects the first form of VCP with a constructivist position, and includes Russell’s no class theory within the latter camp. This points to a tension in the Principia, since classical mathematics (which makes use of impredicative definitions) can be built up on the basis laid out in the Principia, yet – according to Gödel – the philosophical presuppositions of the Principia motivate a strong form of VCP which disallows impredicative definitions. Gödel himself prefers to conclude ‘that the vicious circle principle is false [rather] than classical mathematics is false’ (93/455).
The next section of Gödel’s paper (96-104/460-467) deals with Russell’s attempts at formulating a theory of types. Type theory seeks to avoid VCP by proposing that everything that exists is partitioned into logical types. Individuals (you, me, a hydrogen atom, the table etc.) are of type 0, classes of individuals are of type 1, classes of these classes are of type 2, and so on. There are prohibitions (‘type restrictions’) on classes containing members from different types. Gödel’s discussion of this topic is relatively technical.7 He show that the constructivist ‘no-class theory’ (adopted after some hesitation in the second edition of Principia) cannot provide concepts and classes with ‘all the properties required for their use in mathematics’. He defends his own approach to the domain of constructible sets as an extension of the hierarchy of orders (presented within the framework of classical, i.e. impredicative mathematics) to the domain of the transfinite. He also offers a partial defence of the theory of simple types (a theory which adds certain simplifying symbolic conventions to the theory of types) as independent of the ‘constructive standpoint’, and thus as a ‘stepping stone for a more satisfactory system’ (103/465). Given the assumption of ‘meaningfulness’ which accompanies it, this theory of simple types may allows us to ‘assume every concept to be significant everywhere except for certain “singular points” or “limiting points”, so that the paradoxes [of set theory] would appear as something analogous to dividing by zero’ (104/466). (Jacques-Alain Miller, in his 1967 article ‘U ou “Il n’y a pas de méta-langage”’, refers to Gödel’s speculation here [leaving aside the question of Gödel’s own understanding of the implications of his work] in order to describe the ‘aberrant’ or ‘utopic points’ at which conscious discourses and knowledges are interrupted by the ‘unique’ and ‘ultimate’ language of the unconscious).8
The last issue considered by Gödel is crucial to an assessment of whether the Principia is a successful execution of the logicist project. Are the axioms of the Principia analytic? Use of the term ‘analytic’ along these lines dates back to Kant; here it serves to describe statements that have truth-value in virtue of meaning (rather than in virtue of the extralinguistic world). Analytic truths, it might be thought, have a good claim to be logical, and thus be admissible within a mathematical system for which pure logicality is claimed. Moreover, in the mid-1940s, when Gödel wrote this article, the notion of analyticity was at the foreground of Anglo-American philosophy, thanks to the influence of logical positivism. This movement, a form of empiricism, called into question the meaningfulness of statements which were not susceptible to empirical investigation but made an exception for analytic truths (‘tautologies’), on the assumption that mathematical truth might be understood as analytic.
In his concluding pages, Gödel explores two senses of analyticity. In the first sense, the statements of the Principia would be analytic were it the case that ‘the terms occurring can be defined [...] in such a way that the axioms and theorems become special cases of the law of identity’, the law x = x. Given some minimal assumptions, settled results of mathematical logic show that the Principia’s axioms and theorems cannot be analytic in this first sense. A second sense of analyticity holds simply that analytic truths are true ‘owing to the meaning of the concepts occurring in’ them. The Principia scores better on this sense, at least on some interpretations of the concepts it utilises. Unfortunately, those concepts are indistinct, as is evidenced by the paradoxes. In the years following the publication of Gödel’s article, the very suggestion that any truth might be true in virtue of meaning fell into disrepute in the wake of Quine’s work (1951). More recently, however, the so-called neo-Fregeans have sought to revive the suggestion that mathematical truths possess something like analyticity in Gödel’s second sense of the term.
References to this text in other articles in the Cahiers pour l’Analyse:
- Gödel, Kurt. ‘Russell’s Mathematical Logic’. In The Philosophy of Bertrand Russell, ed. Paul A. Schilpp. New York: Tudor, 1944. 125-153. Reprinted in Philosophy of Mathematics: Selected Readings, ed. Paul Benacerraf and Hilary Putnam. Second edition. Cambridge: Cambridge University Press., 1983. 447-469. English page numbers here refer to the Benacerraf and Putnam edition.
- Gödel, Kurt. Collected Works, ed. Soloman Feferman. 2 vols. Oxford: Oxford University Press, 1986, 1990.
- Quine, Willard V.O. ‘New Foundations for Mathematical Logic’. American Mathematical Monthly 44 (1937): 70-80.
- Quine, Willard V.O. ‘Two Dogmas of Empiricism’. Philosophical Review 60 (1951): 20-43.
- Quine, Willard V.O ‘On Denoting’. Mind 15 (1905): 479-493.
- Quine, Willard V.O Introduction to Mathematical Philosophy. London: Allen & Unwin, 1920.
- Russell, Bertrand, and Alfred North Whitehead. Principia Mathematica. 3 vols. Cambridge: Cambridge University Press. 1910, 1912, 1913.
Selected secondary sources:
- Enderton, Herbert B. Elements of Set Theory. Academic Press: Burlington, MA. 1977.
- Forster, Thomas. Set Theory with a Universal Set. Oxford: Clarendon Press, 1992.
- Hofstader, Douglas. Godel, Escher, Bach : An Eternal Golden Braid. London: Harvester, 1979.
- Maddy, Penelope. ‘Believing the Axioms I’. Journal of Symbolic Logic 53:2 (1988): 488-511.
- Maddy, Penelope. Realism in Mathematics. Oxford: Oxford University Press, 1990.
- Miller, Jacques-Alain. Un Début dans la vie. Paris: Gallimard, 2002.
- Nagel, Ernest and Newman, James. Gödels Proof. New York: NYU Press. 2001
- Neale, Stephen. Descriptions. Cambridge, MA.: MIT Press, 1990.
- Parsons, Charles. ‘Introductory Note to 1944’. In Kurt Gödel, Collected Works Volume II, Publications 1938-1974, ed. Solomon Feferman. Oxford: Oxford University Press, 1990. 103-118.
- Roitman, Judith. Introduction to Modern Set Theory. New York: Wiley, 1990.
- Shapiro, Stewart. Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press, 1997.
- Shapiro, Stewart. Thinking About Mathematics. Oxford: Oxford University Press, 2000
1. Gödel’s celebrated incompleteness theorems, published in 1931, state that any formalised deductive system (or at least one rich enough to express basic arithmetic) will contain undecidable statements, and that no such system can demonstrate or prove its own consistency.Gödel’s incompleteness theorems are too technical to discuss in any detail here but in brief, he established his result by using number theory effectively to ‘talk about’ number theory. Gödel made number theory ‘introspective’, to use Hofstadter’s phrase, by assigning ‘numbers’ (so-called ‘Gödel numbers’) to the various parts of number theory itself (Hofstadter, Gödel, Escher, Bach, 17-19). He then demonstrated that it is possible to formulate, in a kind of re-working of Epimenides’ paradox, an obviously true statement in number theory that number theory itself cannot confirm – such a statement effectively asserts its own unprovability (Gödel, ‘On Formally Undecidable Propositions’, Collected Works, i, 149). In short, Gödel’s result establishes that ‘there are questions of mathematics that mathematics cannot answer; one of these undecidable questions is whether set theory itself is consistent’ (Roitman, Introduction to Modern Set Theory, 32). For more information, see ‘Kurt Gödel’, Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/goedel/; ‘Gödel’s Incompleteness Theorems’, Wikipedia, http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems ↵
2. Parsons, ‘Note on 1944’, 103-104. ‘The essay as a whole might be seen as a defence of these [fundamentally realist] attitudes of Russell against the reductionism prominent in his philosophy and implicit in much of his actual logical work’ (104) ↵
3. For more information see ‘Principia Mathematica’, Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/principia-mathematica/ ↵
5. See in particular Michael Hallett, Cantorian Set Theory and Limitation of Size (Oxford: Oxford University Press, 1984). ↵
6. For more information see the section on ‘Ramified Hierarchy and Impredicative Principles’, in ‘Type Theory’, Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/type-theory/; and ‘Impredicativity’, Wikipedia, http://en.wikipedia.org/wiki/Impredicativity. On realism and impredicativity, see Shapiro, Philosophy of Mathematics, chapter 1. ↵
7. See Shapiro’s Thinking About Mathematics, chapter 5, section 2; Parsons provides a more involved discussion in his ‘Introductory Note to 1944’, pp. 110-115. ↵
8. Gödel’s ‘zero is Lacan’s subject, I would say, and U [the unique language] is made up only of singular points [...]. I imagine the following: the Freudian rule has no other function than to introduce the subject to the U dimension. An analysis is nothing more than a crossing of the unique language’ (Jacques-Alain Miller, ‘U ou “Il n’y a pas de méta-langage”’ [first published in Ornicar? in 1975] in Miller, Un Début dans la vie, 134-135). ↵