Mark van Atten

On the philosophical development of Kurt Gödel

Godel first advocated the philosophy of Leibniz and then, since 1959, that of Husserl. Based on research in Godel’s archive, from which a number of unpublished items are presented, we argue that (1) Godel turned to Husserl in search of a means to make Leibniz’ monadology scientific and systematic, and (2) This explains Godel’s specific turn to Husserl’s transcendental idealism as opposed to the realism of the earlier Logical Investigations. We then give three examples of concrete influence from Husserl on Godel’s writings.

One Hundred Years of Intuitionism (1907-2007)

With logicism and formalism, intuitionism is one of the main foundations for mathematics proposed in the twentieth century; and since the seventies, notably its views on logic have become important also outside foundational studies, with the development of theoretical computer science. The aim of the book is threefold: to review and complete the historical account of intuitionism; to present recent philosophical work on intuitionism; and to give examples of new technical advances and applications of intuitionism. This volume brings together 21 contributions by todays leading authors on these topics, and surveys the philosophical, logical and mathematical implications of the approach initiated in 1907 in L.E.J. Brouwers dissertation.

The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem

Brouwer’s demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

Monads and Sets: On Gödel, Leibniz, and the Reflection Principle

Godel once offered an argument for the general reflection principle in set theory that took the form of an analogy with Leibniz’ monadology. I discuss the mathematical and philosophical background to Godel’s argument, reconstruct the proposed analogy in detail, and argue that it has no justificatory force. The paper also provides further support for Godel’s idea that the monadology needs to be reconstructed phenomenologically, by showing that the unsupplemented monadology is not able to found mathematics directly.

Gödel and Intuitionism

After a brief survey of Godel’s personal contacts with Brouwer and Heyting, examples are discussed where intuitionistic ideas had a direct influence on Godel’s technical work. Then it is argued that the closest rapprochement of Godel to intuitionism is seen in the development of the Dialectica Interpretation, during which he came to accept the notion of computable functional of finite type as primitive. It is shown that Godel already thought of that possibility in the Princeton lectures on intuitionism of Spring 1941, and evidence is presented that he adopted it in the same year or the next, long before the publication of 1958. Draft material for the revision of the Dialectica paper is discussed in which Godel describes the Dialectica Interpretation as being based on a new intuitionistic insight obtained by applying phenomenology, and also notes that relate the new notion of reductive proof to phenomenology. In an appendix, attention is drawn to notes from the archive according to which Godel anticipated autonomous transfinite progressions when writing his incompleteness paper.

Gödel, Mathematics, and Possible Worlds

Hintikka has claimed that Godel did not believe in possible worlds and that the actualism this induces is the motivation behind his Platonism. I argue that Hintikka is wrong about what Godel believed, and that, moreover, there exists a phenomenological unification of Godel’s Platonism and possible worlds theory. This text was written for a special issue of Axiomathes on the philosophy of Nicolai Hartmann, which explains the two introductory paragraphs.

Construction and Constitution in Mathematics

I argue that Brouwer’s notion of the construction of purely mathematical objects and Husserl’s notion of their constitution by the transcendental subject coincide. Various objections to Brouwer’s intuitionism that have been raised in recent phenomenological literature (by Hill, Rosado Haddock, and Tieszen) are addressed. Then I present objections to Godel’s project of founding classical mathematics on transcendental phenomenology. The problem for that project lies not so much in Husserl’s insistence on the spontaneous character of the constitution of mathematical objects, or in his refusal to allow an appeal to higher minds, as in the combination of these two attitudes.

A Note on Leibniz’s Argument Against Infinite Wholes

The part-whole axiom is also referred to as the ‘Aristotelian principle’ (e.g. Benci et al., 2006) or ‘Euclid’s Axiom’. The former label is justified as it follows from what Aristotle says at Metaphysics 1021a4: ‘That which exceeds, in relation to that which is exceeded, is ‘‘so much’’ plus something more’ (Aristotle 1933, 263); the latter, to the extent that it figures as Common Notion 5 in Book I of Euclid’s Elements from (at the latest) Proclus on (Euclid 1956, 232). See also Leibniz’s ‘Demonstratio Axiomatum Euclidis’ (1679), A VI, 4, 167. This contributes to showing that there is no intrinsic obstacle in Leibniz’s philosophy to combining it with Cantorian set theory, as Kurt Gödel wished to do. For further details on this aspect of Gödel’s thought, which provided the motivation for writing the present note, see van Atten (2009). Neither Gödel’s published papers, nor, as far as I can tell from the currently existing partial transcriptions from Gabelsberger shorthand, his notebooks contain a direct comment on Leibniz’s argument. However, in his paper on Russell from 1944, he wrote: Nor is it self-contradictory that a proper part should be identical (not merely equal) to the whole, as is seen in the case of structures in the abstract sense. The structure of the series of integers, e.g., contains itself as a proper part. (Gödel 1944, 139) Among other things, Gödel says here that it is consistent that an equality relation holds between a proper part and the whole. This entails a rejection of Leibniz’s argument. British Journal for the History of Philosophy 19(1) 2011: 121–129Leibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. The refutation of this argument by Russell and others is equally well known. In this note, I argue (against positions recently defended by Arthur, Breger, and Brown) for the following three claims: (1) Leibniz himself had all the means to devise and accept this refutation; (2) This refutation does not presuppose the consistency of Cantorian set theory; (3) This refutation does not cast doubt on the part-whole axiom. Hence, should there be an obstacle to Godel’s wish to integrate Cantorian set theory within Leibniz’ philosophy, it will not be this famous argument of Leibniz’.

“Gödel’s Modernism: On Set-Theoretic Incompleteness,” Revisited

On Friday, November 15, 1940, Kurt Godel gave a talk on set theory at Brown University. The topic was his recent proof of the consistency of Cantor’s Continuum Hypothesis, henceforth CH, with the axiomatic system for set theory ZFC. His friend from their days in Vienna, Rudolf Carnap, was in the audience, and afterward wrote a note to himself in which he raised a number of questions on incompleteness:

On Gödel's awareness of Skolem's Helsinki lecture

Godel always claimed that he did not know Skolems Helsinki lecture when writing his dissertation. Some questions and doubts have been raised about this claim, in particular on the basis of a library slip showing that he had requested Skolems paper in 1928. It is shown that this library slip does not constitute evidence against Godels claim, and that, on the contrary, the library slip and other archive material actually corroborate what Godel said.Gödel always claimed that he did not know Skolems Helsinki lecture when writing his dissertation. Some questions and doubts have been raised about this claim, in particular on the basis of a library slip showing that he had requested Skolems paper in 1928. It is shown that this library slip does not constitute evidence against Gödels claim, and that, on the contrary, the library slip and other archive material actually corroborate what Gödel said.