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**Essays on Gödel’s reception of Leibniz, Husserl, and Brouwer.**
*(English)*
Zbl 1436.03014

Logic, Epistemology, and the Unity of Science 35. Cham: Springer (ISBN 978-3-319-10030-2/hbk; 978-3-319-10031-9/ebook). xiv, 328 p. (2015).

The author presents in this volume a collection of essays on Kurt Gödel’s work, authored or co-authored by him. His aim is to analyse historical and systematic aspects of Gödel’s philosophical program, consisting of two stages: “1. Use Husserl’s transcendental phenomenology to reconstruct and develop Leibniz’ monadology into an axiomatic metaphysics, 2. Apply the metaphysics thus obtained to develop a Platonistic foundation for classical mathematics” (p. 2).

The material is organized in four parts. After an “Introduction” (pp.1–20) which discusses “Gödel’s commitment to phenomenology from about 1959 to the end of his life, the religious component in phenomenology, and the pragmatic value of Husserl’s and Gödel’s historical turns in philosophy” (p.1), Part I “Gödel and Leibniz” contains three essay. In “A note on Leibniz’s argument against infinite wholes” (pp.23–32), the author argues that Leibniz had “all the means to device and accept” (p.24) B.Russell’s refutation of Leibniz’s argument against the existence of infinite wholes based on the part-whole axiom. The author shows furthermore that this refutation does not presuppose the consistency of Cantorian set theory and it does not concern the part-whole axiom as such. In “Monads and sets: on Gödel, Leibniz, and the reflection principle” (pp.33–64), the author relates Leibniz’s monadology to set theory as Gödel did, and discusses the role of Gödel’s version of the reflection principle “A structural property, possibly involving \(V\), which applies only to elements of \(V\), determines a set; or, a subclass of \(V\) thus definable is a set” (p.41). In “Gödel’s Dialectica interpretation and Leibniz” (pp.65–74), the author reports on a shorthand note by Gödel on Leibniz that can be found among material related to the translation and revision of his Dialectica paper of 1968. It shows that Leibniz’s theory of truth served as a source of inspiration for Gödel.

Part II “Gödel and Husserl” contains four essays. The paper “Phenomenology of mathematics” (pp.77–94) is intended as an introduction to the phenomenology of mathematics. The relation between phenomenology and mathematics is discussed in several respects: mathematics as motivating factor for Husserl in developing phenomenology, the phenomenology of mathematics, and transcendental phenomenology as foundation of mathematics. The author’s examples are intuitionistic logic, choice sequences, the bar theorem, Hilbert’s program, and Gödel’s Dialectica interpretation. In the long and important joint paper with Juliette Kennedy, “On the philosophical development of Kurt Gödel” (pp.95–145) Gödel’s turn from Leibniz’s methodological monadology to Husserl’s transcendental idealism which became manifest in 1959 is discussed. The context of this turn is given by sketching Gödel’s philosophical position in the 1950s, in particular concerning his realism and his conviction of “epistemological parity”, i.e., “the idea that, regarding physical objects on the one hand and abstract or mathematical objects on the other, from the point of view of what we know about them, there is no reason to be more (or less) committed to the existence of one than of the other” (p.103). Gödel’s turn to Husserl’s transcendental idealism is described, his criticism towards aspects of this position mentioned. The conceptions of Husserl and Leibniz are compared, and finally Husserl’s influence on Gödel’s writings is shown. In “Gödel, mathematics, and possible worlds” (pp.147–155), the author argues against J.Hintikka’s claim that Gödel did not belief in possible worlds. The paper “Two draft letters from Gödel on self-knowledge of reason” (pp.157–162) gives transcriptions with comments of two draft letters to Time Inc.and to P.Tillich from the 1960s which can be found in the Gödel archives. In the comment, the question of how knowledge of consistency can be gained is central: “This is through self-knowledge (or reflection), in particular reflection which leads to knowledge of essential properties of reason” (p.161).

Part III “Gödel and Brouwer” contains three papers. In “Gödel and Brouwer: two rivalling brothers” (pp.165–171), the different attempts of Gödel and L. E. J. Brouwer to found mathematics are related to each other although their views are connected to different kinds of mathematics. Gödel’s views are bound to classical mathematics, whereas Brouwer’s philosophy leads to intuitionistic mathematics. In “Mysticism and mathematics: Brouwer, Gödel, and the common core thesis” (pp.173–187), the author parallels Brouwer’s and Gödel’s different conceptions of some absolute instances and compares them. “For Gödel, doing mathematics is a way of accessing the absolute.” For Brouwer, doing mathematics precisely prohibits access to the absolute (p.182). The paper “Gödel and intuitionism” (pp.189–234) discusses Gödel’s position towards intuitionism. It documents the relation between Gödel, Brouwer and Brouwer’s follower A.Heyting. Using material from the Gödel papers the author shows that intuitionistic ideas inspired Gödel’s work in respect to the incompleteness theorem, weak counterexamples, intuitionistic logic as modal logic, and the continuity argument in set theory. Special emphasis is laid on the context of the Dialectica interpretation and its revision, representing Gödel’s conception of constructibility and “by far the closest rapprochement of Gödel to intuitionism” (p.196).

In Part IV “A partial assessment” in the final paper “Construction and constitution in mathematics” (pp.237–310), the author argues that “L. E. J. Brouwer’s notion of the construction of purely mathematical objects and Edmund Husserl’s notion of their constitution coincide” (p.237), claiming that “Brouwer’s intuitionistic mathematics should be considered part of Husserl’s transcendental-phenomenological foundations of pure mathematics” (p.237), and that “transcendental phenomenology cannot provide a foundation of pure mathematics that would go beyond intuitionism” (p.238). He also presents objections to Gödel’s project of showing that “Husserl’s phenomenology can provide a foundation also for nonconstructive, classical mathematics” (p.268).

In sum, this impressive volume represents the author’s excellent scholarship in the history and philosophy of mathematics. The relations in the philosophical triangle of Gödel, Husserl, Brouwer are drawn in a historically sensible way, using material from the Gödel papers, but also regarding systematic issues, still under discussion today.

The material is organized in four parts. After an “Introduction” (pp.1–20) which discusses “Gödel’s commitment to phenomenology from about 1959 to the end of his life, the religious component in phenomenology, and the pragmatic value of Husserl’s and Gödel’s historical turns in philosophy” (p.1), Part I “Gödel and Leibniz” contains three essay. In “A note on Leibniz’s argument against infinite wholes” (pp.23–32), the author argues that Leibniz had “all the means to device and accept” (p.24) B.Russell’s refutation of Leibniz’s argument against the existence of infinite wholes based on the part-whole axiom. The author shows furthermore that this refutation does not presuppose the consistency of Cantorian set theory and it does not concern the part-whole axiom as such. In “Monads and sets: on Gödel, Leibniz, and the reflection principle” (pp.33–64), the author relates Leibniz’s monadology to set theory as Gödel did, and discusses the role of Gödel’s version of the reflection principle “A structural property, possibly involving \(V\), which applies only to elements of \(V\), determines a set; or, a subclass of \(V\) thus definable is a set” (p.41). In “Gödel’s Dialectica interpretation and Leibniz” (pp.65–74), the author reports on a shorthand note by Gödel on Leibniz that can be found among material related to the translation and revision of his Dialectica paper of 1968. It shows that Leibniz’s theory of truth served as a source of inspiration for Gödel.

Part II “Gödel and Husserl” contains four essays. The paper “Phenomenology of mathematics” (pp.77–94) is intended as an introduction to the phenomenology of mathematics. The relation between phenomenology and mathematics is discussed in several respects: mathematics as motivating factor for Husserl in developing phenomenology, the phenomenology of mathematics, and transcendental phenomenology as foundation of mathematics. The author’s examples are intuitionistic logic, choice sequences, the bar theorem, Hilbert’s program, and Gödel’s Dialectica interpretation. In the long and important joint paper with Juliette Kennedy, “On the philosophical development of Kurt Gödel” (pp.95–145) Gödel’s turn from Leibniz’s methodological monadology to Husserl’s transcendental idealism which became manifest in 1959 is discussed. The context of this turn is given by sketching Gödel’s philosophical position in the 1950s, in particular concerning his realism and his conviction of “epistemological parity”, i.e., “the idea that, regarding physical objects on the one hand and abstract or mathematical objects on the other, from the point of view of what we know about them, there is no reason to be more (or less) committed to the existence of one than of the other” (p.103). Gödel’s turn to Husserl’s transcendental idealism is described, his criticism towards aspects of this position mentioned. The conceptions of Husserl and Leibniz are compared, and finally Husserl’s influence on Gödel’s writings is shown. In “Gödel, mathematics, and possible worlds” (pp.147–155), the author argues against J.Hintikka’s claim that Gödel did not belief in possible worlds. The paper “Two draft letters from Gödel on self-knowledge of reason” (pp.157–162) gives transcriptions with comments of two draft letters to Time Inc.and to P.Tillich from the 1960s which can be found in the Gödel archives. In the comment, the question of how knowledge of consistency can be gained is central: “This is through self-knowledge (or reflection), in particular reflection which leads to knowledge of essential properties of reason” (p.161).

Part III “Gödel and Brouwer” contains three papers. In “Gödel and Brouwer: two rivalling brothers” (pp.165–171), the different attempts of Gödel and L. E. J. Brouwer to found mathematics are related to each other although their views are connected to different kinds of mathematics. Gödel’s views are bound to classical mathematics, whereas Brouwer’s philosophy leads to intuitionistic mathematics. In “Mysticism and mathematics: Brouwer, Gödel, and the common core thesis” (pp.173–187), the author parallels Brouwer’s and Gödel’s different conceptions of some absolute instances and compares them. “For Gödel, doing mathematics is a way of accessing the absolute.” For Brouwer, doing mathematics precisely prohibits access to the absolute (p.182). The paper “Gödel and intuitionism” (pp.189–234) discusses Gödel’s position towards intuitionism. It documents the relation between Gödel, Brouwer and Brouwer’s follower A.Heyting. Using material from the Gödel papers the author shows that intuitionistic ideas inspired Gödel’s work in respect to the incompleteness theorem, weak counterexamples, intuitionistic logic as modal logic, and the continuity argument in set theory. Special emphasis is laid on the context of the Dialectica interpretation and its revision, representing Gödel’s conception of constructibility and “by far the closest rapprochement of Gödel to intuitionism” (p.196).

In Part IV “A partial assessment” in the final paper “Construction and constitution in mathematics” (pp.237–310), the author argues that “L. E. J. Brouwer’s notion of the construction of purely mathematical objects and Edmund Husserl’s notion of their constitution coincide” (p.237), claiming that “Brouwer’s intuitionistic mathematics should be considered part of Husserl’s transcendental-phenomenological foundations of pure mathematics” (p.237), and that “transcendental phenomenology cannot provide a foundation of pure mathematics that would go beyond intuitionism” (p.238). He also presents objections to Gödel’s project of showing that “Husserl’s phenomenology can provide a foundation also for nonconstructive, classical mathematics” (p.268).

In sum, this impressive volume represents the author’s excellent scholarship in the history and philosophy of mathematics. The relations in the philosophical triangle of Gödel, Husserl, Brouwer are drawn in a historically sensible way, using material from the Gödel papers, but also regarding systematic issues, still under discussion today.

Reviewer: Volker Peckhaus (Paderborn)

### MSC:

03-06 | Proceedings, conferences, collections, etc. pertaining to mathematical logic and foundations |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03-03 | History of mathematical logic and foundations |

01A60 | History of mathematics in the 20th century |

00A30 | Philosophy of mathematics |

03A05 | Philosophical and critical aspects of logic and foundations |