Logic, Epistemology, and the Unity of Science
Reviewed by: Prof. Dr. Manuel Bremer (University of Düsseldorf)
]Van Atten’s book presents itself as an investigation on Gödel’s philosophy of mathematics.
Gödel is almost considered a cult figure in popular science accounts of the history of logic in the 20th century. His reclusive lifestyle and mental disorder, leading, at its worst stages to self-starvation, have certainly added to the picture. Therefore one could think people might be interested in his greater philosophy of mathematics beyond his more technical papers. And this would prove enticing to any academic philosopher of mathematics, as one might suspect that such an important logician would provide critical insights to this field. Expanding on Gödel’s ideas on philosophy of mathematics could thus open new perspectives, notably through his emphasis on forms of Platonism and mathematical ‘intuition’. Unfortunately, to this date, no essay publications of Gödel in relation to philosophy of mathematics are noted. Neither did he produce a finished manuscript, but his notes, discussions with peers, and some of his published papers left a number of insights on the philosophy of mathematics. Van Atten considers these pieces as evidence on which he bases himself to provide foundations of a reading of Gödel’s philosophy of mathematics. As they are sparse and underdeveloped van Atten will tend to interpret the same short remark or thrown in sentence over and over again in the book. Pieces of the puzzle are added by reference to memories collected through people’s recollections of conversation with Gödel, and to passages (e.g. in Husserl’s books) that Gödel read and might have used in in own thought process.
The book is divided in three parts, each bearing the name of the philosophers Gödel reflected upon: Leibniz, Husserl, and Brouwer. The book consists mostly of previously published papers by van Atten, thus one might be faced with considerable overlap and repetition. Moreover, it must be noted that Gödel’s formal work is not discussed in detail.
In the first part of the book, Gödel explores Leibniz’s account of monads a metaphysics resembling his own ideas. The focus is twofold here: on the one hand, he considers the idea of objectivity being guaranteed by concepts in God’s mind, and on the other hand the idea of reflection. Reflection as a principle of set theory roughly argues that if some structural condition is true for sets, then it is also true in a part of the set theoretical universe (i.e. one can have an example without access to the whole set theoretical universe). The importance of the principle also resides in its equivalence to the Axiom of Replacement, which is needed to guarantee the fact that sets of higher infinite cardinality exist. Following this thought, Gödel toys with an analogy to Leibniz’s idea of reflection between the single monad and the relations within the universe of monads, taken as an argument one cannot proceed from Leibniz’ metaphysics to any specific statement of mathematical structural truth, as van Atten shows. (Gödel also showed interest in Leibniz’s notion of reductive proof, which inspired more than one logician as a model of building definitions or proofs on a sound basis.)
Moreover, Gödel considered Leibniz’s account of the subjective consciousness of the monad and its access to knowledge as underdeveloped, and it is here that he turned (in the 1950s) to Husserl. Husserl’s phenomenology provides, as Gödel at least remarked to several people, an approach capable of solving the problem of intuitive access to mathematical (categorical) entities. Phenomenological descriptions may elucidate how we intuit concepts like ‘possibility’. Indeed, Husserl’s notion of intuition fits better to Gödel’s agenda than Kant’s notion of intuition and its role in mathematics, since Husserl aims at the essence of intuition, a form of intuition shared by any mind. If a phenomenology of this type succeeded, it would intuit mathematical objects as they are given in the mind of God, and existence in His mind guarantees – this being Gödel’s tenet – their objectivity. Following Husserl’s claim on categorical intuition of individual mathematical entities, Gödel is focussed on our intuitive grasp of concepts. He considers the example of the concept of ‘powerset’. If our grasp of this concept of powerset can be secured, then we have secured all its applications, especially its role in generating non countable infinities and the Continuum. Gödel thus aims at justifying the axioms (of set theory), an approach that we find today, for instance, in George Boolos’ work on the iterative hierarchy.
In distinction to his attempt at appropriating parts of Leibniz’ and Husserl’s philosophy, Brouwer provides Gödel with a challenge to his view in the philosophy of mathematics. They express contrasting ideas of mathematical reality and the very worth of mathematics, which Brouwer at times derided as aberration of pure subjective thought, whereas Gödel revered mathematics as our access to the absolute. Both share some mystical and illuminational tendencies.
An essay on Gödel’s ‘Dialectica-interpretation’ of intuitionism (so called because it appeared in the journal Dialectica) is at the centre of this part of the book, bringing together Gödel’s reflection on intuitionism and his approval of relying on some form of (phenomenological) intuition of basic concepts. The interpretation is founded on the concept of a ‘computable function of finite type’ that extends in elucidation (i.e. not fixed in a formalism or mechanical algorithm). Our grasp of this concept is taken to be revealed by a priori psychology (this being Brouwer’s intuitionism in Gödel’s eyes) or something resembling phenomenological psychology. Given this foundation Gödel and Brouwer share a rejection of a mechanization of mind (ala Turing), but Gödel, of course, claims our grasp of further concepts, way beyond what basic computable functions are. Even Gödel’s reading and interpretation of intuitionism are not the intended ones by Brouwer. Gödel substitutes his notion of ‘reductive proof’ (going back to definitions, somewhat in the way of Leibniz) for the intuitionist’s general reference to ‘proofs’, taken by Brouwer to be based in individual mental acts.
Thus, in the main part of the book, we learn how Gödel dealt with parts of Leibniz’ and Husserl’s philosophy, and how he tried to partially reconcile or deal with Brouwer’s intuitionism as an alternative philosophy of mathematics. This would belong to an intellectual biography of Gödel, more than to an academic essay setting out any new contribution to phenomenology by Gödel. Here, no new arguments in Gödel’s philosophy of mathematics are exposed, beyond the known desiderata. Gödel praises phenomenology and hints at the discipline in his reflection on intuitionism, but detailed phenomenological analyses are missing. Furthermore, his reference to Leibniz reads more as an analogy than as new foundational argument.
After reading this part, one can understand Gödel’s somewhat surprising turn to Husserl. That even Gödel – typically associated with modern formal logic and part-time member of the Vienna Circle – could not make substantial progress, from Husserl to the philosophical foundations of set theory, may justify that one should not expect further contributions to a realist philosophy of mathematics from that direction.
The last part of the book features a systematic essay in which van Atten defends Brouwer against Gödel, on Husserl’s ground. Indeed, Brouwer and Husserl share many of their foundational thoughts and some of Brouwer’s claims can best be understood within Husserl’s phenomenology. This applies also to specific theses, for example, that of the restriction of mathematics to the potentially infinite only.
If van Atten is right on this, and he sets out a strong case for it, then the combination of Gödel’s ideas and phenomenology was nonetheless still born. If Husserl and Brouwer see mathematical objects as constructions, it only limits their approach to some form of constructive mathematics in the end, and then, unfortunately, this would mean that Gödel’s turn to Husserl must have been in vain.
Students of Gödel may thus find interconnections between Gödel’s scattered remarks on the philosophy of mathematics in van Atten’s book, but no unified Gödelian philosophy of mathematics. This could have been put forth in a longer comprehensive essay, in itself much shorter than the present collection of essays.