- After Gödel: Platonism and Rationalism in Mathematics and Logic
Tieszen’s new book offers a synthesis and extension of his longstanding project of bringing the method of Husserl’s phenomenology to bear on fundamental questions—both epistemological and ontological—in the philosophy of mathematics. Gödel held Husserl’s philosophy in high regard, and thought that it offered the most promising avenue for an adequate understanding of the foundations of mathematical thought and practice, but Gödel himself did not give a full systematic statement of how phenomenology might carry out that project. Tieszen takes up the task in this book, which is structured around two main guiding themes. The first is to locate the sources of Gödel’s philosophical views—principally [End Page 303] in Plato, Leibniz, and of course Husserl, along with Kant as a sort of backcloth—and the second is to argue that from these materials, a promising rationalism in mathematics and logic can be consistently articulated. Along the way we also encounter a good many of Gödel’s arguments against an array of reductive positions, Hilbert’s formalism and Carnap’s “logical syntax” project among them, as well as arguments against the prospects of a mechanical or purely computational understanding of the human mind. Many of the arguments (both negative and positive) stem from a reflection on the consequences of Gödel’s landmark incompleteness results of 1931.
The core of Tieszen’s positive proposals comes in chapters 4, 5, and 6: here is where we meet with “constituted Platonism,” Tieszen’s name for the rationalist, realist synthesis of Husserl’s method and Gödel’s mathematical insights. Methodologically, the starting point is the actual practice of mathematics as a human undertaking, and its conditions of possibility. At the most general level, it is characteristic of thought to be intentional, that is, to be (or try to be) about something, to have a target, to be a bearer of meaning. So it goes for thought about items in sensory awareness, about fictional objects, or about mathematical subject-matter. Concerning the latter, what we encounter in mathematical experience are frequently “phenomena that resist our will, so that we are ‘forced’ in certain directions” (170); what accounts for this resistance to our will are invariants, which have the further characteristics of repeatability—both for a single subject, and among many subjects—and availability for convergence (221), which, I take it, means availability as structures or concepts upon which many independent rational investigators will come to rest. Thinking mathematical thoughts is quite unlike thinking thoughts about fictional items, because fictions are indefinitely plastic in the face of human will. It is also unlike thought about sensory items, in that the intentional objects at issue are not available to outer sense; but from another angle it is more like thought about real material items than imaginary beings, because our mathematical thoughts are exposed to the possibility of error or misrepresentation in various ways. (An elementary example: most people think that 1 is distinct from 0.9999... repeating: this is a misrepresentation, because they are in fact identical.)
The phenomenology of mathematical practice is such that we assay various conjectures and proposals about invariants already given to consciousness by previous practice, and discover what further invariant structures or concepts are revealed as necessary constraints upon the various possibilities that we can experimentally float for ourselves. Sometimes our conjectures or intentions are fulfilled, sometimes they are frustrated; but it is the very fact of fulfillment and frustration which points us to a Platonic reality. It is not the noumenal realm of the “naive” Platonist, in Tieszen’s phrase, but nevertheless an objective reality marked off “on the basis of what stabilizes or becomes invariant in our experience, on the basis of what has a history and a sedimentation that permits of further elaboration, development, constraints, surprises, and so on” (101). In something like this way, human reason and consciousness “constitute the meaning of being” of mathematical objects...