Review of Metamathematics and the Philosophical Tradition

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by William Boos, ed. Florence S. Boos. Berlin: De Gruyter. 2018

William Boos (1943-2014) was a polymath and polyglot, with PhDs in set theory (University of Wisconsin, 1971) and philosophy (‘Metalogical doubt’, University of Chicago, 1981). This posthumous volume edited by his wife, the literature scholar Florence S. Boos, presents his central philosophical claim, and argues it in detailed chapters on Plato, the stoics, Anselm, Leibniz, Berkeley, Hume, Kant, free will and determinism, running to 475 pages including indexes. Boos’ point of departure is the thought that the most effective ancient sceptical arguments worked by diagonalization. Whenever a criterion is given for some cognitive good, be it truth, meaning, consistency, goodness or whatever, the criterion can be turned on itself. This invariably sets off a regress that can only be stopped by some act of dogmatic fiat. Trained as he was in set theory and formal logic, this suggested to him that the famous metamathematical proofs and theorems of the twentieth century that rely on some variant of diagonalization have a bearing on the philosophical topics that dominate the canonical western tradition from antiquity onwards. With Gödel as his guide, he pushes a rigorized, formalised version of the sceptical challenge through selected works from the history of philosophy. He discerns two attitudes: one that finds liberation in the openness and endlessness of enquiry, moral development and the giving of meaning to experience, and another that seeks to impose limits, to put caps on metatheoretic ascents, and to insist on the intended interpretations of terms. Boos is firmly of the former party. His treatments of canonical figures follow a pattern: first, show that the philosophical enquiry sets an ascent going, and then find the moment when the canonical philosopher tries to stop it after an iteration or two by insisting on an axiom, a distinction or some other source of infallible authority. To put his thought in terms of a Platonic myth, when the prisoners escape from the cave, they should not imagine themselves to be standing in sunlight, and they should not believe anyone who tells them that they are. Contemplation of Gödel’s results should show them that they are in a bigger, lighter cave and will be climbing forever through successive caves. That’s fine, though, because climbing is what we’re good for and we have some freedom to choose our route.

Boos’ writing is terse, and follows the mathematical practice of giving the reader just enough information to reconstruct the argument without writing it out in detail. Like a mathematician but unlike a philosopher, he does not anticipate and defuse objections. His treatment of canonical philosophers is textually well grounded, in the sense that he supplies plenty of quotations and references. He evidently knows these works intimately. However, he does not expound their views, consider alternative readings or engage with historians of philosophy. He seems untroubled by the thought that we might not be entirely sure what these great dead thinkers thought, and perhaps in places neither were they. It would have been more methodologically consistent to extend the sceptical lesson to the history of philosophy. Enquiry in the history of ideas is also, like metaphysics, science, and ethics, endlessly open unless closed off by fiat. However, it might not have made a better book to try to work that thought in. The book we have does not explain metamathematics or the canonical philosophers. It requires its readers to know the theorems and have a scholarly understanding of the canonical texts it focusses on. Adding tentative notes and alternative readings would have made it denser and longer without adding to its value. It offers two things, both well worth having: provocative-yet-plausible readings of canonical philosophy (and on the freewill-determinism question), and examples of the art of exposing acts of table-thumping fiat hidden in philosophical arguments.
Original languageEnglish
Pages (from-to)292-293
JournalPhilosophia Mathematica
Issue number2
Early online date5 May 2021
Publication statusPublished - 1 Jun 2021


  • Metamathematics
  • Mathematics
  • Philosophy


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