Michael Detlefsen

(1948 - 2019)

There is a long line of thinkers in the philosophy of mathematics who have sought to base an account of proof on what might be called a 'metaphysical ordering' of the truths of mathematics. Use the term 'metaphysical' to describe these orderings is intended to call attention to the fact that they are regarded as objective and not subjective and that they are conceived primarily as orderings of truths and only secondarily as orderings of beliefs. -/- I describe and consider two models for such orderings, (i) an implicational model and something I call (ii) a mosaic model. -/- Both models fail to do what such orderings should do. Moreover, in both cases, the trouble arises because the class of truths to be ordered is taken to be closed under various weakening implications. This is a result of their employing a global conception of logic.

The aims of this paper are twofold: firstly, to say something about that philosophy of mathematics known as 'intuitionism' and, secondly, to fit these remarks into a more general message for the philosophy of mathematics as a whole. What I have to say on the first score can, without too much inaccuracy, be compressed into two theses. The first is that the intuitionistic critique of classical mathematics can be seen as based primarily on epistemological rather than on meaning-theoretic considerations. The second is that the intuitionist's chief objection to the classical mathematician's use of logic does not center on the use of particular logical principles (in particular, the law of excluded middle and its ilk). Rather on the role the classical mathematician assigns (or at least extends) generally (i.e. regardless of the particular principles used) to the use of logic in the production mathematical proofs. Thus, the intuitionist critique of logic that we shall be presenting is far more radical than that which has commonly been presented. Concerning the second, more general, theme, my claim is this: some restriction of the role of logical inference in mathematical proof such as that mentioned above is necessary if one is to account for the seeming difference in the epistemic conditions of provers whose reasoning is based on genuine insight into the subject-matter being investigated, and would-be provers whose reasoning is based not on such insight, but rather on principles of inference which hold of every subject-matter indifferently.

We argue that Löb's Theorem implies a limitation on mechanism. Specifically, we argue, via an application of a generalized version of Löb's Theorem, that any particular device known by an observer to be mechanical cannot be used as an epistemic authority (of a particular type) by that observer: either the belief-set of such an authority is not mechanizable or, if it is, there is no identifiable formal system of which the observer can know (or truly believe) it to be the theorem-set. This gives, we believe, an important and hitherto unnoticed connection between mechanism and the use of authorities by human-like epistemic agents.