Maths & Logic Workshop: Michael Potter

Michael Potter presented the outline of a theory of arithmetic for which a priori knowledge is not trivial (contra empiricism) and which takes a qualified attitude towards absolutely unknowable truths (contra full-blooded transcendentalism). He started with a short appraisal of Kant’s contribution to arithmetic, holding fast to the Kantian notion of synthetic a priori truths as a genuine addition to monadic and polyadic logic while rejecting Kant’s idealist account of these truths.

Potter proceeded to recall that Gödel’s first incompleteness theorem leads to the disjunctive conclusion that either human minds are not machines or there are absolutely unknowable truths about arithmetic. Both disjuncts are not empiricist (provided the rejection of physicalism about the mind results in a non-empiricist position). However, it is not clear from Gödel’s argument whether both are transcendental. Potter turned to Wittgenstein to motivate a transcendental interpretation of Gödel’s first disjunct. Wittgenstein remarked that there is no space between our capacity for recognising the limits of our cognitive abilities – so that it makes sense to say that there are absolutely unknowable truths about the mind – and the absence of this capacity – so that all a priori truths about the mind turn out trivial. If this is true, the rejection of empiricism about minds implies transcendentalism.

Combining Wittgenstein’s views about cognitive limits with Gödel’s disjunction results in a further disjunction. Potter concluded that the introduction of synthetic a priori truths in arithmetic implies that there are either absolutely unknowable truths about the mind or absolutely unknowable truths about arithmetic. This disjunctive conclusion makes it unnecessary to commit to absolutely unknowable truths across the board (in analogy to Dummett’s non-systematic conception of antirealism). Committing to either transcendentalism about the mind or about arithmetic is unavoidable, but there are no external criteria to adjudicate on which side our commitments go. It is in this sense that Potter’s (tentative) transcendental position on arithmetic is a qualified position and not a full-blooded one. We are committed to transcendentalism about arithmetic if human minds are machines. We may be committed to it if that is not the case.