Following Kant, Frege took the idea that there is such a thing as bona fide a priori knowledge of a large range of necessary propositions for granted. In particular he assumed that such is the character of our knowledge of basic logic and arithmetic. This view is no longer orthodoxy. The idea that pure (for Frege, logical) intellection can provide for substantial knowledge of necessary features of the world is widely regarded with suspicion. However it is fair to say that most recent scepticism about it has been driven either by abstract background theoretical commitments—for instance, by a thoroughgoing empiricism, as in Mill and Quine, or by epistemological externalism1—or by the conviction that the concept of a priori justification allows of no stable, theoretically interesting characterisation.2 The present discussion focuses on the example of elementary arithmetic to develop and explore a different, relatively neglected and, it may be suggested, more fundamental kind of sceptical challenge, one prefigured in the writings on mathematics of the later Wittgenstein but independent of the discussion of following a rule to which his generally deflationary or conventionalist cast of thinking about mathematical knowledge is, after Kripkenstein,3 nowadays usually attributed. Elementary arithmetical truths are normally taken to be justifiable a priori if any truths are. They are also normally taken to be both necessary and substantial—essentially applicable to our dealings with the world and empirically predictive, yet good for counterfactual reasoning about however far-fetched and exotic scenarios. Yet, I will argue, scrutiny of the kind of methods—simple informal cognitive routines involving counting and pictures—whereby such judgements are initially apt to win our confidence serves to make it puzzling how they can justify what they are supposed to at all: how such procedures can merit either the very high levels of confidence we standardly place in the judgements they lead to, or the modal (counterfactual) significance we standardly attach to those judgements. The present discussion elaborates these apparent shortfalls (§§2–4) and reviews four proposals (§§5–8) for redressing them, arguing that each is ineffective. The sceptical challenge accordingly stands unanswered. §9 elaborates its relation to one strand in the Remarks on the Foundations of Mathematics. §10 summarises the resulting dialectical position. The upshot, I believe, is a deepened understanding of an important aspect of Wittgenstein’s later philosophy and a major intellectual challenge to those—possibly still a majority—who incline to side with Frege’s view of the epistemological status of arithmetic.
