The paper is a critical examination of Peacocke’s pioneering work on concepts as grounding the possibility of a priori knowledge. It focuses upon his more recent turn to reference and referential domain, and the two enlargements of the purely conceptual bases for apriority, namely appeal to conceptions and to direct referential sensitivity. I argue that the two are needed, but they produce more problem for the strategy as a whole than they solve. I conclude by suggesting that they point to (...) a possible Benacerraf-like dilemma for conceptualist accounts of armchair knowledge: if concepts are akin to representational contents and/or conceptions, they certainly do not metaphysically determine anything. At best, they fallibly guide our inquiry and get corrected almost by each new important discovery about the nature of their referents. If what is meant by “concept” is a Fregean, objectively correct and metaphysically potent entity, there is little doubt in its power to determine its referent(s), but there is a huge epistemic problem of how we grasp such Platonic concepts. Peacocke’s early metaphysics of concept, which offered beginnings of an answer, is put in jeopardy by the new referential turn, and his valiant attempts to pass between the multiple horns of this dilemma seem to face a lot of difficulties. (shrink)

In this paper I will argue that (principled) attempts to ground a priori knowledge in default reasonable beliefs cannot capture certain common intuitions about what is required for a priori knowledge. I will describe hypothetical creatures who derive complex mathematical truths like Fermat’s last theorem via short and intuitively unconvincing arguments. Many philosophers with foundationalist inclinations will feel that these creatures must lack knowledge because they are unable to justify their mathematical assumptions in terms of the kind of basic facts (...) which can be known without further argument. Yet, I will argue that nothing in the current literature lets us draw a principled distinction between what these creatures are doing and paradigmatic cases of good a priori reasoning (assuming that the latter are to be grounded in default reasonable beliefs). I will consider, in turn, appeals to reliability, coherence, conceptual truth and indispensability and argue that none of these can do the job. (shrink)

I catalog several concepts associated with finite cardinals, and then invoke them to interpret and evaluate several passages in Rips et al.'s target article. Like the literature it discusses, the article seems overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians).