Raphael Anderson

Kant famously rejected an empiricist account of causal claims, because it cannot account for the necessity and universality of causal laws. He then concludes that causal claims must have an a priori basis:1the concept of cause cannot arise in this [empiricist] way at all, but must either be grounded in the understanding completely a priori or else be entirely surrendered as a mere fantasy of the brain. For this concept always requires that something A be of such a kind that something else B follows from it necessarily and in accordance with an absolutely universal rule. Appearances may well offer cases from which a rule is possible in accordance with which something usually happens, but never a rule in accordance with which the succession is necessary; thus to the synthesis of cause and effect there belongs a dignity that can never be expressed empirically, namely that the effect does not merely come along with the cause, but is posited through it and follows from it. [A 91/B 123-4]So much is clearly Kant’s view, and as long as we remain with the vague formulation that causal generalizations have some a priori ground or other, commentators agree.

Officially, for Kant, judgments are analytic iff the predicate is "contained in" the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of containment: genera are contained in the species formed from them. Kant's thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like '7+5=12.' Kant is correct. Operation concepts ( ) bear two relations to number concepts: and are inputs, is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic.

One of the central and most distinctive theses of Kant’s philosophy of mathematics is that mathematical knowledge is synthetic. In this context, synthetic judgments are defined in opposition to analytic ones, whose predicate concept is “contained in” the subject. Kant’s thesis has often been attacked as indefensible, but just as frequently critics have complained that the thesis itself, and even the analytic/synthetic distinction on which it rests, are simply unintelligible. Thus, even prior to questions of its correctness, the Kantian doctrine has been rejected as unclear, based on general skepticism that there is a real distinction of “containment”—or indeed any tolerably clear distinction at all—between analytic and synthetic judgments.