A Philosophical Task in our times
Abstract
Starting from Ian Hacking’s diagnosis of a task for philosophy in our times, I address a neglected problem in the philosophy of mathematics: the stability of reference across historically and culturally situated mathematical practices. Mathematical knowledge appears strongly cumulative, yet its objects, methods, and descriptions vary locally, sometimes even among mathematicians who take themselves to be working in the same domain. Examples drawn from algebraic geometry, Greek geometry, infinitesimal analysis, symbolic algebra, group theory, and set theory show that the difficulty cannot be reduced to controversy, incommensurable paradigms, or changing epistemic values. Nor can the semantic externalism developed for the empirical sciences be transferred directly, since mathematical objects are not available through causal interaction with exemplars. Against both a purely descriptive account of reference and a vague appeal to practice, I propose to analyze the modes by which mathematical reference is stabilized. Leibniz’s theory of blind or symbolic knowledge offers a guiding thread: stabilization may depend on conceptual analysis, on genetic characterizations of objects through operations and constructions, or on the material anchoring of reasoning in symbolic and diagrammatic systems. These plural modes make room for an internal realism in mathematics, able to account at once for historical variability, trans-theoretical identity, and the cumulative character of mathematical knowledge.
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