Virtues of Priority
Abstract
Priority disputes in mathematics are not merely quarrels about dates. They reveal the standards by which mathematical communities distinguish a merely new statement from a contribution that counts as original, influential, or decisive. The dispute over the origin and naming of the Modularity Conjecture supplies a particularly illuminating case. A conjecture occupies a liminal position between idea and text: it can circulate informally, guide research for decades, and only later receive a precise statement. The competing attributions to Taniyama, Shimura, and Weil are therefore best understood not as mutually exclusive historical claims, but as appeals to different mathematical virtues. Taniyama’s 1955 problems exemplify the virtue of a cartesian insight: an imperfect but compelling imaginative leap linking elliptic curves and modular forms through Hasse’s conjecture and Hecke’s theorem. The case for Shimura combines Lang’s realist search for evidence with Shimura’s phenomenological accumulation of facts through the theory of modular and Shimura varieties. The case for Weil, as defended by Serre, lies in the precision added by the conductor and in making the conjecture numerically checkable, hence vulnerable to counterexample. The controversy persisted because these virtues neither exclude nor subsume one another. Its force is finally explained by the role of elliptic curves, the Birch-Swinnerton-Dyer Conjecture, and Fermat’s Last Theorem in shaping the collective desires and prevailing psychology of number theorists.
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