Indefinite Divisibility
نویسندگان
چکیده
Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a “definite totality” but rather is “indefinitely extensible”. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are “potential” rather than “actual”. Besides the intrinsic interest of this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that “indefinite extensibility” arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. ... endure not yet A breach, but an expansion. John Donne, “A Valediction: Forbidding Mourning” 1 Extensibility and Divisibility Sets are supposed to be plenitudinous. It’s tempting to express this plenitude like this: Sets. For any things, there is some set whose members are just those things. But this is inconsistent. The sets are some things, so There is some set whose members are just the sets. This article descends from an earlier paper I presented at the 2009 Arché/CSMN graduate conference; this version doesn’t share much with the original besides its title. Thanks to Andrew Bacon, Einar Bohn, John Hawthorne, Hartry Field, Kit Fine, Shieva Kleinschmidt, Colin Marshall, Ted Sider, Gabriel Uzquiano, and participants at the California Metaphysics Workshop on the Relation Between Logic and Metaphysics, for comments along the way.
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