Confronting the threat of a Spinozistic necessitarianism, Leibniz insists that not all possible substances are compossible—that they can't all be instantiated together—and thus that not all possible worlds are compossible—that they can't all be instantiated together. While it is easy to appreciate Leibniz's reasons for embracing this view, it has proven difficult to see how his doctrine of incompossibility might be reconciled with the broader commitments of his larger philosophical system. This essay develops, in four sections, a novel solution to the “puzzle of incompossibility.” The first section frames the difficulty more carefully and briefly argues that the two dominant strategies developed by Leibniz's commentators fail to solve it fully insofar as they require simply abandoning one or another of its motivating commitments. The second and third sections show how Leibniz's guiding analogy of a geometrical packing or tiling problem may be applied to solve the puzzle of incompossibility in the context of finite and infinite worlds composed of extended corporeal substances. Finally, the fourth section shows how the strategy of Leibniz's packing analogy might be applied even in the context of a thoroughly idealist metaphysics in which the only true substances are nonextended, mindlike “monads.” The essay concludes by drawing some connections between Leibniz's thinking about the puzzle of incompossibility and the development of his views concerning the status of corporeal substances and extended bodies.

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