Infinite versus Indefinite

Summary

Descartes relied heavily on the idea of infinity, most crucially in his first cosmological argument for God's existence in the Third Meditation. He argues that my idea of God's infinity is not derived by negating my own finitude nor by extrapolating from my own limited intellectual growth; on the contrary, my perception of the actually infinite is prior to my perception of the finite and the merely potential. Moreover, this idea is “utterly clear and distinct” and contains maximal objective reality (AT VII 46, CSM II 31) (see clarity and distinctness). So it can only have come from a substance that contains infinite formal reality – that is, God (see being, formal versus objective).

God alone is strictly infinite. Yet there are created beings, like extension and numbers, which are unlimited in certain respects. So Descartes introduces a technical distinction between the infinite, “that in which no limits of any kind can be found,” and the indefinite, that in which “there is merely some respect in which I do not recognize a limit” (AT VII 113, CSM II 81; cf. AT V 355–56, CSMK 377). The distinction is ontological in the first instance; it concerns the degree of limitlessness in things. An ordinary body, for example, is limited in size but unlimited in divisibility. The extension of the universe beyond the earth and stars, which Descartes sometimes calls “imaginary space,” is unlimited in size but limited in power, intelligence, and the like (AT III 274–75, CSMK 166). But Descartes also emphasizes an epistemic side to the distinction. Infinite things are those that I “understand” to be absolutely unlimited (in all respects), while indefinite things are those in “which, from some point of view, we are unable to discover a limit.” Extension, for example, is indefinite because “no imaginable extension is so great that we cannot understand the possibility of an even greater one” (AT VIIIA 15, CSM I 202). But I understand God to be actually infinite “so that nothing can be added to his perfection” (AT VII 47, CSM II 32).