Leibniz and the rationality of the infinite

Abstract

Part 1 Historically the term 'infinite' has had two apparently contrary meanings. On the one hand, it was taken by metaphysicians like Plotinus to mean that which " ... has never known measure and stands outside number, and so is under no limit either in regard to anything external or internal ... " (Branford 1949, V.5.11). 'Infinite' in this sense means 'irrevocably complete'. On the other hand, Aristotle defined it in this way: "A quantity is infinite if it is such that we can always take a part outside what has already been taken." (Hardie and Gaye 1941, 207a, 5-10) 'Infinite', in this second sense means, 'irrevocably incomplete'. Leibniz is someone who uses both these meanings. In particular, he identifies the irrevocably complete with God and the irrevocably incomplete with the world (as we know it). Given, firstly, that what is irrevocably complete includes everything and, secondly, that it excludes anything incomplete, the following conclusion can be drawn: Leibniz's philosophy of the infinite makes of the-world-as-we-know-it something that is logically dependent on God, but also something that exists in contradiction to 'him'. Leibniz cannot escape a kind of contradiction in what he says about God and the world but this is not a straightforward case of self-refutation. The reason turns on the consideration that to divorce the concept of the irrevocably complete from its object is to deprive this concept of its sense, specifically of its sense of completeness. For if the two are distinct, then there is something beyond the irrevocably complete, namely, how this is independently of its concept. It follows that to deny the irrevocably complete is, in the same breath, to affirm that very thing. Yet if we cannot quite deny the irrevocable complete, neither can we as human beings quite affirm it either-for the human mind is, we do not doubt, a limited one. Thus the irrevocably complete can neither be affirmed nor denied without contradiction. There is a strong resemblance between this paradox and the paradox of the liar: in both cases there is a thesis that says of itself that it is untrue and, in both cases, thesis and antithesis tum out to be equivalent. Part 2 Kant offers some powerful reasons to think that the paradox discussed in Part 1 involves no real contradiction. The critical philosophy suggests that the apparent contradiction is real, only if, per impossibile, we have some way to positively employ the concept of the world as it is independently of our conceptions of it. Kant's view of the infinite shares with Leibniz's the vice (if it is one) that it is paradoxical: both philosophers make use of a concept that cannot, strictly speaking, be possessed by the human mind. However each view has the significant virtue that it shows the difference between the irrevocably complete and the irrevocably incomplete to be not simply a logical difference. My overall conclusion is based on a synthesis of the Leibnizian and the Kantian philosophies of the infinite. According to Leibniz, neither the irrevocably complete, nor the irrevocably incomplete, can be eliminated from philosophy. According to Kant, infinity is-from a human perspective at least-something prior to conception; putting Leibniz and Kant together, I conclude that these modes of infinity combine to produce finitude, that they are the joint conditions under which difference, and therefore finitude, is possible. In particular, I argue that the irrevocably complete is the infinity of fullness, and that the irrevocably incomplete is the infinity of emptiness, and that logic is blind to any difference there might be between these, since both are, by definition, undifferentiated. Given that ethics, as well as logic, is dependent on finitude, I conclude, finally, that the perennial ambition to eliminate either the irrevocably complete or the irrevocably incomplete from philosophy is, not merely unrealisable, but potentially dangerous.