Leibniz on the Indefinite as Infinite

Article excerpt

I

Consider the natural numbers: 1, 2, 3,. ... Unreflectively, we typically take there to be infinitely many natural numbers. If pressed, we might offer something like the following reasoning. If there were only finitely many natural numbers, then there would be a largest natural number. If, however, we denote this supposed largest natural number by the symbol `n', then n+1 is also a natural number, contradicting the fact that n is assumed to be the largest natural number. Consequently there are not finitely many, but rather infinitely many, natural numbers.

It is not my intention to assess the status of this reasoning, which in any case is only meant to describe what I take to be (in the late twentieth century) some of our informal preconceptions about the nature of the finite and the infinite. Instead, I would like to set the picture suggested above against an argument that Leibniz uses to show that the number of finite (whole) numbers cannot be infinite. The text from which this argument is drawn, "On the Secrets of the Sublime, or on the Supreme Being," dates from early 1676, a period of tremendous intellectual upheaval in the life of Leibniz. Leibniz was living in Paris and had just invented the infinitesimal calculus.(1) He was engaged in an intensive reading of the Cartesians, especially Descartes and Malebranche, and was to meet Spinoza later that year on his way back to Hannover, where, apart from extended periods of travel, he would reside for the rest of his life. Here is the argument that Leibniz gives:

If the numbers are assumed to exceed each other continuously by one, the

number of such finite numbers cannot be infinite, for in that case the

number of numbers is equal to the greatest number, which is assumed to

be finite. It has to be replied that there is no greatest number. But

even if they were to increase in some way other than by ones, yet if

they always increase by finite differences, it is necessary that the

number of all numbers always has a finite ratio to the last number;(2)

further, the last number will always be greater than the number of all

numbers. From which it follows that the number of numbers is not

infinite; neither, therefore, is the number of units.(3)

A full analysis of this proof must wait for another time. The point I want to make here is that Leibniz distinguishes between there being no greatest number and the number of finite numbers being infinite. That is, Leibniz distinguishes between the indefinite progression of finite numbers, which he here takes to be the case, and the finite numbers being infinite in number, which he here takes to be impossible. This is made particularly clear when, directly following the passage cited above, Leibniz goes on to add to the conclusion he has just reached: "Therefore there is no infinite number, or, such a number is not possible."(4)

Leibniz discusses this distinction between the indefinitely progressing and the infinite explicitly earlier in the same writing, before reaching the conclusion that an infinite number is impossible. Here, focusing on the infinitely small, Leibniz remarks:

One must see if it can be proved that there exists something infinitely

small, but not indivisible. If this exists, wonderful consequences about

the infinite would follow. Namely: if one imagines creatures of another

world, which is infinitely small, we would be infinite in comparison with

them. ... From which it is evident that the infinite is--as indeed we

commonly suppose--something other than the unlimited.(5)

Leibniz goes on to remark, too, that since "the hypothesis of infinites and of infinitely small things is admirably consistent and is successful in geometry, this also increases the probability that they really exist."(6) As this last remark indicates, the metaphysical outpourings of 1676 should in large part be understood as motivated by the antecedent success of the newly invented infinitesimal calculus, which had washed over Leibniz's metaphysics with the force of a tidal wave. …