The Origin of Species—Part III:
Leibniz and Speculation on the

At the end of the chapter of Buffon's Histoire naturelle (p.250) that contains the investigations of the problem of infinite we find the date February 6, 1746; the volume was published in 1750. For further details concerning his ideas, Buffon refers the reader to the foreword of his translation of Newton (not available to me), which appeared in 1740, and thus to the larger mathematical context of the problems associated with the analysis of infinity in his time.

The most significant and most concise fundamental formulations of the problem are found in some letters from Leibniz to Bernoulli (published in 1745). For our purposes the letter of July 29, 1698, is especially important because it contains a statement referring to the problem of preformation. There Leibniz speaks about the division of matter, arguing that no indivisible elements or smallest particles can ever be arrived at, only ever smaller ones that can be split into yet smaller ones. By the same token, increasing a dimension will never lead to the largest one or to infinitely large ones or to ones whose dimensions cannot be increased further.

Applying these principles to the problem of preformation, Leibniz concedes that the germs may be encapsulated but denies that it is possible to arrive at an infinitely small one, much less at an ultimate one. The regression of encapsulation is thus extended into infinity, and the act of creation loses its significance as an absolute beginning.

Another passage in a letter of August, 1698, shows even more clearly that in the infinite regression for each member we must necessarily envision another one, and thus the concept of an absolute infinity is a contradiction in terms. The passage is formulated with particular felicity because it distinctly shows the connection between the problem of the infinitely large with that of the infinitely small; from the vantage point of empirical finite facts, speculation on infinity leads to meaninglessness in both directions.[Fn]

    Fn. August, 1698: [Original Latin text omitted from this webpage.]"Since I have denied arriving at minimal portions, it was easy to judge that I was not speaking of our divisions, but also about those actually occurring in nature. Therefore, although I certainly hold that any part of matter whatsoever is actually subdivided again, still I do not think it therefore follows from this that there exists an infinitely small portion of matter, and still less do I concede that it follows that there exists any altogether minimal portion. If anyone wishes to pursue the consequence formally, he will sense the difficulty.

    But you will inquire: If nothing infinitely small exists, then single parts are finite (I concede); if singular parts are finite, therefore all taken together at once constitute an infinite magnitude. I do not concede this conclusion. I would concede, if there existed some finitude which would be smaller than all others or certainly not greater than any other; for then I confess that on such assumptions, by as many as any given number you like there arises a quantity as large as you like. But it holds true that by any part you like another smaller finite magnitude exists."

The problem of the given fact of an actual infinite again plays an important role in the history of modern mathematics, especially in the construction of Cantor's set theory and theory of transfinite cardinal numbers, sets of sets, and so forth. This development in mathematics is essentially based on the same false reasoning Leibniz discussed in his letters to Bernoulli and Buffon addressed in the context of his criticism of the theory of preformation.

Lately Felix Kaufmann has tried in his works to resolve this false reasoning of set theory and the mathematical theory based on it. His argumentation is essentially the same as Buffon's, cited earlier in the text.

I quote from Kaufmann's book, Das Unendliche in der Mathematik und seine Ausschaltung (Vienna, 1930), 147: "We have established that the natural numbers are logical abstracts of the counting process and that the concept of the 'number series' includes an 'idealization' in addition to this abstraction. It consists of the presupposition of the nonexistence of a fixed upper limit, so that 'number series' comes to mean the abstraction of an infinite counting process." He points out that we must avoid the error "of seeing a self-contained totality of natural numbers in the number series" (148). We must start with the counting process and determine its logical structure; the series of natural numbers must be defined by the law of their formation and not conversely the general form of the process by its product, assumed to be real.

In his "Bemerkungen zum Grundlagenstreit in Logik und Mathematik" ( Erkenntnis , II, 1931) Kaufmann summarized the problem most concisely in the sentence (285): "The circularity (namely, of the concept of the infinite series of natural numbers) lies in the fact that in general where no final limit exists for the number of the function values, the value trend [ Wertverlauf ] of a function can be defined only as a general form, and therefore it is not possible to define this general form by the value trend. "

Chapter 11
Infinite Series and Finitization,
pp 119-121.