Review of du Bois-Reymond's Die allgemeine Functionentheorie
This was an important book when it first appeared (Laupp, Tübingen, 1882) but it was a difficult book and its real importance was only recognized slowly. Thus, although it marked an important stage in the development of the concepts of number and function in the latter part of the 19th century, many of the ideas (such as those involved in the "Infinitärkalkül") were not fully understood until they were taken up by later authors, e.g., by G H Hardy in Orders of infinity, University Press, Cambridge, 1910.
The newer theories of irrational numbers seek to hold fast to the foundation supplied by arithmetic by using a method of forming mathematical concepts that one encounters already in the fifth book of Euclid's Elements. Based on the assumptions of the Ancients concerning the straight line, one can then easily show that to each segment a number can be assigned; while the converse theorem - namely, that to each number there corresponds a segment - has to be viewed as an axiom, or as equivalent to the assumption of continuity of the line. Already on some previous occasions the irrational numbers were derived from geometry, but the real significance of this procedure was not recognized.
Du Bois-Reymond considers this latest method to be consistent. Indeed, arithmetic, just like geometry, is a natural science; it originated from the concept of linear magnitudes (or quantities), i.e., from the idea of magnitudes that have the same properties as do the segments of a bounded straight line, so that they can be put into a one-to-one correspondence with the points of the line. Thus the basic problem of analysis will be to show the existence of the magnitude that corresponds to a specified arithmetical passage to the limit. This problem is formulated by means of the "general convergence principle", i.e. the following theorem: If the difference f(x_{1}) - f(x), with x_{1} > x assumed, from a sufficiently large value of x onward and for arbitrary values of the difference x_{1} - x, remains below an arbitrarily chosen small number, then the function f(x) has a definite limit Y, i.e., there exists a magnitude Y such that, starting from a sufficiently large value of x, the difference Y - f(x) is smaller than an arbitrarily small positive number.
However, the author does not place this problem at the top of his list, but rather the problem - contained in the basic problem - of an unending non-periodic decimal fraction. He shows that the problem of the limit of a decimal fraction can be treated and solved from two points of view that are drastically opposed to one another: the idealistic and the empirical one. In order to emphasize the irreconcilable contrast between these two points of view, he develops them quite independently of each other, taking turns between letting the representative of one view or the other speak. Their discussion leads to the following result.
The idealist believes in the reality of extensions of concepts that go beyond the imagination but are necessitated by our thought processes. Among these are the basic geometric concepts: point, line, surface; the perfect (or complete) straight line and the exact measure, and finally also the infinitely large and the infinitely small. While the empiricist stops at the fact that the unit segment contains unlimitedly many points, the idealist proceeds to the statement: the number of these points is infinitely large. From this he deduces the existence of the infinitely small, since the segment in accordance with its infinite number of points must decompose into infinitely many subsegments, none of which can be finite. The infinitely small segments, compared to the finite ones, are a new species of quantities but, among themselves, have the same properties as do the finite quantities. The zero to the idealist must appear dispensable in analysis.
The empiricist always remains within the limits of the natural domain of imagination. He concedes and acknowledges the arbitrarily exact in geometry but calls the ideally exact an axiom. For him, a point is a space arbitrarily small in all directions, and a line is a thread of arbitrarily small cross section. In his view, only finite magnitudes exist, among them those that grow beyond any measure, however large, or that shrink below any measure, however small. The idealist solves the limit problem by ultimately removing the difference, between the limit and the segments determining it, from the domain of imagination, but the empiricist solves the problem in the end by letting the determining segments and the limit merge into one idea (or image). So far, the systems of idealism and empiricism have hardly been kept strictly apart. (Even in geometry, where individual occasions of empiricism have been occurring for a long time, empiricism as such was developed consistently only in Pasch's lectures on the newer geometry (1882, cf. Vol. 3, Part VIII, Chapter 5A).)
For the theory of functions, the author adopts a neutral manner of presentation, i.e. one which will contradict neither of the two points of view. The idealist metaphysics, the infinitely large and the infinitely small, is not allowed to be used, but neither is the empiricist idea of the arbitrarily exact, so that concepts such as magnitude, value and segment can be used in the idealistic sense. The just described, mainly epistemological investigation of the limit problem forms the content of the first two chapters. The third chapter, "On the argument", provides the theory of distribution of points, as well as the theory of denumerability and the relative cardinalities of unlimited manifolds {i.e. sets, in later terminology} of G Cantor. In the author's terminology, the infinite systems of points on the segment over which the argument varies are divided into {classified as} pantachies { = dense sets}, apantachies { = nowhere dense sets}, and those that cannot be decomposed into a finite collection of pantachic and apantachic portions. Among the apantachies one finds the integrable systems, but the converse implication is false.
The chapter concludes with an idealistic-empirical critique of this theory, which makes it clear that the number continuum (0, 1) exists only for the idealist; i.e., a variable finite point set which, as it increases, would approximate without bounds the set of all possible points, does not exist. In the fourth chapter, "On the function", the general concept of function is explained, the relations of a single function value to the values at nearby points are discussed and subsequently the most important theorems concerning continuous functions are given. From the last of these topics we mention the concept of degree of continuity, r(x, D), at a point x. If f(x) is continuous everywhere in the interval (0, 1) then there exists a function r(v, D) which, without ever increasing, shrinks below every bound together with D and which is related to f(x) by
f(x) - f(x + D) = r(x, D).K, K ≤ 1.
In Chapter 5, "On the final behavior of functions", one finds the proof of the above-mentioned convergence principle and of theorems derived from it, the theory of limits of indeterminacy and envelopes of indeterminacy, and the proof of the general divergence principle: "If the difference f(x_{1}) - f(x), for all x and x_{1} above some sufficiently large value, does not remain below an arbitrary specified small value, then f(x) has no fixed limit." At the conclusion of the chapter, the complete pantachy of linear magnitudes, i.e. the number continuum, is confronted with the infinitary pantachy.
JOC/EFR December 2005
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