Arch. Hist. Exact Sci. 60 (2006) 1–121 Digital Object Identifier (DOI) 10.1007/s00407-005-0102-4 The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes1,2 Philip Ehrlich Communicated by J. J. Gray As a matter of fact, it is by no means impossible to build up a consistent “non- Archimedean” theory of magnitudes in which the axiom of Eudoxus (usually named after Archimedes) does not hold. Hermann Weyl Philosophie der Mathematik und Naturwissenschaft 1927, p. 36 Introduction In his paper RecentWork On The Principles of Mathematics, which appeared in 1901, Bertrand Russell reported that the three central problems of traditional mathematical 1 Portions of this work were presented at the Pittsburgh Center for Philosophy of Science in November 1998 as part of a conference on Philosophical Problems in the Historiography of Science jointly sponsored by the Center and the Division of Logic, Methodology, and Philosophy of Science of the International Union of History and Philosophy of Science, at Northwestern University in September 1999 as part of the Northwestern University History and Philosophy of Science Seminar Series, and at the International History and Philosophy of Mathematics Meeting in Seville, Spain in September 2003. We are very grateful to the various organizers for affording us those opportunities. Thanks are owed also to the National Science Foundation for supporting the early development of this research (Scholars Award #SBR 9602154). We also wish to express our gratitude to Scott Carson and Kathleen Evans-Romaine, who from time to time served as knowl- edgeable and helpful sounding boards for our translations of the German, French and Italian texts. Finally, we are especially grateful to Jeremy Gray and Henk Bos, each of whom provided valuable suggestions for improving the exposition. Throughout the text we follow the convention of only providing the original German, French and Italian texts in the cases of translations of substantial quotations. 2 The companion to the present paper referred to in the text is a continuation of the present work that will focus on the emergence of non-Archimedean geometry. It is the author’s intention to submit the latter paper for publication in the Archive in the not too distant future. 2 P. Ehrlich philosophy – the nature of the infinite, the nature of the infinitesimal, and the nature of the continuum – had all been “completely solved” [1901, p. 89]. Indeed, as Russell went on to add: “The solutions, for those acquainted with mathematics, are so clear as to leave no longer the slightest doubt or difficulty” [1901, p. 89]. According to Russell, the struc- ture of the infinite and the continuum were completely revealed by Cantor and Dedekind, and the concept of an infinitesimal had been found to be incoherent and was “banish[ed] from mathematics” through the work of Weierstrass and others [1901, pp. 88, 90]. These themes were reiterated in Russell’s often reprinted Mathematics and the Metaphysician [1918],3 and further developed in both editions of Russell’s The Principles of Mathe- matics [1903; 1937], the works which perhaps more than any other helped to promulgate these ideas among historians and philosophers of mathematics. In the two editions of the latter work, however, the banishment of infinitesimals that Russell spoke of in 1901 was given an apparent theoretical urgency. No longer was it simply that “nobody could dis- cover what the infinitely little might be,” [1901, p. 90] but rather, according to Russell, the kinds of infinitesimals that had been of principal interest to mathematicians were shown to be either “mathematical fictions” whose existence would imply a contradiction [1903, p. 336; 1937, p. 336] or, outright “self-contradictory,” as in the case of an infinitesimal line segment [1903, p. 368; 1937, p. 368]. In support of these contentions Russell could cite no less an authority than Georg Cantor, the founder of the theory of infinite sets.4 Having accepted along with Russell that infinitesimals had indeed been shown to be incoherent, and that (with the possible exception of constructivist alternatives) the nature of the infinite and the continuum had been essentially laid bear by Cantor and Dedekind, following the development of nonstandard analysis in 1961, a good number of historians and philosophers of mathematics (as well as a number of mathematicians and logicians) readily embraced the now commonplace view that is typified by the following remarks: In the nineteenth century infinitesimals were driven out of mathematics once and for all, or so it seemed. [P. Davis and R. Hersh 1972, p. 78] But ... The German logician Abraham Robinson (1918–1974), who invented what is known as non-standard analysis, thereby eventually conferred sense on the notion of an infinitesimal greater than 0 but less than any finite number. [Moore 1990; 2001, p. 69] Indeed ... nonstandard analysis ..., created by Abraham Robinson in the early 1960s, used tech- niques of mathematical logic and model theory to introduce a rigorous theory of both 3 Russell’s Mathematics and the Metaphysician is a reprinting of [Russell 1901] with newly added footnotes that were introduced because “some points in this essay require modification in view of later work” [Russell 1918, p. 7]. For additional historical remarks on this paper, see [Russell 1993, pp. 363–365]. 4 Russell was willing to accept that “lengths of bounded straight lines are infinitesimal as compared to areas, and these again as compared to volumes of polyhedra.” However, according to Russell, “such genuine cases of infinitesimals ... are always regarded by mathematicians as magnitudes of a different kind ...” [1903, p. 337]. The Emergence of non-Archimedean Systems of Magnitudes 3 [non-Cantorian] infinite and infinitesimal numbers. This, in turn, required a reevaluation of the long-standing opposition, historically, among mathematicians to infinitesimals in particular. [Dauben 1992a, pp. 113–114]5 All of this is of course well known to historians and philosophers, of mathemat- ics alike. What is not so well known in these communities, however, is that whereas most late nineteenth- and pre-Robinsonian twentieth-century mathematicians banished infinitesimals from the calculus, they by no means banished them from mathematics. Indeed, contrary to what the above remarks plainly suggest, between the early 1870s and the appearance of Abraham Robinson’s work on nonstandard analysis in 1961 there emerged a large, diverse, technically deep and philosophically pregnant body of con- sistent non-Archimedean mathematics of the (non-Cantorian) infinitely large and the infinitely small. Unlike non-standard analysis, which is primarily concerned with pro- viding a treatment of the calculus making use of infinitesimals, most of the former work either emerged from the study of the rate of growth of real functions, or is concerned with geometry and the concepts of number and of magnitude or grew out of the natu- ral evolution of such discussions. What may surprise many historians and philosophers even more than the existence of these two bodies of literature is that the latter such body contains constructions of systems of finite, infinite and infinitesimal numbers that are not only sophisticated by contemporary mathematical standards but which are rich enough to embrace the corresponding number systems employed in nonstandard analysis.6 5 These references only begin to indicate just how widespread this misconception is in the literature–in the English-language literature, in particular. A further indication that this view has reached the status of orthodoxy in the English-language literature is evidenced by the follow- ing remarks from Graham Priest’s otherwise informative article “Number” from the Rutledge Encyclopedia of Philosophy. Despite the fact that Cauchy possessed the notion of a limit, he mixed both infinitesimal and limit terminology, and it was left to Weierstrass, later in the century, to replace all appeals to infinitesimals by appeals to limits. At this point infinitesimal numbers disap- peared from mathematics (though they would return, as we shall see). Indeed, as a result of the development ... due to Robinson ... called ‘nonstandard analysis’ ... infinitesimals have been rehabilitated as perfectly good numbers ... the existence of nonstandard models of analysis made infinitesimals legitimate. [Priest 1998, pp. 51–52] 6 We are not, of course, referring to the sophisticated hyperreal number systems that are frequently employed in non-standard analysis [cf. Keisler 1994], but merely to their underlying non-Archimedean ordered “number” fields and to the systems of infinitesimals contained therein. In particular, we are referring to the fact that each non-Archimedean ordered field is isomorphic to a subfield of a Hahn field, the latter being a distinguished type of non-Archimedean ordered field introduced by Hans Hahn in his great pioneering work Über die nichtarchimedischen Grössen- systeme [1907]. Moreover, since this embedding theorem had already emerged as a folk theorem among knowledgeable field-theorists by the early 1950s [cf. Conrad 1954, p. 328], it is quite likely that Robinson, who was thoroughly conversant with the literature on ordered algebraic systems, was well aware of this when he published his pioneering work on non-standard analysis [1961]. 4 P. Ehrlich This of course is not to deny that there were late nineteenth-century mathematicians whose calls for the banishment of infinitesimals