Journal of Humanistic Mathematics Volume 8 | Issue 1 January 2018 What Makes a Theory of Infinitesimals Useful? A View by Klein and Fraenkel Vladimir Kanovei Karin Katz Mikhail Katz Bar-Ilan University Thomas Mormann Follow this and additional works at: http://scholarship.claremont.edu/jhm Part of the Analysis Commons, History of Science, Technology, and Medicine Commons, and the Logic and Foundations Commons ©2018 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ Recommended Citation Kanovei, V. Katz, K. Katz, M. and Mormann, T. "What Makes a Theory of Infinitesimals Useful? A View by Klein and Fraenkel," Journal of Humanistic Mathematics, Volume 8 Issue 1 ( January 2018), pages 108-119. . Available at: http://scholarship.claremont.edu/ jhm/vol8/iss1/7 What Makes a Theory of Infinitesimals Useful? A View by Klein and Fraenkel Vladimir Kanovei V. Kanovei, IPPI, Moscow, and MIIT, Moscow, RUSSIA kanovei@googlemail.com Karin U. Katz Department of Mathematics, Bar Ilan University, Ramat Gan 52900, ISRAEL katzmik@math.biu.ac.il Mikhail G. Katz Department of Mathematics, Bar Ilan University, Ramat Gan 52900, ISRAEL katzmik@macs.biu.ac.il Thomas Mormann Department of Logic and Philosophy of Science, University of the Basque Country UPV/EHU, 20080 Donostia San Sebastian, SPAIN ylxmomot@sf.ehu.es Abstract Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein. Keywords: infinitesimal; Felix Klein; Abraham Fraenkel; hyperreal; Mean Value Theorem Journal of Humanistic Mathematics Vol 8, No 1, January 2018 V. Kanovei, K.U. Katz, M.G. Katz, and T. Mormann 109 1. Introduction Historians often take for granted a historical continuity between the calculus and analysis as practiced by the 17th–19th century authors, on the one hand, and the arithmetic foundation for classical analysis as developed starting with the work of Cantor, Dedekind, and Weierstrass around 1870, on the other. We extend this continuity view by exploiting the Mean Value Theorem (MVT) as a case study to argue that Abraham Robinson's framework for analysis with infinitesimals constituted a continuous extension of the procedures of the historical infinitesimal calculus. Moreover, Robinson's framework provided specific answers to traditional preoccupations, as expressed by Klein and Fraenkel, as to the applicability of rigorous infinitesimals in calculus and analysis. This paper is meant as a modest contribution to the prehistory of Robinson's framework for infinitesimal analysis. To comment briefly on a broader picture, in a separate article by Bair et al. [1] we address the concerns of those scholars who feel that insofar as Robinson's framework relies on the resources of a logical framework that bears little resemblance to the frameworks that gave rise to the early theories of infinitesimals, Robinson's framework has little bearing on the latter.1 For an analysis of Klein's role in modern mathematics see Bair et al-. [2]. For an overview of recent developments in the history of infinitesimal analysis see Bascelli et al. [3]. 2. Felix Klein In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be successful. Namely, one must be able to prove an MVT for arbitrary intervals (including infinitesimal ones). Klein writes: "there was lacking a method for estimating . . . the increment of the function in the finite 1 Such a view suffers from at least two misconceptions. First, a hyperreal extension results from an ultrapower construction exploiting nothing more than the resources of a serious undergraduate algebra course, namely the existence of a maximal ideal (see Section 5). Furthermore, the issue of the ontological justification of infinitesimals in a set-theoretic framework has to be distinguished carefully from the issue of the procedures of the early calculus which arguably find better proxies in modern infinitesimal theories than in a Weierstrassian framework; see further in B laszczyk et al. [5]. 110 What Makes a Theory of Infinitesimals Useful? interval. This was supplied by the mean value theorem; and it was Cauchy's great service to have recognized its fundamental importance and to have made it the starting point accordingly of differential calculus" [17, page 213]. A few pages later, Klein continues: The question naturally arises whether . . . it would be possible to modify the traditional foundations of infinitesimal calculus, so as to include actually infinitely small quantities in a way that would satisfy modern demands as to rigor; in other words, to construct a non-Archimedean system. The first and chief problem of this analysis would be to prove the mean-value theorem f(x+ h)− f(x) = h * f ′(x+ θh) [where 0 ≤ θ ≤ 1] from the assumed axioms. I will not say that progress in this direction is impossible, but it is true that none of the investigators have achieved anything positive. [17, page 219] (emphasis added) See also Kanovei et al. [14, Section 6.1]. Klein's sentiment that the axioms of the traditional foundations need to be modified in order to accommodate a true infinitesimal calculus were right on target. Thus, Dedekind completeness needs to be relaxed; see Section 5.2. The MVT was still considered a research topic in Felix Klein's lifetime. Thus, in 1884 a controversy opposed Giuseppe Peano and Louis-Philippe Gilbert concerning the validity of a proof of MVT given by Camille Jordan; see Luciano [19], Mawhin [20], Besenyei [4], Smoryński [25] for details. 3. Abraham Fraenkel Robinson noted in his book that in 1928, Abraham Fraenkel formulated a criterion similar to Klein's, in terms of the MVT. Robinson first mentions the philosopher Paul Natorp of the Marburg school: "during the period under consideration attempts were still being made to define or justify the use of infinitesimals in Analysis (e.g. Geissler [1904], Natorp [1923])" [23, page 278]. Robinson goes on to reproduce a lengthy comment from Abraham Fraenkel's 1928 book [7, pages 116–117] in German. We provide a translation of Fraenkel's comment: V. Kanovei, K.U. Katz, M.G. Katz, and T. Mormann 111 . . . With respect to this test the infinitesimal is a complete failure. The various kinds of infinitesimals that have been taken into account so far and sometimes have been meticulously argued for, have contributed nothing to cope with even the simplest and most basic problems of the calculus. For instance, for [1] a proof of the mean value theorem or for [2] the definition of the definite integral. . . . There is no reason to expect that this will change in the future." (Fraenkel as quoted in Robinson [23, page 279]; translation ours; numerals [1] and [2] added) Thus Fraenkel formulates a pair of requirements: [1] the MVT and [2] definition of the definite integral. Fraenkel then offers the following glimmer of hope: Certainly, it would be thinkable (although for good reasons rather improbable and, at the present state of science, situated at an unreachable distance [in the future]) that a second Cantor would give an impeccable arithmetical foundation of new infinitely small number that would turn out to be mathematically useful, offering perhaps an easy access to infinitesimal calculus. (ibid., emphasis added) Note that Fraenkel places such progress at unreachable distance in the future. This is perhaps understandable if one realizes that Cantor–Dedekind–Weierstrass foundations, formalized in the Zermelo–Fraenkel (the same Fraenkel) settheoretic foundations, were still thought at the time to be a primary point of reference for mathematics (see Section 1). Fraenkel concludes: But as long this is not the case, it is not allowed to draw a parallel between the certainly interesting numbers of Veronese and other infinitely small numbers on the one hand, and Cantor's numbers, on the other. Rather, one has to maintain the position that one cannot speak of the mathematical and therefore logical existence of the infinitely small in the same or similar manner as one can speak of the infinitely large.2 (ibid.) 2The infinities Fraenkel has in mind here are Cantorian infinities. 112 What Makes a Theory of Infinitesimals Useful? An even more pessimistic version of Fraenkel's comment appeared a quartercentury later in his 1953 book Abstract Set Theory, with MVT replaced by Rolle's theorem [8, page 165]. 4. Modern infinitesimals Fraenkel's 1953 assessment of "unreachable distance" notwithstanding, only two years later Jerzy Loś in [18] (combined with the earlier work by Edwin Hewitt [12] in 1948) established the basic framework satisfying the Klein– Fraenkel requirements, as Abraham Robinson realized in 1961; see [22]. The third, 1966 edition of Fraenkel's Abstract Set Theory makes note of these developments: Recently an unexpected use of infinitely small magnitudes, in particular a method of basing analysis (calculus) on infinitesimals, has become possible and important by means of a nonarchimedean, non-standard, proper extension of the field of the real numbers. For this surprising development the reader is referred to the literature. [9, page 125] (emphasis added) Fraenkel's use of the adjective unexpected is worth commenting on at least briefly. Surely part of the surprise is a foundational challenge posed by modern infinitesimal theories. Such theories called into question the assumption that the Cantor–Dedekind–Weierstrass foundations are an inevitable primary point of reference, and opened the field to other possibilities, such as the IST enrichment of ZFC developed by Edward Nelson; for further discussion see Katz–Kutateladze [15] and Fletcher et al. [6]. This comment of Fraenkel's is followed by a footnote citing Robinson, Laugwitz, and Luxemburg. Fraenkel's appreciation of Robinson's theory is on record: my former student Abraham Robinson had succeeded in saving the honour of infinitesimals although in quite a different way than Cohen and his school had imagined. [10] (cf. [11, page 85]) Here Fraenkel is referring to Hermann Cohen (1842–1918), whose fascination with infinitesimals elicited fierce criticism by both Georg Cantor and Bertrand Russell. For an analysis of Russell's critique see Katz–Sherry [16, Section 11.1]. For more details on Cohen, Natorp, and Marburg neoKantianism, see Mormann–Katz [21]. V. Kanovei, K.U. Katz, M.G. Katz, and T. Mormann 113 5. A criterion Both Klein and Fraenkel formulated a criterion for the usefulness of a theory of infinitesimals in terms of being able to prove a mean value theorem. Such a Klein–Fraenkel criterion is satisfied by the framework developed by Hewitt, Loś, Robinson, and others. Indeed, the MVT (∀x ∈ R)(∀h ∈ R)(∃θ ∈ R) ( f(x+ h)− f(x) = h * g(x+ θh) ) where g(x) = f ′(x) and θ ∈ [0, 1], holds also for the natural extension ∗f of every real smooth function f on an arbitrary hyperreal interval, by the Transfer Principle; see Section 5.1. Thus we obtain the formula (∀x ∈ ∗R)(∀h ∈ ∗R)(∃θ ∈ ∗R) (∗f(x+ h)− ∗f(x) = h * ∗g(x+ θh)), valid in particular for infinitesimal h. 5.1. Transfer The Transfer Principle is a type of theorem that, depending on the context, asserts that rules, laws or procedures valid for a certain number system, still apply (i.e., are "transferred") to an extended number system. In this sense it is a formalisation of the Leibnizian Law of Continuity ; such a connection is explored in Katz–Sherry [16]. Thus, the familiar extension Q ↪→ R preserves the property of being an ordered field. To give a negative example, the extension R ↪→ R ∪ {±∞} of the real numbers to the so-called extended reals does not preserve the field properties. The hyperreal extension R ↪→ ∗R (see Section 5.2) preserves all first-order properties. The result in essence goes back to Loś [18]. For example, the identity sin2 x+cos2 x = 1 remains valid for all hyperreal x, including infinitesimal and infinite inputs x ∈ ∗R. Another example of a transferable property is the property that for all positive x, y, if x < y then 1 y < 1 x . The Transfer Principle applies to formulas like that characterizing the continuity of a function f : R→ R at a point c ∈ R: (∀ε > 0)(∃δ > 0)(∀x) [ |x− c| < δ ⇒ |f(x)− f(c)| < ε ] ; namely, formulas that quantify over elements of the field. 114 What Makes a Theory of Infinitesimals Useful? An element u ∈ ∗R is called finite if −r < u < r for a suitable r ∈ R. Let hR ⊆ ∗R be the subring consisting of finite elements of ∗R. There exists a function st : hR → R called the standard part (sometimes referred to as the shadow) that rounds off each finite hyperreal u to its nearest real number u0 ∈ R, so that u0 = st(u) and u ≈ u0, where a ≈ b is the relation of infinite proximity (i.e., a− b is infinitesimal). 5.2. Extension The hyperreal extension R ↪→ ∗R is the only modern theory of infinitesimals that satisfies the Klein–Fraenkel criterion. Here ∗R can be obtained as the quotient of the ring of sequences RN by a suitable maximal ideal. The fact that it satisfies the criterion is due to the transfer principle. In this sense, the transfer principle can be said to be a "powerful new principle of reasoning". Note that ∗R is not Dedekind-complete. One could object that the classical form of the MVT is not a key result in modern analysis. Thus, in Lars Hörmander's theory of partial differential operators [13, p. 12–13], a key role is played by various multivariate generalisations of the following Taylor (integral) remainder formula: f(b) = f(a) + (b− a)f ′(a) + ∫ b a (b− x)f ′′(x)dx. (1) Denoting by D the differentiation operator and by I = I(f, a, b) the definite integration operator, we can state (1) in the following more detailed form for a function f : (∀a ∈ R)(∀b ∈ R) f(b) = f(a) + (b− a)(Df)(a) + I ( (b− x)(D2f), a, b ) (2) Applying the transfer principle to the elementary formula (2), we obtain (∀a ∈ ∗R)(∀b ∈ ∗R) ∗f(b) = ∗f(a) + (b− a)(∗D ∗f)(a) + ∗I ( (b− x)(∗D2 ∗f), a, b ) (3) for the natural hyperreal extension ∗f of f . The formula (3) is valid on every hyperreal interval of ∗R. Multivariate generalisations of (1) can be handled similarly. V. Kanovei, K.U. Katz, M.G. Katz, and T. Mormann 115 5.3. Mean Value Theorem We have focused on the MVT (and its generalisations) because, historically speaking, it was emphasized by Klein and Fraenkel. The transfer principle applies far more broadly, as can be readily guessed from the above. The mean value theorem is immediate from Rolle's theorem, which in turn follows from the extreme value theorem. For the sake of completeness we include a proof of the extreme value theorem exploiting infinitesimals; see Robinson [23, page 70, Theorem 3.4.13]. Theorem 1. A continuous function f on [0, 1] ⊆ R has a maximum. Proof. The idea is to exploit a partition into infinitesimal subintervals, pick a partition point xi0 where the value of the function is maximal, and take the shadow (see below) of xi0 to obtain the maximum. In more detail, choose infinite hypernatural number H ∈ ∗N \ N. The real interval [0, 1] has a natural hyperreal extension ∗[0, 1] = {x ∈ ∗R : 0 ≤ x ≤ 1}. Consider its partition into H subintervals of equal infinitesimal length 1 H , with partition points xi = i H , i = 0, . . . , H. The function f has a natural extension ∗f defined on the hyperreals between 0 and 1. Among finitely many points, one can always pick a maximal value: (∀n ∈ N) (∃i0 ≤ n) (∀i ≤ n) (f(xi0) ≥ f(xi)). By transfer we obtain (∀n ∈ ∗N) (∃i0 ≤ n) (∀i ≤ n) (∗f(xi0) ≥ ∗f(xi)) , (4) where ∗N is the collection of hypernatural numbers. Applying (4) to n = H ∈ ∗N \N, we see that there is a hypernatural i0 such that 0 ≤ i0 ≤ H and (∀i ∈ ∗N) [ i ≤ H =⇒ ∗f(xi0) ≥ ∗f(xi) ] . (5) Consider the real point c = st(xi0) where st is the standard part function; see Section 5.1. Then c ∈ [0, 1]. By continuity of f at c ∈ R, we have ∗f(xi0) ≈ ∗f(c) = f(c), and therefore st (∗f(xi0)) = ∗f (st(xi0)) = f(c). An arbitrary real point x lies in an appropriate sub-interval of the partition, namely x ∈ [xi, xi+1], so that st(xi) = x, or xi ≈ x. Applying the function st to the inequality in formula (5), we obtain st(∗f(xi0)) ≥ st(∗f(xi)). Hence f(c) ≥ f(x), for all real x, proving c to be a maximum of f (and by transfer, of ∗f as well). 116 What Makes a Theory of Infinitesimals Useful? The partition into infinitesimal subintervals (used in the proof of the extreme value theorem) similarly enables one to define the definite integral as the shadow of an infinite Riemann sum, fulfilling Fraenkel's second requirement, as well; see Section 3. The difficulty of the Klein–Fraenkel challenge was that it required a change in foundational thinking, as we illustrated. 6. Acknowledgments V. Kanovei was partially supported by the RFBR grant no. 17-01-00705. M. Katz was partially supported by the Israel Science Foundation grant no. 1517/12. References [1] Jacques Bair, Piotr B laszczyk, Robert Ely, Valérie Henry, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Taras Kudryk, Semen S. Kutateladze, Thomas McGaffey, Thomas Mormann, David M. Schaps, and David Sherry, "Cauchy, infinitesimals and ghosts of departed quantifiers," Matematichnı Studıı, Volume 47 (2017), Number 2, pages 115– 144. 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