arXiv:1205.0174v1 [math.HO] 1 May 2012 30 . 03A05 a fhmgniy ebi;Rbno;Sei;TeAnalyst. The Stevin; Robinson; Leibniz; homogeneity; of law OSFO EKLYT USL N BEYOND AND RUSSELL TO BERKELEY FROM FOES EBI’ NIIEIAS HI FICTIONALITY, THEIR : LEIBNIZ’S HI OENIPEETTOS N THEIR AND IMPLEMENTATIONS, MODERN THEIR .Peiiaydvlpet 5 7 methodologies Leibnizian of pair 4. A developments 3. Preliminary 2. Introduction 1. ..Ciiu fNewnit9 Nieuwentijt of Critique 4.1. 2000 e od n phrases. and words Key u Prodiisset Cum fhstasedna a fhomogeneity. means of the by law strengthens transcendental assignable Leibniz’s his argument to of of Our inassignable development relating a of as fallacies. strategy calcu- infinitesimals logical modern differential of of for conception thereof. free system criticism was Leibniz’s infinites- Berkeley’s that lus of than moreover, defense grounded show, Leibniz’s firmly syncategore- We that more some argue is by We imals eliminable fiction not paraphrase. pure are rather matic which but fictions, proposed, imaginaries, are Ishiguro like infinitesimals as Leibniz’s fictions, logical not Leibniz. re- to philosophical available and sources mathematical c the Berkeley’s underestimating of by force that icisms, the argue overestimates We others, among of magnitudes. Robinson, inconsistency infinitesimal the historical Leib- the with demonstrating his of reasoning aptly of criticisms as Inspite Berkeley’s light. regards infinitesimal favorable Robinson more sympathies, a histor nizian in the infinitesimal consider to the tradi- others of Leibnizian and Laugwitz, the Lakatos, with inspired tion identification is whose exception are notable himself, commentators A many Robinson continuity. thus historical re- ; a infinitesimals, denying modern comfortable of of theory resources hy- consistent the quire Robinson’s a providing theory. while Robinson’s 20th perreals, as and century such 17th developments the century of calculus infinitesimal between tinuity Abstract. ahmtc ujc Classification. Subject IHI .KT N AI SHERRY DAVID AND KATZ G. MIKHAIL ayhsoin ftecluu eysgicn con- significant deny calculus the of historians Many ekly otnu;ifiieia;lwo continuity; of law infinitesimal; continuum; Berkeley; Contents 1 rmr 63;Scnay01A85, Secondary 26E35; Primary rit- s, y 3 9 2 MIKHAILG.KATZANDDAVIDSHERRY

4.2. , with examples 10 4.3. Souverain principe 12 4.4. Status transitus 12 4.5. Mathematical implementation of status transitus 14 4.6. Assignable versus unassignable 16 5. Laws of continuity and homogeneity 18 5.1. From secant to 18 5.2. The arithmetic of infinities in De Quadratura Arithmetica 19 5.3. Transcendental law of homogeneity 20 5.4. Syncategorematics 21 5.5. Punctiform and non-punctiform continua 24 6. “Marvellous sharpness of Discernment” 24 6.1. Metaphysical criticism 25 6.2. Logical criticism 27 7. Berkeley’s critique of the : “The acme of lucidity”? 28 7.1. Berkeley’s critique 28 7.2. Rebuttal of Berkeley’s logical criticism 30 8. “A thorn in their sides” 31 8.1. Felix Klein on infinitesimal calculus 31 8.2. Did George slay the infinitesimal? 32 9. Varieties of continua 33 10. A continuity between Leibnizian and modern infinitesimals? 35 11. Commentators from Russell onward 38 11.1. Russell’s non-sequiturs 38 11.2. Earmanandthest-function 40 11.3. Bos’ appraisal 43 11.4. Serfati’s creationist epistemology 46 11.5. Cauchy and Moigno as seen by Schubring 48 11.6. Bishop and Connes 50 12. Horv´ath’s analysis 51 12.1. Leibniz and Nieuwentijt 51 12.2. Leibniz’s pair of dual methodologies for justifying the calculus 52 12.3. Incomparable quantities and the 52 12.4. From Leibniz’s Elementa to his Nova Methodus 53 12.5. Justification du calcul des infinitesimales 54 13. A Leibniz–Robinson route out of the labyrinth? 54 Acknowledgements 55 Appendix A. Rival continua 55 A.1. Hyperreals via maximal ideals 56 A.2. Example 59 LEIBNIZ’S INFINITESIMALS 3

A.3. Construction of standard part 59 A.4. The 59 References 60

“. No reasoning about things whereof we have no idea. Therefore no reasoning about Infinitesimals.” G. Berkeley, Philosophical commentaries (no. 354).

1. Introduction Many historians of the calculus deny any significant continuity be- tween infinitesimal analysis of the 17th century and non-standard anal- ysis of the 20th century, e.g., the work of Robinson. While Robinson’s non-standard analysis constitutes a consistent theory of infinitesimals, it requires the resources of modern logic;1 thus many commentators are comfortable denying a historical continuity: [T]here is . . . no evidence that Leibniz anticipated the techniques (much less the theoretical underpinnings) of modern non-standard analysis (Earman 1975, [32, p. 250]). The relevance of current accounts of the infinitesimal to issues in the seventeenth and eighteenth centuries is rather minimal (Jesseph 1993, [57, p. 131]). Robinson himself takes a markedly different position. Owing to his formalist attitude toward mathematics, Robinson sees Leibniz as a kin- dred soul: Leibniz’s approach is akin to Hilbert’s original formal- ism, for Leibniz, like Hilbert, regarded infinitary entities as ideal, or fictitious, additions to concrete mathematics. (Robinson 1967, [127, p. 40]). The Hilbert is similarly reiterated by Jesseph in a recent text.2 Like Leibniz, Robinson denies that infinitary entities are real, yet he promotes the development of mathematics by means of infini- tary concepts [128, p. 45], [126, p. 282]. Leibniz’s was a remarkably modern insight that mathematical expressions need not have a refer- ent, empirical or otherwise, in order to be meaningful. The fictional nature of infinitesimals was stressed by Leibniz in 1706 in the following terms: 1It is often claimed that the hyperreals require the resources of . See Appendix A for a more nuanced view. 2See main text at footnote 7. 4 MIKHAILG.KATZANDDAVIDSHERRY

assignable ❀LC inassignable TLH❀ assignable quantities  quantities  quantities

Figure 1. Leibniz’s law of continuity (LC) takes one from assignable to inassignable quantities, while his transcenden- tal law of homogeneity (TLH) returns one to assignable quantities.

Philosophically speaking, I no more admit magnitudes infinitely small than infinitely great . . . I take both for mental fictions, as more convenient ways of speaking, and adapted to calculation, just like imaginary roots are in . (Leibniz to Des Bosses, 11 March 1706; in Gerhardt [46, II, p. 305]) We shall argue that Leibniz’s system for the calculus was free of con- tradiction, and incorporated versatile heuristic principles such as the law of continuity and the transcendental law of homogeneity (see Fig- ure 1) which were, in the fullness of time, amenable to mathematical implementation as general principles governing the manipulation of in- finitesimal and infinitely large quantities. And we shall be particularly concerned to undermine the view that Berkeley’s objections to the in- finitesimal calculus were so decisive that an entirely different approach to infinitesimals was required. Jesseph suggests an explanation for the irrelevance of modern in- finitesimals to issues in the seventeenth and eighteenth centuries: The mathematicians of the seventeenth and eighteen