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The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. André Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve cannot be so treated and involve additional tools-leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat's manuscript but it is Breger who tampers with Fermat's procedure by moving all terms to the left-hand side so as to accord better with Breger's own interpretation emphasizing the double root idea. We provide modern proxies for Fermat's procedures in terms of relations of infinite proximity as well as the standard part function., Author(s): Jacques Bair [sup.1] , Mikhail G. Katz [sup.2] , David Sherry [sup.3] Author Affiliations: (Aff1) 0000 0001 0805 7253, grid.4861.b, HEC-ULG, University of Liege, , 4000, Liège, Belgium (Aff2) [...]

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy's proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy's proof closely and show that it finds closer proxies in a different modern framework., Author(s): Tiziana Bascelli [sup.1] , Piotr Blaszczyk [sup.2] , Alexandre Borovik [sup.3] , Vladimir Kanovei [sup.4] , Karin U. Katz [sup.5] , Mikhail G. Katz [sup.5] , Semen S. Kutateladze [...]

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians' approach to interpreting James Gregory's expression ultimate terms in his paper attempting to prove the irrationality of [Formula omitted]. Here Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an 'unrigorous' practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory's teacher degli Angeli, and Gregory's books at Venice, in the late 1660s., Author(s): Tiziana Bascelli [sup.1] , Piotr Blaszczyk [sup.2] , Vladimir Kanovei [sup.3] [sup.4] , Karin U. Katz [sup.5] , Mikhail G. Katz [sup.5] , Semen S. Kutateladze [sup.6] , Tahl [...]

Abraham Robinson's framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson's framework is more helpful in understanding their procedures than a Weierstrassian framework., Author(s): Piotr Blaszczyk [sup.1] , Vladimir Kanovei [sup.2] [sup.3] , Karin U. Katz [sup.4] , Mikhail G. Katz [sup.4] , Semen S. Kutateladze [sup.5] , David Sherry [sup.6] Author Affiliations: [...]

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on procedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones., Author(s): Piotr Blaszczyk [sup.1] , Vladimir Kanovei [sup.2] [sup.3] , Karin U. Katz [sup.4] , Mikhail G. Katz [sup.4] , Taras Kudryk [sup.5] , Thomas Mormann [sup.6] , David Sherry [...]

The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev's claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev's grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and clearer system (IST). Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals., Author(s): Alexander E. Gutman [sup.1] , Mikhail G. Katz [sup.2] , Taras S. Kudryk [sup.3] , Semen S. Kutateladze [sup.1] Author Affiliations: (1) Sobolev Institute of Mathematics, Novosibirsk State University, [...]

We apply Benacerraf's distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass's ghost behind some of the received historiography on Euler's infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a 'primary point of reference for understanding the eighteenth-century theories.' Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler's own. Euler's use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler's principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro's assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler's work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves., Author(s): Jacques Bair [sup.1] , Piotr Blaszczyk [sup.2] , Robert Ely [sup.3] , Valérie Henry [sup.4] , Vladimir Kanovei [sup.5] , Karin U. Katz [sup.6] , Mikhail G. Katz [sup.6] [...]

Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy's use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy's mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit context of Moigno's flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz., Author(s): Piotr Blaszczyk [sup.1] , Vladimir Kanovei [sup.2] [sup.3] , Mikhail G. Katz [sup.4] , David Sherry [sup.5] Author Affiliations: (1) 0000 0001 2113 3716grid.412464.1Institute of Mathematics, Pedagogical University of [...]

The small, the tiny, and the infinitesimal (to quote Paramedic) have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was [...]

We examine some of Connes' criticisms of Robinson's infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes' own earlier work in functional analysis. Connes described the hyperreals as both a 'virtual theory' and a 'chimera', yet acknowledged that his argument relies on the transfer principle. We analyze Connes' 'dart-throwing' thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being 'virtual' if it is not definable in a suitable model of ZFC. If so, Connes' claim that a theory of the hyperreals is 'virtual' is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren't definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes' criticism of virtuality. We analyze the philosophical underpinnings of Connes' argument based on Godel's incompleteness theorem, and detect an apparent circularity in Connes' logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace - (featured on the front cover of Connes' magnum opus) and the Hahn-Banach theorem, in Connes' own framework. We also note an inaccuracy in Machover's critique of infinitesimal-based pedagogy. Keywords Axiom of choice * Dixmier trace * Hahn-Banach theorem * Hyperreal * Inaccessible cardinal * Godel's incompleteness theorem * Infinitesimal * Klein-Fraenkel criterion * Leibniz * Noncommutative geometry * P-point * Platonism * Skolem's non-standard integers * Solovay models * Ultrafilter Mathematics Subject Classification (2000) Primary 26E35 * Secondary 03A05, 1 Infinitesimals from Robinson to Connes via Choquet 2 Tools and Objects 2.1 Tool/Object Dichotomy 2.2 The Results of Solovay and Shelah 2.3 The Incompleteness Theorem: Evidence for Platonism? 2.4 [...]

The Dirac [delta] function has solid roots in nineteenth century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his infinitesimal definition of [delta]. Keywords Cauchy * Charles Sanders Peirce * Continuum * Delta function * Felix Klein * Dirac * Hyperreals * Infinitesimal, 1 Dirac on Infinity The specialisation of the scientific disciplines since the nineteenth century has led to a schism between the scientists' pragmatic approaches (e.g., using infinitesimal arguments), on the [...]

The widespread idea that infinitesimals were 'eliminated' by the 'great triumvirate' of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d'Alembert, Cauchy, and others. Keywords Abraham Robinson * Adequality * Archimedean continuum * Bernoullian continuum * Cantor * Cauchy * Cognitive bias * Completeness * Constructivism * Continuity * Continuum * du Bois-Reymond * Epsilontics * Felix Klein * Fermat-Robinson standard part * Infinitesimal * Leibniz-Los transfer principle * Limit * Mathematical rigor * Nominalism * Non-Archimedean * Simon Stevin * Stolz * Sum theorem * Weierstrass, 1 Introduction Here are some common claims. The founders of infinitesimal calculus were working in a vacuum caused by an absence of a satisfactory number system. The incoherence of infinitesimals [...]

Cauchy's contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Bjorling, an infinitesimal definition of the criterion of uniform convergence. Cauchy's foundational stance is hereby reconsidered. Keywords Archimedean axiom * Bernoulli * Cauchy * Continuity * Continuum * du Bois-Reymond * Epsilontics * Felix Klein * Hyperreals * Infinitesimal * Stolz * Sum theorem * Transfer principle * Ultraproduct * Weierstrass, 1 Introduction When the second-named author first came across a recent book by Anderson et al. entitled Who Gave You the Epsilon? And Other Tales of Mathematical History (Anderson et [...]

We explore the potential of Simon Stevin's numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis. Keywords Charles Sanders Peirce * Continuity * Fermat-Robinson standard part * Intermediate value theorem * Leibniz-Los transfer principle * Real decimals * Triumvirate nominalistic scholarship, 1 From Discrete Arithmetic to Arithmetic of the Continuum Simon Stevin (1548-1620) initiated a systematic approach to decimal representation of measuring numbers, marking a transition from a discrete arithmetic as [...]

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research. Keywords Abraham Robinson * Adequality * Archimedean continuum * Bernoullian continuum * Burgess * Cantor * Cauchy * Completeness * Constructivism * Continuity * Dedekind * Du Bois-Reymond * Epsilontics * Errett Bishop * Felix Klein * Fermat-Robinson standard part * Infinitesimal * Law of excluded middle * Leibniz-Los transfer principle * Nominalistic reconstruction * Nominalism * Non-Archimedean * Rigor * Simon Stevin * Weierstrass, 1 Introduction Over the course of the past 140 years, the field of professional pure mathematics (analysis in particular), and to a large extent also its professional historiography, have become [...]