arXiv subjects

David Sherry

Publications and source records attributed to David Sherry.

30 records · Page 2Linked to original sources

arXiv2016

A non-standard analysis of a cultural icon: The case of Paul Halmos

We examine Paul Halmos' comments on category theory, Dedekind cuts, devil worship, logic, and Robinson's infinitesimals. Halmos' scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is "certainty" and "architecture" yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lession, namely that the castle is floating in midair. Halmos' realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians' concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson's framework is "unnecessary" but Henson and Keisler argue that Robinson's framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos' criticisms. Keywords: Archimedean axiom; bridge between discrete and continuous mathematics; hyperreals; incomparable quantities; indispensability; infinity; mathematical realism; Robinson.

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arXiv2016

Is Leibnizian calculus embeddable in first order logic?

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. Keywords: First order logic; infinitesimal calculus; ontology; procedures; Leibniz; Weierstrass; Abraham Robinson

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arXiv2016

Interpreting the infinitesimal mathematics of Leibniz and Euler

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 17th and 18th 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 is 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. Keywords: Archimedean axiom; infinite product; infinitesimal; law of continuity; law of homogeneity; principle of cancellation; procedure; standard part principle; ontology; mathematical practice; Euler; Leibniz

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arXiv2016

Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania

Did Leibniz exploit infinitesimals and infinities `a la rigueur, or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Chapter 5 in (Ishiguro 1990) is a defense of the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz's repeated assertions that infinitesimals violate the Archimedean property, viz., Euclid's Elements, V.4. We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro's interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree; but we shall show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. Keywords: Archimedean property; infinitesimal; logical fiction; pure fiction; quantified paraphrase; law of homogeneity

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arXiv2016

Controversies in the foundations of analysis: Comments on Schubring's Conflicts

Foundations of Science recently published a rebuttal to a portion of our essay it published two 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 variousmisconceptions 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. Keywords: Archimedean axiom, Cauchy, Felix Klein, hornangle, infinitesimal, Leibniz, ontology, procedure

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arXiv2015

Euler's lute and Edwards' oud

In a piece published in 1981, H. M. Edwards touts the benefits of reading the masters. A quarter-century later, Edwards takes seriously his own advice by publishing an encomium on Euler's Institutiones (1755). While we agree with Edwards that we shall all do well by studying the masters, we argue that to derive the full benefit, one must read historically important texts without interposing a lens formed by one's own mathematical accomplishments. We show, in particular, that Edwards misses much of the style and substance of Euler's Institutiones by reading the text through a constructivist lens.

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arXiv2014

Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern theories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions.

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arXiv2013

Is mathematical history written by the victors?

We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal mathematics in a new light. We also detail several procedures of the historical infinitesimal calculus that were only clarified and formalized with the advent of modern infinitesimals. These procedures include Fermat's adequality; Leibniz's law of continuity and the transcendental law of homogeneity; Euler's principle of cancellation and infinite integers with the associated infinite products; Cauchy's infinitesimal-based definition of continuity and "Dirac" delta function. Such procedures were interpreted and formalized in Robinson's framework in terms of concepts like microcontinuity (S-continuity), the standard part principle, the transfer principle, and hyperfinite products. We evaluate the critiques of historical and modern infinitesimals by their foes from Berkeley and Cantor to Bishop and Connes. We analyze the issue of the consistency, as distinct from the issue of the rigor, of historical infinitesimals, and contrast the methodologies of Leibniz and Nieuwentijt in this connection.

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arXiv2013

Infinitesimals, Imaginaries, Ideals, and Fictions

Leibniz entertained various conceptions of infinitesimals, considering them sometimes as ideal things and other times as fictions. But in both cases, he compares infinitesimals favorably to imaginary roots. We agree with the majority of commentators that Leibniz's infinitesimals are fictions rather than ideal things. However, we dispute their opinion that Leibniz's infinitesimals are best understood as logical fictions, eliminable by paraphrase. This so-called syncategorematic conception of infinitesimals is present in Leibniz's texts, but there is an alternative, formalist account of infinitesimals there too. We argue that the formalist account makes better sense of the analogy with imaginary roots and fits better with Leibniz's deepest philosophical convictions. The formalist conception supports the claim of Robinson and others that the philosophical foundations of nonstandard analysis and Leibniz's calculus are cut from the same cloth.

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arXiv2012

Leibniz's Laws of Continuity and Homogeneity

We explore Leibniz's understanding of the differential calculus, and argue that his methods were more coherent than is generally recognized. The foundations of the historical infinitesimal calculus of Newton and Leibniz have been a target of numerous criticisms. Some of the critics believed to have found logical fallacies in its foundations. We present a detailed textual analysis of Leibniz's seminal text Cum Prodiisset, and argue that Leibniz's system for differential calculus was free of contradictions.

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arXiv2012

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.

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arXiv2012

Ten Misconceptions from the History of Analysis and Their Debunking

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.

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