Searcharxiv⌕ Search

arXiv subjects

David Sherry

Publications and source records attributed to David Sherry.

At least 19 recordsLinked to original sources
arXiv2025

Formalism 25

Abraham Robinson's philosophical stance has been the subject of several recent studies. Erhardt following Gaifman claims that Robinson was a finitist, and that there is a tension between his philosophical position and his actual mathematical output. We present evidence in Robinson's writing that he is more accurately described as adhering to the philosophical approach of Formalism. Furthermore, we show that Robinson explicitly argued {against} certain finitist positions in his philosophical writings. There is no tension between Robinson's mathematical work and his philosophy because mathematics and metamathematics are distinct fields: Robinson advocates finitism for metamathematics but no such restriction for mathematics. We show that Erhardt's analysis is marred by historical errors, by routine conflation of the generic and the technical meaning of several key terms, and by a philosophical {parti pris}. Robinson's Formalism remains a viable alternative to mathematical Platonism.

math.HO↗
arXiv2025

Of pashas, popes, and indivisibles

The studies of Bonaventura Cavalieri's indivisibles by Giusti, Andersen, Mancosu and others provide a comprehensive picture of Cavalieri's mathematics, as well as of the mathematical objections to it as formulated by Paul Guldin and other critics. An issue that has been studied in less detail concerns the theological underpinnings of the contemporary debate over indivisibles, its historical roots, the geopolitical situation at the time, and its relation to the ultimate suppression of Cavalieri's religious order. We analyze sources from the 17th through 21st centuries to investigate such a relation.

math.HO↗
arXiv2024

A Leibniz/NSA comparison

We present some similarities between Leibnizian and Robinsonian calculi, and address some objections raised by historians. The comparison with NSA facilitates our appreciation of some Leibnizian procedures that may otherwise seem obscure. We argue that Leibniz used genuine infinitesimals and infinite quantities which are not merely stenography for Archimedean Exhaustion and that Leibniz's procedures therefore find better proxies in NSA than in modern Weierstrassian mathematics.

math.HO↗
arXiv2024

Exploring Felix Klein's contested modernism

An alleged opposition between David Hilbert and Felix Klein as modern vs countermodern has been pursued by marxist historian Herbert Mehrtens and others. Scholars such as Epple, Grattan-Guinness, Gray, Quinn, Rowe, and recently Siegmund-Schultze and Mazzotti have voiced a range of opinions concerning Mehrtens' dialectical methodology. We explore contrasting perspectives on Klein's contested modernism, as well as Hilbert's and Klein's views on intuition, logic, and physics. We analyze Jeremy Gray's comment on Klein's ethnographic speculations concerning Jewish mathematicians and find it to be untenable. We argue that Mehrtens was looking for countermoderns at the wrong address.

math.HO↗
arXiv2023

When does a hyperbola meet its asymptote? Bounded infinities, fictions, and contradictions in Leibniz

In his 1676 text De Quadratura Arithmetica, Leibniz distinguished infinita terminata from infinita interminata. The text also deals with the notion, originating with Desargues, of the perspective point of intersection at infinite distance for parallel lines. We examine contrasting interpretations of these notions in the context of Leibniz's analysis of asymptotes for logarithmic curves and hyperbolas. We point out difficulties that arise due to conflating these notions of infinity. As noted by Rodriguez Hurtado et al., a significant difference exists between the Cartesian model of magnitudes and Leibniz's search for a qualitative model for studying perspective, including ideal points at infinity. We show how respecting the distinction between these notions enables a consistent interpretation thereof.

math.HO↗
arXiv2022

Historical infinitesimalists and modern historiography of infinitesimals

In the history of infinitesimal calculus, we trace innovation from Leibniz to Cauchy and reaction from Berkeley to Mansion and beyond. We explore 19th century infinitesimal lores, including the approaches of Simeon-Denis Poisson, Gaspard-Gustave de Coriolis, and Jean-Nicolas Noel. We examine contrasting historiographic approaches to such lores, in the work of Laugwitz, Schubring, Spalt, and others, and address a recent critique by Archibald et al. We argue that the element of contingency in this history is more prominent than many modern historians seem willing to acknowledge.

math.HO↗
arXiv2022

Three case studies in current Leibniz scholarship

We examine some recent scholarship on Leibniz's philosophy of the infinitesimal calculus. We indicate difficulties that arise in articles by Bassler, Knobloch, and Arthur, due to a denial to Leibniz's infinitesimals of the status of mathematical entities violating Euclid V Definition 4.

math.HO↗
arXiv2021

Leibniz on bodies and infinities: rerum natura and mathematical fictions

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part-whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals - as well as imaginary roots and other well-founded fictions - may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality.

math.HO↗
arXiv2021

Two-track depictions of Leibniz's fictions

Leibniz described imaginary roots, negatives, and infinitesimals as useful fictions. But did he view such 'impossible' numbers as mathematical entities? Alice and Bob take on the labyrinth of the current Leibniz scholarship.

math.HO↗
arXiv2019

On mathematical realism and the applicability of hyperreals

We argue that Robinson's hyperreals have just as much claim to applicability as the garden variety reals. In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between techniques that are applicable and those that are merely instrumental. Unfortunately the authors have not shown that their distinction is a clear and fruitful one, as the examples they provide are superficial and unconvincing. Moreover, their analysis is vitiated by a reliance on a naive version of object realism which has long been abandoned by most philosophical realists in favor of truth-value realism. ET's argument against the applicability of hyperreals based on automorphisms of hyperreal models involves massaging the evidence and is similarly unconvincing. The purpose of the ET text is to argue that Robinson's infinitesimal analysis is merely instrumental rather than applicable. Yet in spite of Robinson's techniques being applied in physics, probability, and economics, ET don't bother to provide a meaningful analysis of even a single case in which these techniques are used. Instead, ET produce page after page of speculations mainly imitating Connesian chimera-type arguments `from first principles' against Robinson. In an earlier paper Easwaran endorsed real applicability of the sigma-additivity of measures, whereas the ET text rejects real applicability of the axiom of choice, voicing a preference for ZF. Since it is consistent with ZF that the Lebesgue measure is not sigma-additive, Easwaran is thereby walking back his earlier endorsement. We note a related inaccuracy in the textbook Measure Theory by Paul Halmos.

math.HO↗
arXiv2018

Klein vs Mehrtens: restoring the reputation of a great modern

Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens' conclusions have been picked up by both historians (Jeremy Gray) and mathematicians (Frank Quinn). We argue that Klein and Hilbert, both at Goettingen, were not adversaries but rather modernist allies in a bid to broaden the scope of mathematics beyond a narrow focus on arithmetized analysis as practiced by the Berlin school. Klein's Goettingen lecture and other texts shed light on Klein's modernism. Hilbert's views on intuition are closer to Klein's views than Mehrtens is willing to allow. Klein and Hilbert were equally interested in the axiomatisation of physics. Among Klein's credits is helping launch the career of Abraham Fraenkel, and advancing the careers of Sophus Lie, Emmy Noether, and Ernst Zermelo, all four surely of impeccable modernist credentials. Mehrtens' unsourced claim that Hilbert was interested in production rather than meaning appears to stem from Mehrtens' marxist leanings. Mehrtens' claim that [the future SS-Brigadefuehrer] "Theodor Vahlen ... cited Klein's racist distinctions within mathematics, and sharpened them into open antisemitism" fabricates a spurious continuity between the two figures mentioned and is thus an odious misrepresentation of Klein's position. Keywords: arithmetized analysis; axiomatisation of geometry; axiomatisation of physics; formalism; intuition; mathematical realism; modernism; Felix Klein; David Hilbert; Karl Weierstrass

math.HO↗
arXiv2018

Fermat's dilemma: Why did he keep mum on infinitesimals? and the European theological context

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. Andre 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. Keywords: adequality; atomism; cycloid; hylomorphism; indivisibles; infinitesimal; jesuat; jesuit; Edict of Nantes; Council of Trent 13.2

math.HO↗
arXiv2017

Cauchy, infinitesimals and ghosts of departed quantifiers

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Keywords: historiography; infinitesimal; Latin model; butterfly model

math.HO↗
arXiv2017

Analyzing Benardete's comment on decimal notation

Philosopher Benardete challenged both the conventional wisdom and the received mathematical treatment of zero, dot, nine recurring. An initially puzzling passage in Benardete on the intelligibility of the continuum reveals challenging insights into number systems, the foundations of modern analysis, and mathematics education. A key concept here is, in Terry Tao's terminology, that of an ultralimit. Keywords: real analysis; infinitesimals; decimal notation; procedures vs ontology

math.CA↗
arXiv2017

Cauchy's infinitesimals, his sum theorem, and foundational paradigms

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. Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.

math.HO↗
arXiv2016

Gregory's sixth operation

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 pi. 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.

math.HO↗
arXiv2016

Toward a history of mathematics focused on procedures

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.

math.HO↗