David M. Schaps

The Woman Least Mentioned: Etiquette And Women's Names

‘And if I must make some mention of the virtue of those wives who will now bein widowhood, I will indicate all with a brief word of advice. To be no worse thanyour proper nature, is a great honour for you; andgreat honour is hers, whose reputation among males is least, whether for praise or for blame.’

Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond

We analyze some of the main approaches in the literature to the method of ‘adequality’ with which Fermat approached the problems of the calculus, as well as its source in the παρισότης of Diophantus, and propose a novel reading thereof. Adequality is a crucial step in Fermats method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermats collected works (62, pp. 133–172). We show that at least some of the manifestations of adequality amount to variational techniques exploiting a small, or infinitesimal, variation e. Fermats treatment of geometric and physical applications suggests that an aspect of approximation is inherent in adequality, as well as an aspect of smallness on the part of e. We question the relevance to understanding Fermat of 19th century dictionary definitions of παρισότης and adaequare, cited by Breger, and take issue with his interpretation of adequality, including his novel reading of Diophantus, and his hypothesis concerning alleged tampering with Fermats texts by Carcavy. We argue that Fermat relied on Bachets reading of Diophantus. Diophantus coined the term παρισότης for mathematical purposes and used it to refer to the way in which 1321/711 is approximately equal to 11/6. Bachet performed a semantic calque in passing from parisoo to adaequo. We note the similar role of, respectively, adequality and the Transcendental Law of Homogeneity in the work of, respectively, Fermat (1896) and Leibniz (1858) on the problem of maxima and minima.

Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends 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 (i.e., 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 show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions.

Interpreting the Infinitesimal Mathematics of Leibniz and Euler

Abstract 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.

War and Peace, Imitation and Innovation, Backwardness and Development: The Beginnings of Coinage in Ancient Greece and Lydia

The economic revolution produced by the invention of coinage was both conducive to and facilitated by a number of smaller, incremental innovations, some in the coin itself (the use of silver rather than electrum, the use of bronze for small amounts, fiduciary coinage), some in the ways of doing business (retail commerce, salaried labor, new credit systems such as the small lender, the bank, bottomry loans and transfer orders), some in the management of money (public finance). Paradoxically but not uncharacteristically, it was the more primitive Greek society that was more open to innovation, and that used the new coins in ways undreamed of in the more developed Near Eastern economies. War may have been what engendered the original innovation; peace was what nourished it, so that it grew and prospered.

Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms

AbstractCauchys 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.

Cauchy, Infinitesimals and ghosts of departed quantifiers

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinsons frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibnizs distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinsons framework, while Leibnizs 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 Leibnizs 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 Robinsons framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Eulers own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinsons 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. Cauchys procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinsons framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Keywords: historiography; infinitesimal; Latin model; butterfly model