We apply Benacerraf’s distinction between mathematical ontology and mathematical 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. (shrink)
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. (shrink)
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 \. 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. (shrink)
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.
Professor Grünbaum's much-discussed refutation of Zeno's metrical paradox turns out to be ad hoc upon close examination of the relevant portion of measure theory. Although the modern theory of measure is able to defuse Zeno's reasoning, it is not capable of refuting Zeno in the sense of showing his error. I explain why the paradox is not refutable and argue that it is consequently more than a mere sophism.
Selections from Hume's major writings are grouped under the headings: Reason and Experience, Reason and Sentiment, and Reason and Religion. There is also a short conclusion entitled "Skepticism." A Treatise on Human Nature, An Enquiry Concerning the Human Understanding, and An Enquiry Concerning the Principles of Morals are from the 1962 and 1947 translations by André Leroy. The Dialogues on Natural Religion were translated in 1912 by Maxime David. Part I gives Hume's account of impressions, ideas, and their relations. (...) Also covered are the crucial arguments on causality from the Treatise and the Enquiry Concerning Understanding--including the role of experience of constant conjunction and the role of instinct in our construction and use of the notion of causality. Part II contains the famous statement from the Treatise that moral matters are "more rightly felt than judged of" and a treatment of the natural and artificial virtues. Considering its central place in recent ethics, the English-speaking reader would miss the familiar lines remarking the passage from "is" to "ought." Part III and the Conclusion are drawn entirely from the Dialogues on Natural Religion.--M. B. M. (shrink)
F. A. Hayek is uniquely responsible for his fellow economists grasping the importance of the decentralization of knowledge: as Hayek shows in his pathbreaking “The Use of Knowledge in Society,” knowledge nowhere exists as a coherent whole and to pretend otherwise is a most serious error. Hayek also shares responsibility for the popularity of a strong form of the methodological individualist research program which asserts that since collectives as such have no impact on the choices of individuals, investigators ought to (...) purge any reliance on collectives from our analysis. (shrink)
It used to be that a book on utopia that did not quote Oscar Wilde's homily about a map of the world without utopia was itself not worth glancing at, for it left out the one thing we thought we could all agree on. But what if the world map only serves to reinforce the systems of domination inherent to colonialism, racism, capitalism, and patriarchy? And why should the quest for utopia take us to the high seas anyway, rather than (...) surveying those existing social formations that resist oppression?For David Bell, Wilde was wrong. Utopia is on the world map, but it lies neither in the unexplored place on the horizon for which humanity is forever setting sail nor in some nonexistent linear future: following Tom Moylan's... (shrink)
Wittgenstein's Method: Neglected Aspects By Gordon Baker. Oxford: Blackwell, 2004 pp. 328. £40.00 HB.. Wittgenstein's Copernican Revolution: The Question of Linguistic Idealism By Ilham Dilman. Basingstoke: Palgrave, 2002. pp. 240. £52.50 HB. Wittgenstein: Connections and Controversies By P. M. S. Hacker. Oxford: Oxford University Press,. pp. 400. £45.00 HB; £19.99 PB. Wittgenstein's Philosophical Investigations: An Introduction By David G. Stern. Cambridge: Cambridge University Press, 2004. pp. 224. £40.00 HB; £10.99 PB.
In “The Jesuits and the Method of Indivisibles” DavidSherry criticizes a central thesis of my book Infinitesimal: that in the seventeenth century the Jesuits sought to suppress the method of indivisibles because it undermined their efforts to establish a perfect rational and hierarchical order in the world, modeled on Euclidean Geometry. Sherry accepts that the Jesuits did indeed suppress the method, but offers two objections. First, that the book does not distinguish between indivisibles and infinitesimals, and (...) that whereas the Jesuits might have reason to object to the first, the second posed no problem for them. Second, seeking an alternative explanation for the Jesuits’ hostility to the method, he proposes that its implicit atomism conflicted with the Catholic doctrine of the sacrament of the Eucharist, and was therefore heretical. In response to Sherry’s first criticism I point out that the Jesuits objected to all forms of the method of indivisibles, and that the distinction between indivisibles and infinitesimals consequently cannot explain the fight over the method in the seventeenth century. With regards to the Eucharist, I agree with Sherry that the Jesuits were indeed concerned about the method’s affinity to atomism and materialism, though for a different reason: these doctrines were antithetical to their efforts to impose divine hierarchy and order on the world. In as much as the technical details of the miracle of the Eucharist mattered, they provided no grounds for objecting to a mathematical doctrine. (shrink)