This article explores George Berkeley's philosophy of mathematics, in comparison with his philosophy of religion, with particular attention to his book, The Analyst, and other contemporaneous texts. Through this comparison, it sheds light on his real attitude to the calculus, as well as other mathematical impossibilities such as negative or imaginary numbers. In both mathematics and religion, Berkeley rejected "barren speculation," but he found value in both from their practical benefits in life. Viewed in this way, it turns out that (...) he really wasn't an enemy of the calculus at all, as he has often been portrayed. (shrink)

George Berkeley is known as “The Good Bishop,” a name celebrating his faith, pastoral ministry and earnest commitment to his philosophical views. To mathematicians, he is known for his agitated performance in his 1734 critique of fluxions, The Analyst. That work and its petulant tone were occasioned by (i) his “philo-mathematical” opponents’ alleged admonitions on religious mysteries’ lack of logical respectability and (ii) what Berkeley saw as a related public appetite for reformist and deist religious movements. This paper questions Berkeley’s (...) saintly reputation by outlining the hostility and flippancy in his handling of Jacob Walton’s contributions to the debate. Berkeley’s interactions with Walton bear many of the hallmarks of contemporary “trolling” culture and this paper emphasises and explains these parallels. Finally, given Berkeley’s theological and political agenda in his mathematical interactions, I argue that we should not read his mathematical rhetoric as straightforwardly representing his philosophical views. (shrink)

This paper consists in an exposition of a proof Newton gave in 1666 of the parallelogram law for compounding velocities, and an examination of its implications for understanding his treatment of motion resulting from a continuously acting force in the Principia. I argue that the “moments” invoked in the fluxional proof of the vector resolution and composition of velocities are “virtual times”, a device allowing Newton to represent motions by the linear displacements produced in such a time; the ratio of (...) velocities at an instant can then be represented by assuming the velocities continue for such a virtual time. By the Method of First and Ultimate Ratios, the first ratio of the velocities is then given by the ratio of such lines, under the presupposition that in the limit they will shrink to zero magnitude. I then argue that the same device is implicit in Newton’s appeal to “moments” or “particles of time” in his proof of Kepler’s Area Law in the “Locke Paper” and in Proposition 1 of Book 1 of the Principia, and that the limiting process involved there is therefore the same as that implicit in the Method of First and Ultimate Ratios. (shrink)

If we consider heritage as a process of exhibiting the past and the present to present these for future observers then encyclopedias are good candidates for assessing what constitutes heritage. We propose to study the Fluxion entries in British encyclopaedias over a long period of time. With the help of this corpus of more than thirty articles, it will then be possible to identify several markers that contribute to making mathematics a heritage object—a reference to history, sources of inspiration (...) and bibliographical references. (shrink)

Mate Meršić (Merchich, 1850-1928) sees the origin of Zeno’s paradox ‘Achilles’ in the ambiguities of the concept of the infinity. According to him (and to the tradition started by Gregory St. Vincent), those ambiguities are resolved by the concept of convergent geometric series. In this connection, Meršić proposes a general ontological theory with the priority of the finite over the infinite, and, proceeding from Newton’s concept of fluxion, he develops a modal interpretation of differential calculus.

This collection of specially commissioned essays by leading scholars presents research on Isaac Newton and his main philosophical interlocutors and critics. The essays analyze Newton's relation to his contemporaries, especially Barrow, Descartes, Leibniz and Locke and discuss the ways in which a broad range of figures, including Hume, Maclaurin, Maupertuis and Kant, reacted to his thought. The wide range of topics discussed includes the laws of nature, the notion of force, the relation of mathematics to nature, Newton's argument for universal (...) gravitation, his attitude toward philosophical empiricism, his use of 'fluxions', his approach toward measurement problems and his concept of absolute motion, together with new interpretations of Newton's matter theory. The volume concludes with an extended essay that analyzes the changes in physics wrought by Newton's Principia. A substantial introduction and bibliography provide essential reference guides. (shrink)

The reconstruction of Leibniz’s metaphysics that Deleuze undertakes in The Fold provides a systematic account of the structure of Leibniz’s metaphysics in terms of its mathematical foundations. However, in doing so, Deleuze draws not only upon the mathematics developed by Leibniz—including the law of continuity as reflected in the calculus of infinite series and the infinitesimal calculus—but also upon developments in mathematics made by a number of Leibniz’s contemporaries—including Newton’s method of fluxions. He also draws upon a number of subsequent (...) developments in mathematics, the rudiments of which can be more or less located in Leibniz’s own work—including the theory of functions and singularities, the Weierstrassian theory of analytic continuity, and Poincaré’s theory of automorphic functions. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics. While the Weierstrassian theory of analytic continuity serves to clarify Leibniz’s work, Poincaré’s theory of automorphic functions offers a solution to overcome and extend the limits that Deleuze identifies in Leibniz’s metaphysics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. This essay is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

Wee people at London, are so humbly immersd in slavish business, & taken up wth providing for a wretched Carkasse; yt there's nothing almost, but what is grosse & sensuall to be gotten from us. If a bright thought springs up any time here, ye Mists & Foggs extinguish it again presently, & leaves us no more, yn only ye pain, of seeing it die & perish away from us. Humphrey Ditton to Roger Cotes, ca. 1703THE CALCULUS OF ACCOMPLISHMENTDuring the (...) last decade of his life, Sir Isaac Newton took the measure of achievement. Probably shortly before 1725, Newton scribbled on the undated cover of a letter a brief list of those discoveries he believed belonged entirely ‘to the English’. Included were ‘the variation of the Variation’ ; the circulation of the blood; telescopic sights and the micrometer variously improved by his contemporaries, Robert Hooke and John Flamsteed; and ‘the Libration of the Moon’ likely in reference to Newton's own explanation of lunar eccentricity. Notably, this was not simply a personal calculation. Newton makes no mention of such controversial matters as the fluxional calculus, the refraction of light, or even the measure of universal gravitation, which he otherwise might have claimed as his own efforts. Even the private lights of the solitary genius could still accommodate a distinctly broader sense of the depth of national accomplishment. (shrink)

Much has been made of Deleuze’s Neo-Leibnizianism,3 however not very much detailed work has been done on the specific nature of Deleuze’s critique of Leibniz that positions his work within the broader framework of Deleuze’s own philo- sophical project. The present chapter undertakes to redress this oversight by providing an account of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in The Fold. Deleuze provides a systematic account of the structure of Leibniz’s metaphys- ics in terms of its mathematical underpinnings. (...) However, in doing so, Deleuze draws upon not only the mathematics developed by Leibniz – including the law of continuity as reflected in the calculus of infinite series and the infinitesimal calculus – but also the developments in mathematics made by a number of Leibniz’s contemporaries – including Newton’s method of fluxions – and a number of subsequent developments in mathematics, the rudiments of which can be more or less located in Leibniz’s own work – including the theory of functions and singularities, the theory of continuity and Poincaré’s theory of automorphic functions. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics. While the theory of continuity serves to clarify Leibniz’s work, Poincaré’s theory of automorphic functions offers a solution to overcome and extend the limits that Deleuze identifies in Leibniz’s metaphysics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. What is provided in this chapter is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

Taking Isaac Newton at his own word, historians have long agreed that the decade of the 1660s, when Newton was a young man in his twenties, was the critical period in his scientific career. In the years 1665 and 1666, he has told us, he hit on the ideas of cosmic gravitation, the composition of white light, and the fluxional calculus. The elaboration of these basic ideas constituted his scientific achievement. Nevertheless, the decade of the 1660s has remained a virtual (...) blank in our knowledge of Newton. It need not remain so always. His papers contain a wealth of manuscripts from his undergraduate years and the period immediately following. The first volume of his mathematical papers, which will soon be published, will demonstrate how extensive the information on his early mathematical development is. The development of his non-mathematical studies, especially of what I shall call his scientific studies to distinguish them from the mathematical, can be followed as well—in his reading notes, in his notebooks, above all in the passage in his philosophical notebook labelled Quaestiones quaedam Philosophicae. In this passage we see emerging into consciousness for the first time the questions on which Newton's philosophy of nature was built. (shrink)

When I read your Defence of the British Mathematicians, I could not Sir, but admire your Courage in asserting with such undoubting Assurance things so easily disproved. This to me seemed unaccountable, till I reflected on what you saywhen upon my having appealed to every thinking Reader, whether it be possible to frame any clear Conception of Fluxions, you express your self in the following manner, "Pray sir who are those thinking Readers you appeal to? Are they Geometricians or Persons (...) wholly ignorant of Geometry? If the former I leave it to them: if the latter, I ask how well are they qualified to judge of the Method of Fluxions?" It must be acknowledged you seem by this Dilemma secure in the favour of one Part of your Readers, and the ignorance of the other. I am nevertheless persuaded there are fair and candid Men among the Mathematicians. And for those who are not Mathematicians, I shall endeavour so to unveil this Mystery, and put the Controversy between us in such a Light, as that every Reader of ordinary Sense and Reflection may be a competent Judge thereof. (shrink)

The Bishop of Cloyne, George Berkeley, was the sort of philosopher who, although most genial himself, was quite apt to embroil opponents and critics of his time and of our own in long-lasting and sometimes unresolved controversies. In attacking the “infidel mathematicians”, the “minute philosophers” among the scientists, Berkeley initiated a controversy on behalf of religion by taking to task the theory of fluxions held by Sir Isaac Newton, his friends, and followers which, beginning with Berkeley's Analyst and replies to (...) it by Jurin and Walton, was continued on over one hundred years by subsequent writers. Florian Cajori in his History of Mathematics , commenting on the affair has written: ‘We must not neglect to express our appreciation of the fact that Berkeley withdrew from the controversy after he had said all that he had to say on his subject. Some of the debates that came later were almost interminable, because the participants continued writing even after they had nothing more to say.’. (shrink)

According to the received view, eighteenth-century British mathematicians were responsible for a decline of mathematics in the country of Newton; a decline attributed to chauvinism and a preference for geometrical thinking. This paper challenges this view by first describing the complexity of Newton's mathematical heritage and its reception in the early decades of the eighteenth century. A section devoted to Maclaurin's monumental Treatise of Fluxions describes its attempt to reach a synthesis of the different strands of Newton's mathematical legacy, and (...) compares it with contemporary Continental work. It is shown that in the middle of the eighteenth century academic Continental mathematicians such as Euler and Lagrange were driven by local cultural assumptions in directions which sensibly diverged from the ones followed by Maclaurin and his fellow countrymen. (shrink)

Mach repudiated Newton's argument for absolute time. He denied there is such a thing as time itself that exists independently of any external change. In doing so, Mach failed to appreciate Newton's scientific practice. Absolute time is intrinsically related to Newton's laws of motion and the method of fluxions. Commentators have noted similarities between Mach's rejection of Newtonian time and his rejection of the independent existence of atoms. In this article, it shall be argued that the juxtaposition of absolute time (...) and the atomic theory is unsound. Mach had good reasons to question the existence of substantial time, and he went on to provide an alternative, ontologically relational account. Whereas his dismissal of atoms can be seen as a questionable form of “phenomenalism” or “positivism,” this is not the case regarding his position on time. (shrink)

Según la concepción heredada, los matemáticos británicos del siglo XVIII fueron responsables de una decadencia de las matemáticas en el país de Newton; una decadencia atribuida al chovinismo y a una preferencia por el pensamiento geométrico. Este artículo debate este punto de vista describiendo, primero, la complejidad de la herencia matemática de Newton y su recepción durante las primeras décadas del siglo XVIII. Una sección dedicada al monumental Treatise of Fluxions (1742) de Maclaurin describe el intento de lograr una síntesis (...) de las diferentes corrientes de pensamiento del legado matemático de Newton, y lo compara con el trabajo contemporáneo continental. Se demuestra que a mediados del siglo XVIII los matemáticos académicos continentales tales como Euler y Lagrange estaban inspirados por suposiciones culturales locales en direcciones que divergían sensiblemente de las seguidas por Maclaurin y sus seguidores conterráneos. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 61.2 (2000) 263-283 [Access article in PDF] Contesting Metaphors and the Discourse of Consciousness in William James Jill M. Kress Ah, not to be cut off,not by such slight partitionto be excluded from the stars' measure.What is inwardness?What if not sky intensified,flung through with birds and deepwith winds of homecoming? --Rainer Maria Rilke William James's lifelong attention to questions about human mental experience (...) elucidates the development of the concept of consciousness through its realization in fields as disparate as natural science, radical empiricism, and religious mysticism. 1 Over the course of a career that both establishes and traverses disciplinary boundaries, James's work embodies tensions between scientific explanations for mental phenomena and the inescapability of metaphysical arguments. Jamesian psychology thus alternates between materialist and spiritualist assumptions of scientists and philosophers at the turn of the century, joining their compulsive investigations into the nature of consciousness. 2 Most readers [End Page 263] of James puzzle over the theoretical contradictions within his work, debating central philosophical dilemmas concerning the status of the conscious self. Perhaps the most paradoxical aspect of James's theories, his response to dualism, emerges as he attempts to negotiate ethereal explanations for consciousness with bodily processes. 3 The mind-body problem disrupts James's more delicately balanced theories, a disruption registered explicitly through the metaphors in his texts.Reading The Principles of Psychology (1890) alongside "Does 'Consciousness' Exist" (1904), this essay demonstrates how figurative language directs James's study of consciousness. I argue that metaphor does more than describe consciousness; metaphor constructs James's arguments, governs his conflicting theories of mind through a series of rhetorical configurations and provokes the continual reconstruction of his ideas of human subjectivity. Furthermore, while it is impossible for James to explain and to study consciousness without metaphor, he experiences great anxiety about using language, figurative or otherwise, to represent his object of study. The language becomes more powerful than his intentions or designs and, indeed, raises implications that James cannot control. James possesses a heightened awareness of what he crafts with words; he also, perhaps more penetratingly, acknowledges the inadequacy of any linguistic system to realize concepts thoroughly. 4 Struggling over appropriate "names" and "terms" for consciousness, James presupposes some clearly delineated concept around which we might wrap a verbal expression; yet he also seems painfully aware that every new metaphor launches an entirely new theory. Fluid and unpredictable, consciousness becomes like language itself, yielding its power precisely because it can be so many things at once. The ultimate principle of Jamesian consciousness seems to be its creative capacity; still, once James admits that the problem of consciousness is a problem of language, his carefully constructed designs begin to unravel. Though consciousness will be translated into James's notion of "pure experience" by the time he writes the radical empiricism [End Page 264] essays, James remains simultaneously committed to his figures (especially of the "stream of consciousness") and ambivalent about the account of consciousness that they provide. James helps create the modern self with its enhanced individuality, though his metaphors at once direct us inward to a centered, private self and propel us outward to find consciousness materializing in the fluxional cycle of the natural world.In presenting his initial theory of the stream of our thoughts, James argues for its coherence. His most famous metaphor of the "stream" of consciousness appears to saturate the varied material of the mind, its water washing over and through any distinctions or separations which the mind might present: Consciousness, then, does not appear to itself chopped up in bits. Such words as "chain" or "train" do not describe it fitly as it presents itself in the first instance. It is nothing jointed; it flows. A "river" or a "stream" are the metaphors by which it is most naturally described. In talking of it hereafter, let us call it the stream of thought, of consciousness, or of subjective life. 5James's desire to find words that are more "natural... (shrink)

The concept of molecular structure is fundamental to the practice and understanding of chemistry, but the meaning of this term has evolved and is still evolving. The Born–Oppenheimer separation of electronic and nuclear motions lies at the heart of most modern quantum chemical models of molecular structure. While this separation introduces a great computational and practical simplification, it is neither essential to the conceptual formulation of molecular structure nor universally valid. Going beyond the Born–Oppenheimer approximation introduces new paradigms, bringing fresh (...) insight into the chemistry of fluxional molecules, proteins, superconductors and macroscopic dielectrics, thus opening up new avenues for exploration. But it requires that our ideas of molecular structure need to evolve beyond simple ball-and-stick-type models. (shrink)

It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental propositions of (...) his calculus follow the analytical trend. This appears to have been possible due to a nominalistic conception of variable that allowed him to deal with expressions as names, rather than abstract quantities. Still, Cunha tried to keep the definition of fluxion directly applicable to geometrical magnitudes. According to a friend of Cunha’s, his calculus had an algebraic (analytical) branch and a geometrical branch, and it was because of this that his definition of fluxion appeared too complex to some contemporaries. (shrink)

El estudio estético de las situaciones cotidianas es, fundamentalmente, el lugar de convergencia de modalidades dramáticas y registros retóricos que dan cuenta de los diversos factores que intervienen en los procesos de significación y comunicación. En este sentido, y a partir de la novela El crimen del siglo, de Miguel Torres, el artículo aquí propuesto tiene como fin presentar el análisis de una discusión entre Juan Roa Sierra, quien aparentemente es el asesino de Jorge Eliécer Gaitán, y su exmujer. Al (...) respecto, Katya Mandoki propone un método de análisis en sus libros sobre Prosaica, en los que tiene en cuenta los registros retóricos: acústico, somático, escópico y léxico, dentro de las modalidades dramáticas: proxémica, cinética, enfática y fluxión. Esta propuesta será la base para el análisis de la discusión de los exmaridos, comprendida como una situación común y cotidiana. Este artículo da cuenta, en últimas, de las posibles causas que llevaron a Juan Roa Sierra a actuar de la manera en que lo hizo e intenta explicar cómo cobran sentido, en medio de la contienda, los registros retóricos y las modalidades dramáticas. (shrink)

Cet article a pour but de donner quelques pistes sur les notions d’espace et de temps chez Maclaurin (1698-1746) à la fois en physique et aussi dans leur représentation mathématique. En s’appuyant essentiellement sur deux de ses ouvrages, l’Account of Sir Isaac Newton’s Philosophical Discoveries et le Treatise of Fluxions, nous verrons en quoi il est proche de Newton et nous donnerons en guise d’exemple, l’application à la question de la figure de la Terre.

Cet article a pour but de donner quelques pistes sur les notions d’espace et de temps chez Maclaurin (1698-1746) à la fois en physique et aussi dans leur représentation mathématique. En s’appuyant essentiellement sur deux de ses ouvrages, l’Account of Sir Isaac Newton’s Philosophical Discoveries et le Treatise of Fluxions, nous verrons en quoi il est proche de Newton et nous donnerons en guise d’exemple, l’application à la question de la figure de la Terre.

이 글은 수학과 과학에서 미적분계산법의 원 저작권에 관련된 유무형의 사용대상의 지적 소유권 귀속논의이다. 미적분계산법은 뉴턴과 라이프니츠가 독립적으로 발견한 것이며, 영 국과 대륙의 수학자그룹은 1699년에 시작하여 1714년까지 우위논쟁을 벌였다. 수학적 계산 방법의 지적인 소유의 귀속 사안은 실용적 효용성과 순수한 추상성과 인간지식의 문화적 보 편성에 비추어 결코 심각한 문제는 아니다. 하지만 자연법과 자연현상이 원 저작권자에 의 하여 언제 발견되고 어떻게 설명되는지는 원저자의 지적 권리와 우위에 대한 표준적 규범화 요구와 관련되기 때문에 과학의 진보와 과학과 수학교육과 관련하여 매우 중요하다. 게다가 수학적 발견에 따른 권리와 (...) 표준지식의 규범화는 학문공동체의 도덕적 의무로 위임되면서 원저자의 수학적 대상의 발견의 우위의 정당성확보를 위한 노력은 국가차원의 지지를 받아 온 문화적 우월성의 표현이다. 과학에서 지적 우위의 특권을 얻는다는 것은 표준지역에서 비표준지역에로의 지식 확산을 의미한다. 뉴턴이 뒤늦게 미적분계산의 우위를 주장하고 나 섰지만 오늘날 국제학술지 인용지수의 관점에서 보자면 라이프니츠에게 더 많은 지적인 귀 속이 가야 한다. 뉴턴이 계산방법을 먼저 고안하여 사용하고도 출간은 뒤로 미루었던 것은 17세기 바로크과학 연구관행의 정체성 때문이다. 미적분계산법의 우위에 관한 액면논쟁에 서 양자는 미적분학의 동등한 발견자로 인정된다. 미적분계산법의 최초의 발견과정은 어떠 하였는지를 평가하는 논증장소에서 당사자와 주변연구진은 원저자의 권리귀속을 수학적으 로만 해결하지 않았다. 그들은 과학기술 분야에서 미적분계산법의 가치와 중요성을 인식하 면서 용어 사용의 기술적 정당화를 시도하였다. 미적분계산법은 수사학을 통하여 발전할 수 있었고 18세기 유럽의 수학과 과학기술 분야에 비약적 발전을 이끌 수 있었다. (shrink)