This is a commentary on the use of squares in elementary statistics. One sees an ubiquitous use of squares in statistics, and the analogy of "distance in a statistical sense" is teased out. We conjecture that elementary statistical theory has its roots in classical Calculus, and preserves the notion of two senses described in this paper. We claim that the senses of the differentials dx/dy hold between classical and modern infinitesimal Calculus and show how this sense becomes (...) cashed out in both models. We suggest that the commonalities of both accounts can be used to find a informational realist account of Calculus. This account involves infinitesimal Calculus as acting as an epistemic mediator that expands our conceptual space and enables us to create new measures. (shrink)

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)

In this paper we introduce a paraconsistent reasoning strategy, Chunk and Permeate. In this, information is broken up into chunks, and a limited amount of information is allowed to flow between chunks. We start by giving an abstract characterisation of the strategy. It is then applied to model the reasoning employed in the original infinitesimal calculus. The paper next establishes some results concerning the legitimacy of reasoning of this kind - specifically concerning the preservation of the consistency of (...) each chunk and concludes with some other possible applications and technical questions. (shrink)

In the above mentioned article [1] unfortunately the names of the translators of Toeplitz's lecture were omitted. The correct title is:The Problem of University Courses on Infinitesimal Calculus and TheirDemarcation from Infinitesimal Calculus in High SchoolsOtto ToeplitzTranslated into English by Michael N. Fried and Hans Niels Jahnke.

I offer a novel solution to Zeno’s paradox of The Arrow by introducing nilpotent infinitesimal lengths of time. Nilpotents are nonzero numbers that yield zero when multiplied by themselves a certain number of times. Zeno’s Arrow goes like this: during the present, a flying arrow is moving in virtue of its being in flight. However, if the present is a single point in time, then the arrow is frozen in place during that time. Therefore, the arrow is both moving (...) and at rest. In “Zeno’s Arrow, Divisible Infinitesimals, and Chrysippus,” White suggests using an infinitesimal value as the length of the present. Contra Zeno, this allows the arrow to be moving in the present, rather than frozen in place. In this paper, I follow the basic outline of White’s solution but argue that his solution suffers from arbitrariness and a related theoretical artificiality in relation to the system of infinitesimals he invokes, viz. in relation to the hyperreal infinitesimals of nonstandard analysis. After arguing that any solution to the paradox must satisfy certain theoretical requirements, I examine White’s solution alongside two nilpotent solutions. One of these solutions is inspired by F.W. Lawvere’s Smooth Infinitesimal Analysis and the other is inspired by Paolo Giordano’s ring of Fermat Reals. I argue that Giordano’s nilpotents supply the best answer to Zeno’s paradox. (shrink)

In Hegel ou Spinoza,1 Pierre Macherey challenges the influence of Hegel’s reading of Spinoza by stressing the degree to which Spinoza eludes the grasp of the Hegelian dialectical progression of the history of philosophy. He argues that Hegel provides a defensive misreading of Spinoza, and that he had to “misread him” in order to maintain his subjective idealism. The suggestion being that Spinoza’s philosophy represents, not a moment that can simply be sublated and subsumed within the dialectical progression of the (...) history of philosophy, but rather an alternative point of view for the development of a philosophy that overcomes Hegelian idealism. Gilles Deleuze also considers Spinoza’s philosophy to resist the totalising effects of the dialectic. Indeed, Deleuze demonstrates, by means of Spinoza, that a more complex philosophy antedates Hegel’s, which cannot be supplanted by it. Spinoza therefore becomes a significant figure in Deleuze’s project of tracing an alternative lineage in the history of philosophy, which, by distancing itself from Hegelian idealism, culminates in the construction of a philosophy of difference. It is Spinoza’s role in this project that will be demonstrated in this paper by differentiating Deleuze’s interpretation of the geometrical example of Spinoza’s Letter XII (on the problem of the infinite) in Expressionism in Philosophy, Spinoza,2 from that which Hegel presents in the Science of Logic.3. (shrink)

When the Association of German Scientists and Physicians last met in Düsseldorf exactly twenty-eight years ago on September 24, a debate took place following lectures by Felix Klein and Alfred Pringsheim on roughly the same topic to which I would like to direct your attention today. The printed report of the Düsseldorf debate only remarked that, “It is not possible to go into details here,” so one can only guess how two of the most powerful teacher personalities among German mathematicians (...) of that time had confronted one another with their diametrically opposed views on this topic and how they did so with their characteristically lively spirit. (shrink)

The foundations of analysis offered by Cauchy and Riemann were not immediately welcomed by the mathematical community. Before 1870 the foundations of mathematics were considered more or less a national affair. In this paper, Dutch ideas of rigour in analysis between 1840 and 1870 will be discussed. These ideas show that Dutch mathematicians were aware of developments abroad but preferred the concept of infinitesimals as a foundation of mathematics.

We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and (...) mathematical. Our main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the “great triumvirate” of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity, and to exploit it in making sense of infinitesimals and related concepts. (shrink)

In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. Against this, we show (...) that by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions—a conception he later characterized as “syncategorematic”. Thus, one cannot infer the existence of infinitesimals from their successful use. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero; and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom. (shrink)

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 pro- cedural 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. (shrink)

During the Enlightenment period the concept of the infinitesimal was developed as a means to solve the mathematical problem of the incommensurability between human reason and the movements of physical beings. In this essay, the author analyzes the metaphysical prejudices subtending Enlightenment Humanism through the lens of the infinitesimal calculus. One of the consequences of this analysis is the perception of a two-fold possibility occasioned by the infinitesimal. On the one hand, it occasions an extreme form (...) of humanism, “transhumanism,” which exhibits limitless confidence in the possibility of human science. On the other hand, the concept of the infinitesimal also contains within itself a source for a critical “posthumanism,” that is to say, a source which initiates the dissolution of the presuppositions of humanism while simultaneously announcing a different ontological organization. In, Tostoy’s novel takes up the problem of the relation between reason and motion and makes the two-fold possibility visible by presenting a contrast between its theoretical presentations and the lived experiences of the characters in the novel. Thus, is the setting in which the author has chosen to conduct this analysis. (shrink)

The infinitesimal methods commonly used in the 17th and 18th centuries to solve analytical problems had a great deal of elegance and intuitive appeal. But the notion of infinitesimal itself was flawed by contradictions. These arose as a result of attempting to representchange in terms ofstatic conceptions. Now, one may regard infinitesimals as the residual traces of change after the process of change has been terminated. The difficulty was that these residual traces could not logically coexist with the (...) static quantities traditionally employed by mathematics. The solution to this difficulty, as it turns out, is to regard these quantities asalso being subject to (a form of) change, for then they will have the same nature as the infinitesimals representing the residual traces of change, and will become,ipso facto, compatible with these latter.In fact, the category-theoretic models which realize the Principle of Infinitesimal Linearity may themselves be regarded as representations of a general concept of variation (cf. Bell (1986)). While the static set-theoretical models represent change or motion by making a detour through the actual (but static) infinite, the varying category-theoretic models enable such change to be representeddirectly, thus permitting the introduction of geometric infinitesimals and, as we have attempted to demonstrate in this paper, the virtually complete incorporation of the methods of the early calculus.It is surely a remarkable — even an ironic — fact that the contradiction between the flux of the objective world and the stasis of mathematical entities has found its resolution in category theory, a branch of mathematics commonly, and, as one now sees, mistakenly, regarded as the summit of gratuitous abstraction. (shrink)

In the paper of Brown and Priest 2004, the authors developed the chunk and permeate method, which they described as a ?paraconsistent reasoning strategy?. There it is suggested that the method of chunk and permeate could apply to the historical infinitesimal calculus. However, no attempt was made to look at actual historical examples. In this paper, I show that the method of chunk and permeate can indeed apply, as a rational reconstruction, to certain of Isaac Newton's arguments that (...) use infinitesimals. This rational reconstruction maintains and uses, rather than sidesteps, the apparent contradictions in Newton's arguments. The applicability of chunk and permeate to other historical arguments, e.g. of Leibniz/L'Hospital and Fermat, has also been investigated and will be communicated in future publications. (shrink)

"The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? This collection of essays, by scholars from Canada, the US, Germany, United Kingdom and Switzerland, gives a comprehensive study of the controversies over the nature and status of the infinitesimal. Aside from Leibniz, the scholars considered are Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. The collection also contains newly discovered marginalia (...) of Leibniz to the writings of Hobbes."--BOOK JACKET. (shrink)

We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated a (...) sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the "great triumvirate" of Cantor, Dedekind, and Weierstrass. Expressed in terms of modern mathematics, the Marburg philosophers saw the introduction of both infinitesimals and limits as completions whose prototype was Dedekind's of the rational number system resulting in the real numbers. At least partially,, this idea of "completions" can be captured in terms of a category-theoretical description of the conceptual development of modern mathematics. The feasibility of such a modern reformuation may be taken as evidence that the philosophical resources of Marburg neo-Kantianism may be of interest even for contemporary philosophy of mathematics. (shrink)

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth 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. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Infinitesimal Differences: Controversies between Leibniz and his ContemporariesFrançoise Monnoyeur-BroitmanUrsula Goldenbaum and Douglas Jesseph, editors. Infinitesimal Differences: Controversies between Leibniz and his Contemporaries. Berlin-New York: Walter de Gruyter, 2008. Pp. vi + 327. Cloth, $109.00.Leibniz is well known for his formulation of the infinitesimal calculus. Nevertheless, the nature and logic of his discovery are seldom questioned: does it belong more to mathematics or metaphysics, and (...) how is it connected to his physics? This book, composed of fourteen essays, investigates the nature and foundation of the calculus, its relationship to the physics of force and principle of continuity, and its overall method and metaphysics. The Leibnizian calculus is presented in its origin and context together with its main contributors: Archimedes, Cavalieri, Wallis, Hobbes, Pascal, Huygens, Bernoulli, and Nieuwentijt.Many of us know and probably have used the Leibnizian formula: to calculate the area under a curve; this book considers the origin, nature, method, and metaphysics of this formula. The most fascinating question discussed in it is the fiction introduced to define the infinitesimal calculation: in his mature period, Leibniz himself referred to his calculus as “a well-founded fiction.” We learn that he gradually gave up the idea of an actual infinite in favor of describing infinitely small quantities. As a result, the quantities became fictive and finite or indefinite instead of real and infinite, and the process of calculation shifted towards mathematics rather than metaphysics.In the first essay, Richard Arthur compares the calculus of Leibniz and Newton and demonstrates how it was based for both on the development of continuously varying quantity. He also shows how the understanding of this quantity derived from the Archimedean axiom that excludes the existence of actual infinite quantity. Ursula Goldenbaum next demonstrates how Hobbes, with his conatus considered as point and instant, had a direct impact on the Leibnizian conception of continuity; to confirm her interpretation, she includes at the end of her article the “Marginalia or Leibniz comments” on Hobbes. Samuel Levey establishes a connection between the notion of the infinitesimal as a “well-founded fiction” and the law of continuity, arguing that the law of continuity, understood as an extension of Archimedes’ principle, gives a finite foundation to the infinitesimal calculation. O. Bradley Bassler discusses the idea of the differential in terms of infinitely small quantities and the use of proportions between infinitely small and finite quantities in order to measure [End Page 527] the infinitely small; here, differentials are also presented as fictions. Fritz Nagel discusses the methodological sophistication of the calculus and its opponents. His main questions concern the different grades of infinity, quantities regarded as zero, and the infinitesimal as a fiction that can be calculated. Douglas Jesseph considers the reality of the infinitesimal and the “well-founded fiction,” showing how Leibniz methodically constructs infinitesimal quantities without reality by returning to Hobbes’ notion of conatus. Hobbes is, for Jesseph, the initiator of these infinitely small quantities without existence.A second group of essays examines the connection between algebra and geometry in the calculus. Philip Beeley demonstrates how John Wallis, on the path of Cavalieri’s method of the indivisibles, found a way to transform geometric problems into sums of arithmetic sequences. Siegmund Probst expresses the different phases in the discovery of the Leibnizian method. He mentions that Leibniz coined the term ‘infinitesimal’ in 1673, and points out that he did not need the method of indivisibles to find a curve from its given arc, but only a series expansion using differences of higher order. Emily Grosholz looks at how, thanks to his algebraic and geometrical notations of curves, Leibniz could formulate the equation of his infinitesimal calculus. According to Grosholz, Leibniz exploits the ambiguity of notations in order to express the geometrical and dynamical aspect of the calculus. Herbert Breger underlines the novelty and abstract nature of Leibniz’s new method of calculation, showing how the theory of infinitesimals became an algebraic calculus independent of geometry.A final group of articles studies the relationship between the calculus and Leibniz’s physics. François Duchesneau shows how the differential and integral calculus provided a model for the integrative laws of Leibniz’s... (shrink)

Leibniz is well known for his formulation of the infinitesimal calculus. Nevertheless, the nature and logic of his discovery are seldom questioned: does it belong more to mathematics or metaphysics, and how is it connected to his physics? This book, composed of fourteen essays, investigates the nature and foundation of the calculus, its relationship to the physics of force and principle of continuity, and its overall method and metaphysics. The Leibnizian calculus is presented in its origin (...) and context together with its main contributors: Archimedes, Cavalieri, Wallis, Hobbes, Pascal, Huygens, Bernoulli, and Nieuwentijt. Many of us know and probably have used the Leibnizian formula: to .. (shrink)

This article studies Leibniz’s treatment of infinitesimals: their application to the calculus and his opinion that they did not exist. In partial agreement with Brunschvicg’s and Ishiguro’s commentaries on the paradoxical status of Leibniz´s infinitesimals, this study proposes a synthesis of both..

The goal of this paper is to investigate the relation between Cohen's approach to differential calculus and his doctrine of pure thinking. We claim that Cohen's logic of origin is firmly based on his interpretation of infinitesimal analysis. More precisely, the transcendental method, when applied to differential calculus, reveals the productive capacity of thinking expressed by the judgment of origin.

The aim of this paper is to analyze Dimitry Gawronsky’s doctrine of actual infinitesimals. I examine the peculiar connection that his critical idealism establishes between transcendental philosophy and mathematics. In particular, I reconstruct the relationship between Gawronsky’s differentials, Cantor’s transfinite numbers, Veronese’s trans-Archimedean numbers and Robinson’s hyperreal numbers. I argue that by means of his doctrine of actual infinitesimals, Gawronsky aims to provide an interpretation of calculus that eliminates any alleged given element in knowledge.

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth (...) class='Hi'>Infinitesimal Analysis (SIA), as propounded by John Bell. Despite many parallels between SIA and Leibniz’s approach —the non-punctiform nature of infinitesimals, their acting as parts of the continuum, the dependence on variables (as opposed to the static quantities of both Standard and Non-standard Analysis), the resolution of curves into infinitesided polygons, and the finessing of a commitment to the existence of infinitesimals— I find some salient differences, especially with regard to higher-order infinitesimals. These differences are illustrated by a consideration of how each approach might be applied to Newton’s Proposition 6 of the Principia, and the derivation from it of the v2/r law for the centripetal force on a body orbiting around a centre of force. It is found that while Leibniz’s syncategorematic approach is adequate to ground a Leibnizian version of the v2/r law for the “solicitation” ddr experienced by the orbiting body, there is no corresponding possibility for a derivation of the law by nilsquare infinitesimals; and while SIA can allow for second order differentials if nilcube infinitesimals are assumed, difficulties remain concerning the compatibility of nilcube infinitesimals with the principles of SIA, and in any case render the type of infinitesimal analysis adopted dependent on its applicability to the problem at hand. (shrink)

Excerpt from The Mathematical Psychology of Gratry and Boole: Translated From the Language of the Higher Calculus Into That of Elementary Geometry Dear Dr. Maudsley, - You have often asked me to explain, for students unaquainted with the Infinitesimal Calculus, certain doctrines expressed in terms of that Calculus by P. Gratry and my late husband. That you permit me to dedicate my attempt to you will, at least, be a guarantee that the main ideas of mathematical (...) psychology are based, not on mystic dreams, but on scientific induction. I am glad of this opportunity of expressing to you my gratitude, both for your published works, and for much personal kindness to myself. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth (...) class='Hi'>Infinitesimal Analysis, as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton’s Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse. “If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.”—(Bos 1974–1975, 82–83). (shrink)

A comienzos deI siglo XVIII se origina una polémica en la Academia de Ciencias de París a propósito de la fundamentación deI calculo infinitesimal. Con motivo de la misma Leibniz presentará los infinitesimales corno ficciones útiles, noción que agrega polémica a la polémica y que habrá que precisar. Leibniz se desmarcará claramente de la idea de infinitesimal mantenida por sus seguidores franceses. Resultado de todo ello es un triunfo en la práctica deI cálculo infinitesimal y un alto (...) en cuanto a su fundamentación.In the beginning of the XVIII century arises a discussion in the Paris Academy of Sciences about the justification of infinitesimal Calculus. In this line, Leibniz will present infinitesimals as useful fictions, a problematic notion that requires an accurate meaning. Leibniz does not accept the infinitesimal concept of his french followers. The result of the controversy was a triumph for Calculus, but a pause in its justification. (shrink)

The early calculus is a popular example of an inconsistent but fruitful scientific theory. This paper is concerned with the formalisation of reasoning processes based on this inconsistent theory. First it is shown how a formal reconstruction in terms of a sub-classical negation leads to triviality. This is followed by the evaluation of the chunk and permeate mechanism proposed by Brown and Priest in, 379–388, 2004) to obtain a non-trivial formalisation of the early infinitesimal calculus. Different shortcomings (...) of this application of C&P as an explication of inconsistency tolerant reasoning are pointed out, both conceptual and technical. To remedy these shortcomings, an adaptive logic is proposed that allows for conditional permeations of formulas under the assumption of consistency preservation. First the adaptive logic is defined and explained and thereafter it is demonstrated how this adaptive logic remedies the defects C&P suffered from. (shrink)

In this paper, we propose a new method to deal with continuously varying quantity in the situation calculus based on the concept of the nonstandard analysis. The essential point of the method is to devise a new model called nonstandard situation calculus, which is an interpretation of the situation calculus in the set of hyperreals. This nonstandard model allows discrete but uncountable (hyperfinite) state transition, so that we can describe and reason about the continuous dynamics which are (...) usually treated with differential equations. In this enlarged perspective of the nonstandard situation calculus, we discuss about an infinitesimal action, infinite plan and an abnormality theory for the temporal prediction based on the differentiability. (shrink)

Current formal mathematics, being divorced from the empirical, is entirely a social construct, so that mathematical theorems are no more secure than the cultural belief in two-valued logic, incorrectly regarded as universal. Computer technology, by enhancing the ability to calculate, has put pressure on this social construct, since proof-oriented formal mathematics is awkward for computation, while computational mathematics is regarded as epistemo-logically insecure. Historically, a similar epistemological fissure between computational/practical Indian mathematics and formal/spiritual Western mathematics persisted for centuries, during a (...) dialogue of civilizations, when texts on "algorismus" and "infinitesimal" calculus were imported into Europe, enhancing the ability to calculate. It is argued here that this epistemological tension should be resolved by accepting mathematics as empirically based and fallible, and by revising accordingly the mathematics syllabus outlined by Plato. (shrink)

Current formal mathematics, being divorced from the empirical, is entirely a social construct, so that mathematical theorems are no more secure than the cultural belief in two-valued logic, incorrectly regarded as universal. Computer technology, by enhancing the ability to calculate, has put pressure on this social construct, since proof-oriented formal mathematics is awkward for computation, while computational mathematics is regarded as epistemo-logically insecure. Historically, a similar epistemological fissure between computational/practical Indian mathematics and formal/spiritual Western mathematics persisted for centuries, during a (...) dialogue of civilizations, when texts on "algorismus" and "infinitesimal" calculus were imported into Europe, enhancing the ability to calculate. It is argued here that this epistemological tension should be resolved by accepting mathematics as empirically based and fallible, and by revising accordingly the mathematics syllabus outlined by Plato. (shrink)

Do there exist real numbers d with d2 = 0? The question is formulated provocatively, to stress a formalist view about existence: existence is consistency, or better, coherence.Also, the provocation is meant to challenge the monopoly which the number system, invented by Dedekind et al., is claiming for itself as THE model of the geometric line. The Dedekind approach may be termed “arithmetization of geometry”.We know that one may construct a number system out of synthetic geometry, as Euclid and followers (...) did : “geometrization of arithmetic”..Starting from the geometric side, nilpotent elements are somewhat reasonable, although Euclid excluded them. The sophist Protagoras presented a picture of a circle and a tangent line; the apparent little line segment D which tangent and circle have in common, are, by Pythagoras' Theorem, precisely the points, whose abscissae d have d2 = 0. Protagoras wanted to use this argument for destructive reasons: to refute the science of geometry.A couple of millenia later, the Danish geometer Hjelmslev revived the Protagoras picture. His aim was more positive: he wanted to describe Nature as it was. According to him, the Real Line, the Line of Sensual Reality, had many nilpotent infinitesimals, which we can see with our naked eyes. (shrink)

In Bertrand Russell's 1903 Principles of Mathematics, he offers an apparently devastating criticism of the neo-Kantian Hermann Cohen's Principle of the Infinitesimal Method and its History (PIM). Russell's criticism is motivated by his concern that Cohen's account of the foundations of calculus saddles mathematics with the paradoxes of the infinitesimal and continuum, and thus threatens the very idea of mathematical truth. This paper defends Cohen against that objection of Russell's, and argues that properly understood, Cohen's views of (...) limits and infinitesimals do not entail the paradoxes of the infinitesimal and continuum. Essential to that defense is an interpretation, developed in the paper, of Cohen's positions in the PIM as deeply rationalist. The interest in developing this interpretation is not just that it reveals how Cohen's views in the PIM avoid the paradoxes of the infinitesimal and continuum. It also reveals some of what is at stake, both historically and philosophically, in Russell's criticism of Cohen. (shrink)

Despite his commitment to freedom, Leibniz’ philosophy is also founded on pre-established harmony. Understanding the life of the individual as a spiritual automaton led Leibniz to refer to the puzzle of the way out of determinism as the Labyrinth of Freedom. Leibniz claimed that infinite complexity is the reason why it is impossible to prove a contingent truth. But by means of Leibniz’ calculus, it actually can be shown in a finite number of steps how to calculate a summation (...) of infinite parts. It appears that the analogy Leibniz drew between the mathematics of infinite series and the logic of contingent truths did more harm than good. A solution consistent with Leibniz’ perception of infinity is proposed. Alongside the existence of the aforementioned analogy, it is based on a disanalogy between the mathematics of infinite series and the logic of infinitely complex truths. This is a disanalogy that Leibniz had already used to solve the Labyrinth of Continuum which he declared more than once could, like the Labyrinth of Freedom, be solved by means of the nature of infinity. The solution of both labyrinths is based on the fictitiousness of the infinitesimal. (shrink)

In this article, we argue that the application of mathematics in the construction of physical theories constrains the form of our scientific understanding. Specifically, we discuss the constraints that the mathematical structure of the differential calculus imposes on the understanding of the structure of the world within a Newtonian worldview. In the first section of the paper, we develop the formal structure of the differential calculus. In the second section, we provide a discussion of the constraints that the (...) differential calculus imposes on the application of Newton's second law. In the final section, we present a case study of a thought experiment by John Norton, simply called `the dome'. We argue that the pathological nature of Norton's dome cannot serve as an argument to sustain the claim that Newton's second law is indeterministic because the constraints of the differential calculus dictate that Newton's second law cannot be well-defined in the infinitesimal neighbourhood of the apex of Norton's dome. (shrink)

It is well known that the history of Calculus in the nineteenth century coincides with the process of substitution of infinitesimals by the notion of limit. But it is adviseable to keep in mind the ontological implications of that process.We can find a background for this ontological approach in Abraham Robinson’s Non-Standard AnaIysis and “The Metaphysics of the Calculus”. Indeed, by the choice of the word “metaphysics” and by the several recalls of the ontological nature of the arguments, (...) Robinson claims for a filiation which is at the same time fruitful in the mathematical register and necessary for a true philosophical reflection about the infinite. (shrink)

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example, they actually reveal the behavior of a real function not “close to” a point but “in” the point. We are interested in our research in the (...) diagrams which play an optical role –microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role, and an unveiling role. In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations. (shrink)

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile (...) Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (shrink)