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- Richard Arthur, Leibniz's Syncategorematic Infinitesimals, Smooth Infinitesimal Analysis, and Newton's Proposition.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 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.
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| | Scholar | At my library | More options ... We discuss various ways, which have been plainly justified in the secondhalf of the twentieth century, to introduce infinitesimals, and we considerthe new style of reasoning in mathematical analysis that they allow.
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| | Other links: jstor.org | Scholar | At my library | More options ... A través de una reconstrucción de la evolución de su pensamiento, en este artículo se estudia la utilización de infinitesimales por parte de Newton. Se distingue entre dos concepciones sucesivas de lo que denominó momento. A la primera de estas entidades la caracterizó como un infinitesimal, pero a la segunda (un indivisible generador de magnitudes finitas, que interviene en su método de las primeras y últimas razones) no la consideró como tal. Se entiende así su manifestación de rechazo a los infinitesimales, formulada en una segunda etapa, y se ve que las dudas arrojadas por algunos investigadores sobre la veracidad de tal manifestación se deben a una determinada interpretación de esta última concepción de momento.This paper discusses Newton’s recourse to infinitesimals through the reconstruction of the evolution of his thought. Two successive concepts of what he termed “moment” are told apart. The first of those entities was characterized by him as an infinitesimal, while the second -an indivisible generating finite magnitudes, present in his method of first and last reasons- was not considered such an entity. This move makes understandable his express rejection of infinitesimals in the second stage, and exposes the doubts of some scholars about the sincerity of Newton’s rejection as due to a peculiar interpretation of his last concept of “moment”.
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| | Scholar | At my library | More options ... The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity —a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit.
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| | Scholar | More options ... A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The approach developed has a pronounced applied character and is based on the principle “The part is less than the whole” introduced by the ancient Greeks. This principle is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in this paper uses strongly physical ideas emphasizing interrelations that hold between a mathematical object under observation and the tools used for this observation. It is shown how a new numeral system allowing one to express different infinite and infinitesimal quantities in a unique framework can be used for theoretical and computational purposes. Numerous examples dealing with infinite sets, divergent series, limits, and probability theory are given.
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| | Scholar | At my library | More options ... 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.
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| | Scholar | At my library | More options ... A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis (‘SIA’), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis (‘CA’) without resort to the method of limits. Formally, however, unlike Robinsonian ‘nonstandard analysis’, SIA conflicts with CA, deriving, e.g., ‘not every quantity is either = 0 or not = 0.’ Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this ‘change of logic’, arguing that standard arguments based on ‘smoothness’ requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism.
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| | Other links: tc.umn.edu jstor.org | Scholar | At my library | More options ... Before establishing his mature interpretation of infinitesimals as fictions, Gottfried Leibniz had advocated their existence as actually existing entities in the continuum. In this paper I trace the development of these early attempts, distinguishing three distinct phases in his interpretation of infinitesimals prior to his adopting a fictionalist interpretation: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no gaps between them; (iii) (1672- 75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus). In 1676, finally, Leibniz ceased to regard infinitesimals as actual, opting instead for an interpretation of them as fictitious entities which may be used as compendia loquendi to abbreviate mathematical reasonings.
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| | Scholar | More options ... Newton and Leibniz had profound disagreements concerning metaphysics and the relationship of mathematics to natural philosophy, as well as deeply opposed attitudes towards analysis. Nevertheless, or so I shall argue, despite these deeply held and distracting differences in their background assumptions and metaphysical views, there was a considerable consilience in their positions on the status of infinitesimals. In this paper I compare the foundation Newton provides in his Method Of First and Ultimate Ratios (sketched at some time between 1671 and 1684, and published in the Principia of 1687) with that provided independently by Leibniz in his unpublished manuscript De quadratura arithmetica (1675-6) as well as in later writings. I argue that both appeal to a version of the Archimedean Axiom to underwrite their use of infinitesimal techniques, which must be interpreted as a shorthand for rigorously finitist methods.
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| | Scholar | More options ... In this paper I attempt to trace the development of Gottfried Leibniz’s early thought on the status of the actually infinitely small in relation to the continuum. I argue that before he arrived at his mature interpretation of infinitesimals as fictions, he had advocated their existence as actually existing entities in the continuum. From among his early attempts on the continuum problem I distinguish four distinct phases in his interpretation of infinitesimals: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no gaps between them; (iii) (1672-75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus); (iv) (1676 onward) infinitesimals are fictitious entities, which may be used as compendia loquendi to abbreviate mathematical reasonings; they are justifiable in terms of finite quantities taken as arbitrarily small, in such a way that the resulting error is smaller than any pre-assigned margin. Thus according to this analysis Leibniz arrived at his interpretation of infinitesimals as fictions already in 1676, and not in the 1700's in response to the controversy between Nieuwentijt and Varignon, as is often believed.
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| | Scholar | More options ... Discussion of Richard Arthur, Leibniz's syncategorematic infinitesimals, smooth infinitesimal analysis, and Newton's proposition
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