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..

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)

Es indudable la importancia de la noción de función para la matemática y la lógica actuales y es sabido que es G. W. Leibniz quien utiliza por vez primera el término función en un sentido matemático, un término que, además, es introducido en el marco de su cálculo infinitesimal. Puesto que el pensador alemán es, junto con I. Newton, uno de los descubri­dores del cálculo, suele pensarse que también debemos a él el concepto de función. Sin embargo, (...) poco se ha escrito sobre la manera específica en la que Leibniz utilizó el término función, evadiendo con frecuencia la pregunta de si acaso el término y el concepto han correspondido siempre. En el presente trabajo exploramos el significado del término en el contexto de su surgimiento: el manuscrito de 1673 De functionibus plagulae quattuor. Con tal objetivo, hacemos, en primer lugar, una breve reconstrucción de la historia del instinto de funcionalidad que sirva de marco para comprender el significado del término función. Este significado es obtenido, en segundo lugar, atendiendo a los usos que el autor hace del término en dicho manuscrito. (shrink)

In his 1676 text De Quadratura Arithmetica, Leibniz distinguished infinita terminata from infinita interminata. The text also deals with the notion, originating with Desargues, of the point of intersection at infinite distance for parallel lines. We examine contrasting interpretations of these notions in the context of Leibniz’s analysis of asymptotes for logarithmic curves and hyperbolas. We point out difficulties that arise due to conflating these notions of infinity. As noted by Rodríguez Hurtado et al., a significant difference exists between the (...) Cartesian model of magnitudes and Leibniz’s search for a qualitative model for studying perspective, including ideal points at infinity. We show how respecting the distinction between these notions enables a consistent interpretation thereof. (shrink)

O verdadeiro infinito, afirma Leibniz em seus Novos ensaios, não é um modo da quantidade, é anterior a qualquer composição e não é formado pela adição de partes. O infinito, para Leibniz, é atual e é propriedade de todas as coisas. Como criaturas finitas conhecem o infinito? Neste artigo, investigamos que tipo de relação pode ter o infinito matemático, quantitativo, para o conhecimento da infinitude divida e do infinito atual que existe no mundo. A ordem ideal da matemática instrui sobre (...) a ordem real do mundo? E quais os limites da ordem ideal na explicação do infinito atual que caracteriza Deus e o mundo? (shrink)

Com este trabalho pretende dar-se um panorama do que foi a actividade científica, no domínio da Matemática e das ciências que lhe andavam ligadas (Geografia, Astronomia, Mecânica), no grande centro cultural que foi o Museu e a Biblioteca de Alexandria, nos tempos áureos e até ao declínio definitivo da ciência nesse espaço. Pretende-se igualmente sublinhar as descobertas e progressos que abriram caminho para posteriores estádios de desenvolvimento da Matemática, assim como dar conta do cruzamento de saberes e da grande mobilidade (...) de cientistas matemáticos e relações entre si, alguns provenientes de muitos espaços e sediados em Alexandria, outros formados em Alexandria e regressados ao seu espaço de origem. (shrink)

El objetivo de este trabajo es examinar el concepto leibniziano de ficción matemática, con especial énfasis en la tesis de Leibniz acerca del carácter ficcional de las nociones infinitarias. Se propone en primer lugar, como marco general de la investigación, un conjunto de cinco condiciones que una ficción tiene que cumplir para ser matemáticamente admisible. Sobre la base de las concepciones de Leibniz acerca del conocimiento simbólico, se propone la ficción matemática como la clase de nociones confusas que carecen de (...) denotación a raíz de la imposibilidad de su objeto. Tomando como punto de partida el análisis de la imposibilidad en términos de inconsis tencia, se muestra que Leibniz admite otras formas de imposibilidad, que afectan especialmente a las nociones infinitarias. Proponemos así la imposibilidad como irrepresentabilidad geométrica y la imposibilidad por incompatibilidad con principios arquitectónicos. Así, el resultado de nuestro examen fundamenta la admisión de tres tipos de ficcionalidad matemática, la ficción 1, que se corresponde con la noción inconsistente, la ficción 2 que incluye las nociones geométricamente irrepresentables y la ficción 3, que se aplica a las nociones “arquitectónicamente” imposibles. En conclusión, los conceptos infinitarios, sin ser por sí mismos inconsistentes, corresponden al tipo de ficcion 2 y ficción 3. Finalmente, se concluye que Leibniz está más preocupado por la imposibilidad por incompatibilidad con principios arquitectónicos que por la cuestión de la irrepresentabilidad geométrica. (shrink)

We show that infinitesimal probabilities are much too small for modeling the individual outcome of a countably infinite fair lottery.

The primary purpose of this paper is to analyze the relationship between the familiar non-Archimedean field of hyperreals from Abraham Robinson’s nonstandard analysis and Paolo Giordano’s ring extension of the real numbers containing nilpotents. There is an interesting nontrivial homomorphism from the limited hyperreals into the Giordano ring, whereas the only nontrivial homomorphism from the Giordano ring to the hyperreals is the standard part function, namely, the function that maps a value to its real part. We interpret this asymmetry to (...) mean that the nilpotent infinitesimal values of Giordano’s ring are “smaller” than the hyperreal infinitesimals. By viewing things from the “point of view” of the hyperreals, all nilpotents are zero, whereas by viewing things from the “point of view” of Giordano’s ring, nonnilpotent, nonzero infinitesimals register as nonzero infinitesimals. This suggests that Giordano’s infinitesimals are more fine-grained. (shrink)

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, I advance an original view of the structure of space called Infinitesimal Gunk. This view says that every region of space can be further divided and some regions have infinitesimal size, where infinitesimals are understood in the framework of Robinson’s nonstandard analysis. This view, I argue, provides a novel reply to the inconsistency arguments proposed by Arntzenius and Russell, which have troubled a more familiar gunky approach. Moreover, it has important advantages over the alternative views (...) these authors suggested. Unlike Arntzenius’s proposal, it does not introduce regions with no interior. It also has a much richer measure theory than Russell’s proposal and does not retreat to mere finite additivity. (shrink)

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. _1_ Introduction _2_ The Limits of Classical Probability Theory _2.1_ Classical probability functions _2.2_ Limitations _2.3_ Infinitesimals to the rescue? _3_ NAP Theory _3.1_ First four axioms of NAP _3.2_ Continuity and conditional probability _3.3_ The final axiom of (...) NAP _3.4_ Infinite sums _3.5_ Definition of NAP functions via infinite sums _3.6_ Relation to numerosity theory _4_ Objections and Replies _4.1_ Cantor and the Archimedean property _4.2_ Ticket missing from an infinite lottery _4.3_ Williamson’s infinite sequence of coin tosses _4.4_ Point sets on a circle _4.5_ Easwaran and Pruss _5_ Dividends _5.1_ Measure and utility _5.2_ Regularity and uniformity _5.3_ Credence and chance _5.4_ Conditional probability _6_ General Considerations _6.1_ Non-uniqueness _6.2_ Invariance Appendix. (shrink)

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. (shrink)

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general.

A lógica I1, um sistema trivalorado de caráter fracamente intuicionista, foi introduzida, via sistema axiomático (Hilbertiano) em 1995 por Sette e Carnielli. O presente artigo tem por objetivo apresentar esse sistema em um formalismo lógico em Cálculo de Sequentes, denominado de GI1, o qual se apresenta como um sistema de prova de teoremas, caracterizado como um algoritmo, sendo mais aplicável do ponto de vista computacional, por meio da dualização do sistema de tableaux analíticos TI1. Ademais, é apresentado a equivalência (...) dedutiva entre o sistema de sequentes GI1 com o sistema Hilbertiano I1. (shrink)

It is natural to think that questions in the metaphysics of chance are independent of the mathematical representation of chance in probability theory. After all, chance is a feature of events that comes in degrees and the mathematical representation of chance concerns these degrees but leaves the nature of chance open. The mathematical representation of chance could thus, un-controversially, be taken to be what it is commonly taken to be: a probability measure satisfying Kolmogorov’s axioms. The metaphysical questions about chance (...) seem to be left open by all this. I argue that this is a mistake. The employment of real numbers as measures of chance in standard probability theory brings with it commitments in the metaphysics of (objective) chance that are not only substantial but also mistaken. To measure chance properly we need to employ extensions of the real numbers that contain infinitesimals: positive numbers that are infinitely small. But simply using infinitesimals alone is not enough, as a number of arguments show. Instead we need to put three ideas together: infinitesimals, the non-locality of chance and flexibility in measurement. Only those three together give us a coherent picture of chance and its mathematical representation. (shrink)

I propose a theory of space with infinitesimal regions called smooth infinitesimal geometry based on certain algebraic objects, which regiments a mode of reasoning heuristically used by geometricists and physicists. I argue that SIG has the following utilities. It provides a simple metaphysics of vector fields and tangent space that are otherwise perplexing. A tangent space can be considered an infinitesimal region of space. It generalizes a standard implementation of spacetime algebraicism called Einstein algebras. It solves the (...) long-standing problem of interpreting smooth infinitesimal analysis realistically, an alternative foundation of spacetime theories to real analysis, 277–392, 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations. Against this, I argue that SIG is such a reformulation. But SIG has an unorthodox mereology, in which the principle of supplementation fails. (shrink)

In a letter of September 1679 to Huygens, Leibniz proposed a calculus situs directly applicable to geometric relations without use of magnitudes. His researehes on this kind of Geometric Calculus were developed along all his life but, unfortunately, only a few Leibniz’ s writings on these matters had been published by Gerhardt and Couturat. They were closely connected to his own researches on Logic Calculus. From a chronological point of view, the unpublished manuscript Circa Geometrica Generalia may be considered as (...) the third most important Leibniz’ contribution on Calculus Situs. CGG summarizes several results obtained by Leibniz from 1679 to 1682 and contains some interesting ideas concerning set theory, geometric axioms, General Topology and Logic Foundations of Geometry.En una carta a Huygens escrita en septiembre de 1679, Leibniz propuso un calculus situs que fuese aplicable al estudio directo de las relaciones geometricas, sin utilizar magnitudes. Sus investigaciones en torno a los Cálculos Geometricos continuaron a lo largo de toda su vida, pero, lamentablemente, Gerhardt y Couturat sólo publicaron una pequeña parte de sus manuscritos sobre estos temas. Estasinvestigaciones estuvieron estrechamente conectadas con los trabajos de Leibniz sobre Cálculos Lógicos. El manuscrito inédito Circa Geometrica Generalia puede ser considerado, desde un punto de vista cronológico, como la tercera más importante contribución de Leibniz al Calculus Situs. En CGG Leibniz resume varios logros obtenidos desde 1679 hasta 1682 y expone ideas interesantes en relación con la Teoría de Conjuntos, los axiomas de la geometria, la Topologia General y los fundamentos logicos de la Geometría. (shrink)

O presente artigo discute as noções de cálculo e medida na obra de Nietzsche, mais especificamente na transição do seu período inicial para o período intermediário. Com isso, nossa intenção é explicitar como tais noções que soam tão pouco dionisíacas – e consequentemente, nietzschianas – podem fazer parte do conjunto da obra de Nietzsche e, mais ainda, serem essenciais para a compreensão de seu pensamento. Para que esse objetivo fosse alcançado, foram necessários os desdobramentos de conceitos como paixões e (...) criação na obra nietzschiana, fazendo reaparecer características do conceito de apolíneo, que são praticamente despercebidos uma vez que o filósofo combina seus dois conceitos anteriores. Porém, ao fazer isso, ele compartilha conosco sua ideia de “criar a si mesmo como obra de arte”. Por fim, tentamos deixar claro que quando introduzimos tais noções como cálculo e medida em sua filosofia, isso jamais o faz parecer um filósofo racionalista ou moralista, isto é, alguém que busque apresentar e estabelecer determinadas regras e normas de conduta consideradas apropriadas. (shrink)

The 'best-system' analysis of lawhood [Lewis 1994] faces the 'zero-fit problem': that many systems of laws say that the chance of history going actually as it goes--the degree to which the theory 'fits' the actual course of history--is zero. Neither an appeal to infinitesimal probabilities nor a patch using standard measure theory avoids the difficulty. But there is a way to avoid it: replace the notion of 'fit' with the notion of a world being typical with respect to a (...) theory. (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)

I compare three sorts of case in which philosophers have argued that we cannot assert the Law of Excluded Middle for statements of identity. Adherents of Smooth Infinitesimal Analysis deny that Excluded Middle holds for statements saying that an infinitesimal is identical with zero. Derek Parfit contended that, in certain sci-fi scenarios, the Law does not hold for some statements of personal identity. He also claimed that it fails for the statement ‘England in 1065 was the same nation (...) as England in 1067’. I argue that none of these cases poses a serious threat to Excluded Middle. My analysis of the last example casts doubt on the principle of the Determinacy of Distinctness. While David Wiggins's ‘conceptualist realism’ provides a metaphysics which can dispense with that principle, it leaves no house-room for infinitesimals. (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)

RESUMO O propósito deste artigo é aproximar o significado do pensar calculador de Heidegger e a teoria sobre a sociedade do risco de Beck, considerando suas interpelações com o significado da técnica na modernidade. Porém, mais que tratar das aproximações entre ambos os pensadores, este estudo pretende também demonstrar a importância da filosofia da técnica de Heidegger para pensar o sentido do cálculo do risco e do risco do cálculo na sociedade do risco. Assim, argumenta-se que a teoria (...) do risco de Beck confirma e atualiza o pensamento de Heidegger sobre a técnica moderna, quando se observa que o pensar calculador, que dirige e controla o modo de ser na era do técnico, manifesta-se hoje com toda claridade na sociedade do risco global. ABSTRACT The purpose of this paper is to bring the meaning of Heidegger’s calculative thinking and Beck’s risk society theory closer, as well as their interpellations with the meaning of technique in modernity. Nevertheless, more than dealing with the convergences between both thinkers, this study also intends to demonstrate the importance of Heidegger’s technique philosophy in order to investigate the meaning of risk calculation and of calculation risk in risk society. Therefore, it argues that Beck’s risk theory confirms and updates Heidegger’s thinking on modern technique, as one observes that calculative thinking, which directs and controls the way of being in the age of technicality, clearly manifests itself nowadays, in the global risk society. (shrink)

It has been recently debated whether there exists a so-called “easy road” to nominalism. In this essay, I attempt to fill a lacuna in the debate by making a connection with the literature on infinite and infinitesimal idealization in science through an example from mathematical physics that has been largely ignored by philosophers. Specifically, by appealing to John Norton’s distinction between idealization and approximation, I argue that the phenomena of fractional quantum statistics bears negatively on Mary Leng’s proposed path (...) to easy road nominalism, thereby partially defending Mark Colyvan’s claim that there is no easy road to nominalism. (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)

In the opening passage of the Breviarium of Festus we read the following: ‘… ac morem secutus calculonum, qui ingentes summas aeris breuioribus exprimunt, res gestas signabo, non eloquar. Accipe ergo quod breuiter dictis breuis conputetur …’ The problem that I should like briefly to discuss in the following study is: Who were the calculones, ‘qui ingentes surnmas aeris breuioribus exprimunt’? This term calculo, and indeed the whole problematic clause can, I suggest, only be fully understood and appreciated in the (...) light of monetary developments of the later fourth century. (shrink)

In the opening passage of the Breviarium of Festus we read the following: ‘… ac morem secutus calculonum, qui ingentes summas aeris breuioribus exprimunt, res gestas signabo, non eloquar. Accipe ergo quod breuiter dictis breuis conputetur …’ The problem that I should like briefly to discuss in the following study is: Who were the calculones, ‘qui ingentes surnmas aeris breuioribus exprimunt’? This term calculo, and indeed the whole problematic clause can, I suggest, only be fully understood and appreciated in the (...) light of monetary developments of the later fourth century. (shrink)

The notion of indivisibles and atoms arose in ancient Greece. The continuum—that is, the collection of points in a straight line segment, appeared to have paradoxical properties, arising from the ‘indivisibles’ that remain after a process of division has been carried out throughout the continuum. In the seventeenth century, Italian mathematicians were using new methods involving the notion of indivisibles, and the paradoxes of the continuum appeared in a new context. This cast doubt on the validity of the methods and (...) the reliability of mathematical knowledge which had been regarded as established by the axiomatic method in geometry expounded by Aristotle’s younger contemporary Euclid. The teaching of indivisibles was banned within the Society of Jesus, the Jesuits. In England, indivisibles were used by the mathematician John Wallis, and there was an acrimonious and extended feud between Wallis and the philosopher Thomas Hobbes over legitimate methods of argument in mathematics. Notions of the infinitesimal were used by Isaac Newton and Gottfried Leibniz, and were attacked by Bishop Berkeley for the vagueness of the concept and the illegitimate reasoning applied to it. This article discusses aspects of these events with reference to the book Infinitesimal by Amir Alexander and to other sources. Also discussed are wider issues arising from Alexander’s book including: the changes in cultural sensibility associated with the growth of new mathematical and scientific knowledge in the seventeenth century, the changes in language concomitant with these changes, what constitutes valid methods of enquiry in various contexts, and the question of authoritarianism in knowledge. More general aims of this article are to widen the immediate mathematical and historical contexts in Alexander’s book, to bridge a gap in conversations between mathematics and the humanities, and to relate mathematical ideas to wider human and contemporary issues. (shrink)

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.

In the years following Bertrand Russell's visit in China, fragments from his work on mathematical logic and the foundations of mathematics started to enter the Chinese intellectual world. While up...

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.

In this paper, I develop an original view of the structure of space—called infinitesimal atomism—as a reply to Zeno’s paradox of measure. According to this view, space is composed of ultimate parts with infinitesimal size, where infinitesimals are understood within the framework of Robinson’s nonstandard analysis. Notably, this view satisfies a version of additivity: for every region that has a size, its size is the sum of the sizes of its disjoint parts. In particular, the size of a (...) finite region is the sum of the sizes of its infinitesimal parts. Although this view is a coherent approach to Zeno’s paradox and is preferable to Skyrms’s infinitesimal approach, it faces both the main problem for the standard view and the main problem for finite atomism, leaving it with no clear advantage over these familiar alternatives. (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)

En este artículo consideraré los puntos de vista de Newton sobre el método matemático. Newton nunca escribió en extenso sobre este tema, sin embargo, en sus escritos polémicos contra Descartes y Leibniz expresó la idea de que su método era superior a los propuestos por el francés y el alemán. Considerar estos escritos nos puede ayudar a comprender el papel que Newton le atribuyó al álgebra y al cálculo en su pensamiento matemático.

Péraire, Y., Infinitesimal approach to almost-automorphic functions, Annals of Pure and Applied Logic 63 283–297. Thanks to the use of ideal elements of several levels, we are able to give a compact topological characterization of almost-automorphic functions. This new characterization turns out to be equivalent to a geometrical one: the existence of a relatively dense group of “pointwise periods”. However, the more significant result obtained, in our opinion, is a very important lowering of the complexity in characterizations and proofs.

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, 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)

ABSTRACTA probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson’s transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields may have advantages over hyperreals in probabilistic (...) modeling. We show that probabilities developed over such fields are less expressive than hyperreal probabilities. (shrink)

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.

The probability calculus is very often used in the philosophy of science in order to support or to analyse epistemological points of view. The aim of this paper is to present in a summary the usual axioms of this calculus, as weIl as its most common consequences and theorems, which the philosopher of science in his arguments ressorts to.

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.

Es werden zwei Bedeutungen von „Infinitesimal“ unterschieden und zwei Thesen verteidigt: (1) Leibniz glaubte, das Infinitesimale in einer der beiden Bedeutungen sei nicht nur eine nützliche Erdichtung, sondern es sei sogar notwendig fur die Differentialrechnung; (2) die moderne Nichtstand-Analysis rechtfertigt weder Leibniz's Griinde fur die Einführung des Infinitesimalen noch seinen Gebrauch desselben.