This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This is the first English-language intellectual biography of the German-Jewish philosopher Ernst Cassirer (1874-1945), a leading figure on the Weimar intellectual scene and one of the last and finest representatives of the liberal-idealist ...

Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested (...) in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body. (shrink)

: Few texts summarize and at the same time compound the challenges of their author's philosophy so sharply as Hermann Cohen's Das Prinzip der Infinitesimalmethode und seine Geschichte (1883). The book's meaning and style are greatly illuminated by placing it in the scientific, political, and academic context of late-nineteenth century Germany. As this context changed, so did both the reception of the philosophy of the infinitesimal and of the Marburg school more generally. A study of this transformation casts significant light (...) on the political relevance of the philosophy of science in theWilhelmine era. As a means of following this development across time, Cohen's text is read through its changing reception in the philosophy of his closest disciple, Ernst Cassirer. (shrink)

Few texts summarize and at the same time compound the challenges of their author's philosophy so sharply as Hermann Cohen's Das Prinzip der Infinitesimalmethode und seine Geschichte . The book's meaning and style are greatly illuminated by placing it in the scientific, political, and academic context of late-nineteenth century Germany. As this context changed, so did both the reception of the philosophy of the infinitesimal and of the Marburg school more generally. A study of this transformation casts significant light on (...) the political relevance of the philosophy of science in theWilhelmine era. As a means of following this development across time, Cohen's text is read through its changing reception in the philosophy of his closest disciple, Ernst Cassirer. (shrink)

The notion of idealization has received considerable attention in contemporary philosophy of science but less in philosophy of mathematics. An exception was the ‘critical idealism’ of the neo-Kantian philosopher Ernst Cassirer. According to Cassirer the methodology of idealization plays a central role for mathematics and empirical science. In this paper it is argued that Cassirer's contributions in this area still deserve to be taken into account in the current debates in philosophy of mathematics. For extremely useful criticisms on earlier versions (...) I am grateful to B.P. Larvor and another anonymous journal referee. CiteULike Connotea Del.icio.us What's this? (shrink)

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the (...) infinitelysmall. (shrink)

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the (...) infinitelysmall. (shrink)

This article was found among the papers left by Prof. Laugwitz (May 5, 1932–April 17, 2000). The following abstract is extracted from a lecture he gave at the Fourth Austrain Symposion on the History of Mathematics (Neuhofen/ybbs, November 10, 1995).About 100 years ago, the Cantor-Veronese controversy found wide interest and lasted for more than 20 years. It is concerned with “actual infinity” in mathematics. Cantor, supported by Peano and others, believed to have shown the non-existence of infinitely small quantities, and (...) therefore he fought against the infinitely large and small numbers in Veronese’s geometry, but also against the non-archimedean systems of Thomae, du Bois-Reymond, and Stolz. As a positive consequence of the controversy the distinction between Cantor’s transfinite arithmetic and the theory of ordered algebraic structures becomes clear. (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)

Leibniz viewed the principle of continuity, the principle that all natural changes are produced by degrees, as a useful heuristic for evaluating the truth of a theory. Since the Cartesian laws of motion entailed discontinuities in the natural order, Leibniz could safely reject it as a false theory. The principle of continuity has similar implications for analyses of Leibniz's theory of consciousness. I briefly survey the three main interpretations of Leibniz's theory of consciousness and argue that the standard account entails (...) a discontinuity that Leibniz could not allow. I argue that the principle of continuity and the textual data favor an interpretation according to which a conscious mental state just is a perception that is distinct to a sufficient degree. (shrink)

According to Michael Friedman, Ernst Cassirer’s “outstanding contribution [to Neo-Kantianism] was to articulate, for the first time, a clear and coherent conception of formal logic within the context of the Marburg School” (Friedman 2000, p. 30). In his paper “Kant und die moderne Mathematik” (1907), Cassirer argued not only that the new relational logic of Frege1 and Russell was a major breakthrough with profound philosophical implications, but also that the logicist thesis itself was a “fact” of modern mathematics. Cassirer summarizes (...) his evaluation of Russell’s work: Here logic and mathematics have been fused into a true, henceforth indissoluble unity; and from this inner connection there arises for each... (shrink)

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be said to contain (...) “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas $\mathbb{R}$should merely be regarded as constituting an arithmetic continuum, No may be regarded as a sort of absolute arithmetic continuum, and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum. (shrink)

During the first months of 1887, while completing the drafts of his Mitteilungen zur Lehre vom Transfiniten, Georg Cantor maintained a continuous correspondence with Benno Kerry. Their exchange essentially concerned two main topics in the philosophy of mathematics, namely, (a) the concept of natural number and (b) the infinitesimals. Cantor's and Kerry's positions turned out to be irreconcilable, mostly because of Kerry's irremediably psychologistic outlook, according to Cantor at least. In this study, I will examine and reconstruct the main points (...) in the discussion around (a) and (b) and stress some interesting aspects of the philosophical and mathematical thought of Benno Kerry. (shrink)

Giuseppe Veronese (1854-1917) est connu pour ses études sur les espaces à plusieurs dimensions ; moins connus sont les écrits « philosophiques », qui concernent les fondements de la géométrie et des mathématiques et qui expliquent les raisons pour la construction d’une géométrie non-archimédienne (une dizaine d’années avant David Hilbert) et la formulation d’un concept de continu, qui contient des éléments infinis et infiniment petits. L’article esquissera quelques traits saillants de son épistémologie et analysera le rapport entre géométrie et intuition (...) spatiale, en comparant les conceptions de l’espace géométrique dans les œuvres de Veronese, Helmholtz et Poincaré. Selon Veronese l’espace intuitif, de même que l’espace géométrique, n’est pas euclidien et il n’a pas un nombre déterminé de dimensions : on peut donc avoir l’intuition aussi bien d’un espace à quatre dimensions que d’un continu non-archimédien. (shrink)

Giuseppe Veronese (1854-1917) est connu pour ses études sur les espaces à plusieurs dimensions ; moins connus sont les écrits « philosophiques », qui concernent les fondements de la géométrie et des mathématiques et qui expliquent les raisons pour la construction d’une géométrie non-archimédienne (une dizaine d’années avant David Hilbert) et la formulation d’un concept de continu, qui contient des éléments infinis et infiniment petits. L’article esquissera quelques traits saillants de son épistémologie et analysera le rapport entre géométrie et intuition (...) spatiale, en comparant les conceptions de l’espace géométrique dans les œuvres de Veronese, Helmholtz et Poincaré. Selon Veronese l’espace intuitif, de même que l’espace géométrique, n’est pas euclidien et il n’a pas un nombre déterminé de dimensions : on peut donc avoir l’intuition aussi bien d’un espace à quatre dimensions que d’un continu non-archimédien. (shrink)

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...) with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

Edmund Husserl, known as the founder of the phenomenological movement, was one of the most influential philosophers of the twentieth century. A prolific scholar, he explored an enormous landscape of philosophical subjects, including philosophy of math, logic, theory of meaning, theory of consciousness and intentionality, and ontology in addition to phenomenology. This deeply insightful book traces the development of Husserl’s thought from his earliest investigations in philosophy—informed by his work as a mathematician—to his publication of _Ideas_ in 1913. Jitendra N. (...) Mohanty, an internationally renowned Husserl scholar, presents a masterful study that illuminates Husserl’s central concerns and provides a definitive assessment of the first phases of the philosopher’s career. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Excerpt from Substanzbegriff und Funktionsbegriff: Untersuchungen Uber die Grundfragen der Erkenntniskritik Die erste Anregung zu den Untersuchungen, die dieser Band enthalt, ist mir aus Studien zur Philosophie der Mathe matik erwachsen. Indem ich versuchte, von Seiten der Logik aus einen Zugang zu den Grundbegriffen der Mathematik zu gewinnen, erwies es sich vor allem als notwendig, die B e g r i f f s f u n k t i 0 n e'lhsimaher zu zergliedern und auf ihre Voraussetzungen zuruckzufuhren. Hier (...) aber machte sich alsbald eine eigentumliche Schwierigkeit geltend: die her kommliche logische Lehre vom Begriff zeigte sich in ihren bekannten Hauptzugen als unzureichend, die Probleme, zu denen die Prinzipienlehre der Mathematik hinfuhrt, auch nur vollstandig z u b e z e i c h n e n. Die exakte Wissenschaft war hier, wie sich mir immer deutlicher zu ergeben schien, zu Fragen gelangt, fur welche die Formensprache der tradi tionellen Logik kein genaues Correlat besitzt. Der sachliche Gehalt der mathematischen Erkenntnisse wies auf eine Grund form des Begriffs zuruck, die in der Logik selbst nicht zu klarer Bezeichnung und Anerkennung gekommen war. Ins besondere waren es Untersuchungen jiber den Reihenbegriff und den Grenzbegriff (deren spezielles Ergebnis in die allgemeineren Erorterungen dieses Buches nich? 3iffge nommen werden konnte), die diese Uberzeugung in mir be festigten und damit zu einer erneuten Analyse der Prinzipien der Begriffsbildung selbsfhindrangten. 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)

This major publication is a history of the semantic tradition in philosophy from the early nineteenth century through its incarnation in the work of the Vienna Circle, the group of logical positivists that emerged in the years 1925-1935 in Vienna who were characterised by a strong commitment to empiricism, a high regard for science, and a conviction that modern logic is the primary tool of analytic philosophy. In the first part of the book, Alberto Coffa traces the roots of logical (...) positivism in a semantic tradition that arose in opposition to Kant's theory that a priori knowledge is based on pure intuition and the constitutive powers of the mind. In Part II, Coffa chronicles the development of this tradition by members and associates of the Vienna Circle. Much of Coffa's analysis draws on the unpublished notes and correspondence of many philosophers. The book, however, is not merely a history of the semantic tradition from Kant 'to the Vienna Station'. Coffa also critically reassesses the role of semantic notions in understanding the ground of a priori knowledge and its relation to empirical knowledge and questions the turn the tradition has taken since Vienna. (shrink)

It hath been an old remark, that Geometry is an excellent Logic.

This book introduces a new approach to the issue of radical scientific revolutions, or "paradigm-shifts," given prominence in the work of Thomas Kuhn. The book articulates a dynamical and historicized version of the conception of scientific a priori principles first developed by the philosopher Immanuel Kant. This approach defends the Enlightenment ideal of scientific objectivity and universality while simultaneously doing justice to the revolutionary changes within the sciences that have since undermined Kant's original defense of this ideal. Through a modified (...) Kantian approach to epistemology and philosophy of science, this book opposes both Quinean naturalistic holism and the post-Kuhnian conceptual relativism that has dominated recent literature in science studies. Focussing on the development of "scientific philosophy" from Kant to Rudolf Carnap, along with the parallel developments taking place in the sciences during the same period, the author articulates a new dynamical conception of relativized a priori principles. This idea applied within the physical sciences aims to show that rational intersubjective consensus is intricately preserved across radical scientific revolutions or "paradigm-shifts and how this is achieved. (shrink)