Dedekind's major work on the foundations of arithmetic employs several techniques that have left him open to charges of psychologism, and through this, to worries about the objectivity of the natural-number concept he defines. While I accept that Dedekind takes the foundation for arithmetic to lie in certain mental powers, I will also argue that, given an appropriate philosophical background, this need not make numbers into subjective mental objects. Even though Dedekind himself did not provide that background, one can nevertheless (...) be found in the work of the neo-Kantian Ernst Cassirer on mathematical concept formation. (shrink)

This paper has a two-fold objective: to provide a balanced, multi-faceted account of the origins of logicism; to rehabilitate Richard Dedekind as a main logicist. Logicism should be seen as more deeply rooted in the development of modern mathematics than typically assumed, and this becomes evident by reconsidering Dedekind's writings in relation to Frege's. Especially in its Dedekindian and Fregean versions, logicism constitutes the culmination of the rise of ?pure mathematics? in the nineteenth century; and this rise brought with it (...) an inter-weaving of methodological and epistemological considerations. The latter aspect illustrates how philosophical concerns can grow out of mathematical practice, as opposed to being imposed on it from outside. It also sheds new light on the legacy and the lasting significance of logicism today. (shrink)

This paper compares Cohen’s Logic of Pure Knowledge and Cassirer’s Substance and Function in order to evaluate how in these works Cohen and Cassirer go beyond the limits established by Kantian philosophy. In his Logic, Cohen seeks to ground in pure thought all the elements which Kant distinguishes in empirical intuition: its matter (sensation) as well as its form (time and space). In this way, Cohen tries to provide an account of knowledge without appealing to any receptivity. In accordance with (...) Cohen’s project of reformulating the Kantian theory of sensibility, Cassirer undertakes in Substance and Function the task of developing an alternative doctrine of pure and empirical manifolds. But whereas Cohen analyzes the laws of pure thought, Cassirer aims to highlight the functional character of concepts in the development of modern mathematics and physics. I will discuss these two different approaches to the problems raised by Kantian philosophy and I will argue that Cassirer went further than Cohen in the project of critical idealism. (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)

Both Hilbert's axiomatics and Cassirer's philosophy of symbolic forms have their roots in Leibniz's idea of a 'universal characteristic,' and grow on Hertz's 'principles of mechanics,' and Dedekind's 'foundations of arithmetic'. As Cassirer recalls in the introduction to his Philosophy of Symbolic Forms, it was the discovery of the analysis of infinity that led Leibniz to focus on "the universal problem inherent in the function of symbolism, and to raise his 'universal characteristic' to a truly philosophical plane." In Leibniz's view, (...) the logic of 'things' cannot be separated from the logic of 'signs,' as "the sign is no mere accidental cloak of the idea, but its necessary and essential organ."Every 'law' of .. (shrink)

This paper tries to make sense of Kant's scattered remarks about conic sections to see what light they shed on his philosophy of mathematics. It proceeds by confronting his remarks with the source that seems to have informed his thinking about conic sections: the Conica of Apollonius. The paper raises questions about Kant's attitude towards mathematics and the way he understood the cognitive resources available to us to do mathematics.

Until recently, leading work on the philosophy of thought experiments mistakenly credited Mach with coining the term. While Ørsted’s prior use has become more widely acknowledged, there remains a c...

Ernst Cassirer’s book Substanzbegriff und Funktionsbegriff is a difficult book for contemporary readers to understand. Its topic, the theory of concept formation, engages with debates and authors that are largely unknown today. And its “historical” style violates the philosophical standards of clarity first propounded by early analytic philosophers. Cassirer, for instance, never says explicitly what he means by “substance-concept” and “function-concept.” In this article, I answer three questions: Why did Cassirer choose to focus on the topic of concept formation? What (...) did Cassirer mean in contrasting “substance-concepts” and “function-concepts”? How does Cassirer’s polemic against traditional theories of concept formation lead to the distinctive philosophy of mathematics that he defends in the book? I argue that Cassirer’s contrast between substance-concepts and function-concepts includes a series of interrelated contrasts—contrasts that touch on issues in logic, metaphysics, epistemology, and the theory of objectivity. Cassirer’s defense of mathematical structuralism flows out of a progressively unfolding and intricate argument that begins with epistemological problems in the theory of concept formation. (shrink)

This paper offers an introduction to Hermann Cohen’s Das Princip der Infinitesimal-Methode, and recounts the history of its controversial reception by Cohen’s early sympathizers, who would become the so-called ‘Marburg school’ of Neo-Kantianism, as well as the reactions it provoked outside this group. By dissecting the ambiguous attitudes of the best-known representatives of the school, as well as those of several minor figures, this paper shows that Das Princip der Infinitesimal-Methode is a unicum in the history of philosophy: it represents (...) a strange case of an unsuccessful book’s enduring influence. The “puzzle of Cohen’s Infinitesimalmethode,” as we will call it, can be solved by looking beyond the scholarly results of the book, and instead focusing on the style of philosophy it exemplified. Moreover, the paper shows that Cohen never supported, but instead explicitly opposed, the doctrine of the centrality of the ‘concept of function’, with which Marburg Neo-Kantianism is usually associated. (shrink)

This paper offers an introduction to Hermann Cohen’s Das Princip der Infinitesimal-Methode, and recounts the history of its controversial reception by Cohen’s early sympathizers, who would become the so-called ‘Marburg school’ of Neo-Kantianism, as well as the reactions it provoked outside this group. By dissecting the ambiguous attitudes of the best-known representatives of the school, as well as those of several minor figures, this paper shows that Das Princip der Infinitesimal-Methode is a unicum in the history of philosophy: it represents (...) a strange case of an unsuccessful book’s enduring influence. The “puzzle of Cohen’s Infinitesimalmethode,” as we will call it, can be solved by looking beyond the scholarly results of the book, and instead focusing on the style of philosophy it exemplified. Moreover, the paper shows that Cohen never supported, but instead explicitly opposed, the doctrine of the centrality of the ‘concept of function’, with which Marburg Neo-Kantianism is usually associated. (shrink)

L’enjeu de cet article est d’exposer les principaux aspects du débat philosophique contemporain autour de la conception de l’ a priori de Michael Friedman. Ce dernier a défendu l’existence de principes constitutifs a priori permettant de coordonner la structure mathématique des théories physiques avec l’expérience sensorielle. Bien qu’il s’appuie sur la conception cassirérienne de l’ a priori pour défendre une continuité entre les structures mathématiques des théories, Friedman, dans la lignée d’Hans Reichenbach, conçoit ces principes constitutifs comme révisables en fonction (...) des changements de théories physiques. Cette conception des principes constitutifs s’est vu reprocher un certain arbitraire dans le choix de ses principes, mais également son insuffisance et son simplisme pour décrire la manière avec laquelle les théories physiques s’appliquent à l’expérience. Enfin, il ne semble pas que l’on puisse, comme Friedman, réduire l’ a priori d’Ernst Cassirer à son rôle régulateur, puisque celui-ci concevrait également des principes constitutifs alternatifs à ceux de Reichenbach. (shrink)

The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to (...) these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics. (shrink)

The question of the applicability of mathematics is an epistemological issue that was explicitly raised by Kant, and which has played different roles in the works of neo-Kantian philosophers, before becoming an essential issue in early analytic philosophy. This paper will first distinguish three main issues that are related to the application of mathematics: indispensability arguments that are aimed at justifying mathematics itself; philosophical justifications of the successful application of mathematics to scientific theories; and discussions on the application of real (...) numbers to the measurement of physical magnitudes. A refinement of this tripartition is suggested and supported by a historical investigation of the differences between Kant’s position on the problem, several neo-Kantian perspectives, early analytic philosophy, and late 19th century mathematicians. Finally, the debate on the cogency of an application constraint in the definition of real numbers is discussed in relation to a contemporary debate in neo-logicism, in order to suggest a comparison not only with Frege’s original positions, but also with the ideas of several neo-Kantian scholars, including Hölder, Cassirer, and Helmholtz. (shrink)

Hermann von Helmholtz’s geometrical papers have been typically deemed to provide an implicitly group-theoretical analysis of space, as articulated later by Felix Klein, Sophus Lie, and Henri Poincaré. However, there is less agreement as to what properties exactly in such a view would pertain to space, as opposed to abstract mathematical structures, on the one hand, and empirical contents, on the other. According to Moritz Schlick, the puzzle can be resolved only by clearly distinguishing the empirical qualities of spatial perception (...) from those describable in terms of axiomatic geometry. This paper offers a partial defense of the group-theoretical reading of Helmholtz along the lines of Ernst Cassirer in the fourth volume of The Problem of Knowledge of 1940. In order to avoid the problem raised by Schlick, Cassirer relied on a Kantian view of space not so much as an object of geometry, but as a precondition for the possibility of measurement. Although the concept of group does not provide a description of space, the modern way to articulate the concept of space in terms of transformation groups reveals something about the structure and the transformation of spatial concepts in mathematical and natural sciences. (shrink)