Michael Friedman

What was Leibniz’s approach to artisanal knowledge? And how did he consider it with respect to mathematical, and more concretely, to geometrical knowledge? On the one hand, Leibniz emphasizes several times in his writings that one should extract “secrets and inventions” from the artisans. On the other hand, Leibniz points out that such artisans cannot formulate by themselves the geometric principles at the base of their machines. In this paper, we examine these intricate relations between Leibniz’s reflections on artisans, especially clockmakers and textile workers, as well as his thoughts on mechanical and geometric knowledge. Leibniz’s considerations of various artisanal machines, like clocks and looms, lead us to discuss his wish to expand geometry, presenting these machines as embodying a “secret,” “hidden,” or even “deeper” or “more profound” geometry.

This open access book collects the historical and medial perspectives of a systematic and epistemological analysis of the complicated, multifaceted relationship between model and mathematics, ranging from, for example, the physical mathematical models of the 19th century to the simulation and digital modelling of the 21st century. The aim of this anthology is to showcase the status of the mathematical model between abstraction and realization, presentation and representation, what is modeled and what models. This book is open access under a CC BY 4.0 license.

The concept of a “scientific philosophy” first developed in the mid nineteenth century, as a reaction against what was viewed as the excessively speculative and metaphysical character of post-Kantian German idealism. One of the primary intellectual models of this movement was a celebrated address by Hermann von Helmholtz, “Über das Sehen des Menschen,” delivered at the dedication of a monument to Kant at Königsberg in 1855. Helmholtz begins by asking, on behalf of the audience, why a natural scientist like himself is speaking in honor of a philosopher. This question only arises, he says, because of the current deplorable climate of enmity and mutual distrust between the two fields – a climate which is due, in Helmholtz’s opinion, to the entirely speculative system of Naturphilosophie that Schelling and Hegel have erected wholly independent of, and even in open hostility towards, the actual positive results of the natural sciences. What Helmholtz is now recommending, however, is a return to the close cooperation between the two fields exemplified in the work of Kant, who himself made significant contributions to natural science, and, in general, “stood in relation to the natural sciences together with the natural scientists on precisely the same fundamental principles.”1 And it was this recommendation that was enthusiastically embraced within the emerging “back to Kant!” movement, where it led to the idea that all metaphysics should be replaced by the new discipline of “epistemology” or “theory of knowledge”, so that philosophy itself would now become “scientific.”

Rudolf Carnap is increasingly regarded as one of the most important philosophers of the twentieth century. He was one of the leading figures of the logical empiricist movement associated with the Vienna Circle and a central figure in the analytic tradition more generally. He made major contributions to philosophy of science and philosophy of logic, and, perhaps most importantly, to our understanding of the nature of philosophy as a discipline. In this volume a team of contributors explores the major themes of his philosophy and discusses his relationship with the Vienna Circle and with philosophers such as Frege, Husserl, Russell, and Quine. New readers will find this the most convenient and accessible guide to Carnap currently available. Advanced students and specialists will find a conspectus of recent developments in the interpretation of Carnap.

This paper focuses on the relationship between the general metaphysics of the Critique of Pure Reason and the special metaphysics of the Metaphysical Foundations of Natural Science. Kant claims that what distinguishes the latter from the former is that the latter presupposes an empirical concept, namely the concept of matter, whereas the former does not. It is argued that the concept of matter is empirical not in any ordinary sense, but in the sense that it requires actual perceptible objects to be given. Only the system of contingently given, empirical objects of the solar system as described by Newtonian physics, satisfies this requirement. This interpretation suggests that “experience”, whose possibility Kant is exploring in the first Critique, is exclusively “scientific experience”, and that the task of theoretical philosophy thus lies primarily in understanding the conceptual presuppositions of scientific experience.

ABSTRACT This essay considers the relationship between history of science and philosophy of science from Thomas Kuhn to the present. This relationship, of course, has often been troubled, but there is now new hope for an ongoing productive interaction—due to an increasing awareness, among other things, of the mutual entanglement between the development of modern science and the development of modern philosophy on the part of both professional (historically minded) philosophers and professional historians of science. This idea is illustrated with several examples, including the entanglement of Einstein's theory of relativity with logical empiricism and its precursors and the entanglement of the analytic tradition in philosophy first with mathematical logic and then (through mathematical logic) with the cognitive sciences. Against this background, finally, the essay considers the emergence of contemporary social and cultural history of science from the Kantian and neo‐Kantian philosophical traditions.

Does the materiality of a three-dimensional model have an effect on how this model operates in an exploratory way, how it prompts discovery of new mathematical results? Material mathematical models were produced and used during the second half of the nineteenth century, visualizing mathematical objects, such as curves and surfaces—and these were produced from a variety of materials: paper, cardboard, plaster, strings, wood. However, the question, whether their materiality influenced the status of these models—considered as exploratory, technical, or representational—was hardly touched upon. This article aims to approach this question by investigating two case studies: Beltrami’s paper models vs. Dyck’s plaster ones of the hyperbolic plane; and Chisini’s string models of braids vs. Artin’s and Moishezon’s algebraization of these braids. These two case studies indicate that materiality might have a decisive role in how the model was taken into account mathematically: either as an exploratory or rather as a technical or pedagogical object.

This is a very important book. It has already become required reading for researchers on the relation between the exact sciences and Kant’s philosophy. The main theme is that Kant’s continuing program to find a metaphysics that could provide a foundation for the science of his day is of crucial importance to understanding the development of his philosophical thought from its earliest precritical beginnings in the thesis of 1747, right through the highwater years of the critical philosophy, to his last unpublished writings in the Opus postumum. In the course of articulating this theme, Friedman has made extensive use of detailed historical information about their scientific and mathematical background to illuminate Kant’s texts. Over and over again, such information is used to suggest interesting and quite subtle interpretations for texts that may have seemed puzzling or just wrong-headed.

Kant was centrally concerned with issues in the philosophy of natural science throughout his career. The Metaphysical Foundations of Natural Science presents his most mature reflections on these themes in the context of both his 'critical' philosophy, presented in the Critique of Pure Reason, and the natural science of his time. This volume presents a translation by Michael Friedman which is especially clear and accurate. There are explanatory notes indicating some of the main connections between the argument of the Metaphysical Foundations and the first Critique - as well as parallel connections to Newton's Principia. The volume is completed by an historical and philosophical introduction and a guide to further reading.

This volume, originally published in 2002, assembles the historical sequence of writings that Kant published between 1783 and 1796 to popularize, summarize, amplify and defend the doctrines of his masterpiece, the Critique of Pure Reason of 1781. The best known of them, the Prolegomena, is often recommended to beginning students, but the other texts are also vintage Kant and are important sources for a fully rounded picture of Kant's intellectual development. As with other volumes in the series there are copious linguistic notes and a glossary of key terms. The editorial introductions and explanatory notes shed light on the critical reception accorded Kant by the metaphysicians of his day and on Kant's own efforts to derail his opponents.