In this article, I propose an investigation of Ernst Cassirer’s conception of truth, as set out in his inaugural address as rector of Hamburg University, in light of two key concepts of his philosophy, i.e., function and symbolic form. My aim is neither to give an exhaustive exposition of the subject, nor to attempt to elaborate a complete Cassirerian theory of truth. Rather, I want to focus on how the Cassirerian conception of truth can be useful in countering certain contemporary (...) anti-scientific tendencies, particularly those born from the fear that modern science could destroy other cultural forms (especially religion). Specifically, I believe that Cassirer’s conception enables us to identify criteria by which to distinguish science from non-scientific endeavours and to preserve the primacy of science in the cognitive enterprise, without calling social consensus into question (avoiding the risks of relativization), but also without resorting to positivistic dogmatisms that condemn everything that is not science to falsity or nonsense. By tracing back any kind of truth to its own contest of meaning and by identifying this context of meaning on transcendental and semiotic bases, Cassirer’s conception opens to a pluralistic and pragmatic approach that leaves space for the possibility of non-scientific truths, which, while not being entitled to interfere with the scientific work, can still be considered legitimate and valuable. (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)

One of the most important philosophical topics in the early twentieth century and a topic that was seminal in the emergence of analytic philosophy was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of (...) logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant. (shrink)

The quest to understand the natural and the mathematical as well as philosophical principles of dynamics of life forms are ancient in the human history of science. In ancient times, Pythagoras and Plato, and later, Copernicus and Galileo, correctly observed that the grand book of nature is written in the language of mathematics. Platonism, Aristotelian logism, neo-realism, monadism of Leibniz, Hegelian idealism and others have made efforts to understand reasons of existence of life forms in nature and the underlying principles (...) through the lenses of philosophy and mathematics. In this paper, an approach is made to treat the similar question about nature and existential life forms in view of mathematical philosophy. The approach follows constructivism to formulate an abstract model to understand existential life forms in nature and its dynamics by selectively combining the elements of various schools of thoughts. The formalisms of predicate logic, probabilistic inference and homotopy theory of algebraic topology are employed to construct a structure in local time-scale horizon and in cosmological time-scale horizon. It aims to resolve the relative and apparent conflicts present in various thoughts in the process, and it has made an effort to establish a logically coherent interpretation. (shrink)

This paper deals with Natorp’s version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorp’s mathematical philosophy of science is contained in his “knowledge equation” that may be considered as a mathematical model of the “transcendental method” conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, in the Aufbau, contended that this endeavor (...) can be divided into two distinct parts, namely, a finite “constitution” of the object of knowledge and an infinite incompletable empirical description. In contrast, and more in the original spirit of Cohen and Natorp, the physicist and philosopher Margenau in The Nature of Physical Reality (Margenau. 1950) conceived the infinity of this “Aufgabe” as an infinite dialectical process, in which relative “data” and “conceptual constructs” determine each other. This dialectical process eliminates the dichotomy between Anschauung and Begriff that distinguished the Marburg Neo-Kantianism from Kantian orthodoxy, namely, the abandonment of the difference between intuition and concept. Finally, the paper deals with the non-Archimedean geometrical systems that played a central role in Natorp’s defence of Cohen’s “infinitesimal” metaphysics. (shrink)

In spite of Ernst Cassirer’s criticisms of psychologism throughout Substance and Function, in the final chapter he issues a demand for a “psychology of relations” that can do justice to the subjective dimensions of mathematics and natural science. Although these remarks remain somewhat promissory, the fact that this is how Cassirer chooses to conclude Substance and Function recommends it as a topic worthy of serious consideration. In this paper, I argue that in order to work out the details of Cassirer’s (...) psychology of relations in Substance and Function, we need to situate it within two broader frameworks. First, I position Cassirer’s view in relation to the view of psychology and logic endorsed by his Marburg Neo-Kantian predecessors, Hermann Cohen and Paul Natorp. Second, I augment Cassirer’s early account of the psychology of relations in Substance and Function with the more mature view of psychology that he presents in The Philosophy of Symbolic Forms. By placing Cassirer’s account within these contexts, I claim that we gain insight into his psychology of relations, not just as it pertains to mathematics and natural science, but to culture as a whole. Moreover, I maintain that pursuing this strategy helps shed light on one of the most controversial features of his philosophy of mathematics and natural science, viz., his theory of the a priori. (shrink)

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 article discusses an interpretation of Kant's conception of transcendental subjectivity, which manages to avoid many of the concerns that have been raised by analytic interpreters over this doctrine. It is an interpretation put forward by selected C19 and early C20 neo-Kantian writers. The article starts out by offering a neo-Kantian interpretation of the object as something that is constituted by the categories and that serves as a standard of truth within a theory of judgment. The second part explicates transcendental (...) subjectivity as the system of categories, which is self-referential and constitutes objects, in order to then evaluate this conception by means of a comparison with Hegel's absolute subject. Rather than delineating the differences between neo-Kantian writers, the article systematically expounds a shared project, which consists in providing the ultimate foundation for judgments by means of an anti-psychologist and non-metaphysical interpretation of transcendental subjectivity. (shrink)

Introduction to the Special Volume, “Method, Science and Mathematics: Neo-Kantianism and Analytic Philosophy,” edited by Scott Edgar and Lydia Patton. At its core, analytic philosophy concerns urgent questions about philosophy’s relation to the formal and empirical sciences, questions about philosophy’s relation to psychology and the social sciences, and ultimately questions about philosophy’s place in a broader cultural landscape. This picture of analytic philosophy shapes this collection’s focus on the history of the philosophy of mathematics, physics, and psychology. The following essays (...) uncover, reflect on, and exemplify modes of philosophy that are engaged with these allied disciplines. They make the case that, to the extent that analytic philosophers are still concerned with philosophy’s ties to these disciplines, we would do well to pay attention to neo-Kantian views on those ties. (shrink)

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)

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)

Cassirer’s neo-Kantian epistemology has become a classical reference in contemporary history and philosophy of science. However, the historical aspects of his thought are sometimes seen to be in some tension with his defence of a priori elements of knowledge. This paper reconsiders Cassirer’s strategy to address this tension by positing functional dependencies at the core of the notion of objectivity. This requires the epistemologist to account for the determination of the objects of knowledge within given scientific theories, but also for (...) conceptual changes over time. (shrink)