The stage on which our ingenious play is performed: Kant's epistemology of Weltkenntnis

This paper focuses on Kant's account of physical geography and his theory of the Earth. In spelling out the epistemological foundations of Kant's physical geography, the paper examines 1) their connection to the mode of holding-to-be-true, mathematical construction and empirical certainty and 2) their implications for Kant's view of cosmopolitan right. Moreover, by showing the role played by the mathematical model of the Earth for the foundations of Kant's Doctrine of Right, the exact relationship between the latter and physical geography is highlighted. Finally, this paper shows how, in Kant's view, the progress of physical geography can be assured if and only if the free circulation of human beings is established and regulated by law. Therefore, examining the mutual relationship between the theory of Earth and the foundations of right opens new perspectives on the relationship between epistemology and practical philosophy within Kant's system.     

Helmholtz's Kant revisited (Once more). The all-pervasive nature of Helmholtz's struggle with Kant's Anschauung

In this analysis, the classical problem of Hermann von Helmholtz's (1821–1894) Kantianism is explored from a particular vantage point, that to my knowledge, has not received the attention it deserves notwithstanding its possible key role in disentangling Helmholtz's relation to Kant's critical project. More particularly, we will focus on Helmholtz's critical engagement with Kant's concept of intuition [Anschauung] and (the related issue of) his dissatisfaction with Kant's doctrinal dualism. In doing so, it soon becomes clear that both (i) crucially mediated Helmholtz's idiosyncratic appropriation and criticism of (certain aspects of) Kant's critical project, and (ii) can be considered as a common denominator in a variety of issues that are usually addressed separately under the general header of (the problem of) Helmholtz's Kantianism. The perspective offered in this analysis can not only shed interesting new light on some interpretive issues that have become commonplace in discussions on Helmholtz's Kantianism, but also offers a particular way of connecting seemingly unrelated dimensions of Helmholtz's engagement with Kant's critical project (e.g. Helmholtz's views on causality and space). Furthermore, it amounts to the rather surprising conclusion that Helmholtz's most drastic revision of Kant's project pertains to his assumption of free will as a formal condition of experience and knowledge.     

Kant and the scope of the analytic method

The paper investigates Kant's pre-critical views on the use of analytic and synthetic methods in Newtonian science and in philosophical reasoning. In his 1755/56 writings, Kant made use of two variants of the analytic method, i.e., conceptual analysis in a Cartesian (or Leibnizean) sense, and analysis of the phenomena in a Newtonian sense. His Prize Essay (1764) defends Newton's analytic method of physics as appropriate for philosophy, in contradistinction to the synthetic method of mathematics. A closer look, however, shows that Kant does not identify Newton's method with conceptual analysis, but just suggests a methodological analogy between both methods. Kant’s 1768 paper on incongruent counterparts also fits in with his pre-critical use of conceptual analysis. Here, Kant criticizes Leibniz’ relational concept of space, arguing that it is incompatible with the phenomenon of chiral objects. Since this result was in conflict with his pre-critical views about space, Kant abandoned the analytic method of philosophy in favour of his critical method. The paper closes by comparing Kant's pre-critical analytic method and the way in which he once again took up the methodological analogy between Newtonian science and metaphysics, in the preface B to the Critique of Pure Reason, in the context of his thought experiment of pure reason.     

Ernst Cassirer's transcendental account of mathematical reasoning

Cassirer's philosophical agenda revolved around what appears to be a paradoxical goal, that is, to reconcile the Kantian explanation of the possibility of knowledge with the conceptual changes of nineteenth and early twentieth-century science. This paper offers a new discussion of one way in which this paradox manifests itself in Cassirer's philosophy of mathematics. Cassirer articulated a unitary perspective on mathematics as an investigation of structures independently of the nature of individual objects making up those structures. However, this posed the problem of how to account for the applicability of abstract mathematical concepts to empirical reality. My suggestion is that Cassirer was able to address this problem by giving a transcendental account of mathematical reasoning, according to which the very formation of mathematical concepts provides an explanation of the extensibility of mathematical knowledge. In order to spell out what this argument entails, the first part of the paper considers how Cassirer positioned himself within the Marburg neo-Kantian debate over intellectual and sensible conditions of knowledge in 1902–1910. The second part compares what Cassirer says about mathematics in 1910 with some relevant examples of how structural procedures developed in nineteenth-century mathematics.     

Kant on anatomy and the status of the life sciences

This paper contributes to recent interest in Kant's engagement with the life sciences by focusing on one corner of those sciences that has received comparatively little attention: physical and comparative anatomy. By attending to remarks spread across Kant's writings, we gain some insight into Kant's understanding of the disciplinary limitations but also the methodological sophistication of the study of anatomy and physiology. Insofar as Kant highlights anatomy as a paradigmatic science guided by the principle of teleology in the Critique of the Power of Judgment, a more careful study of Kant's discussions of anatomy promises to illuminate some of the obscurities of that text and of his understanding of the life sciences more generally. In the end, it is argued, Kant's ambivalence with regard to anatomy gives way to a pessimistic conclusion about the possibility that anatomy, natural history, and, by extension, the life sciences more generally might one day become true natural sciences.     

Analogical reflection as a source for the science of life: Kant and the possibility of the biological sciences

In contrast to the previously widespread view that Kant's work was largely in dialogue with the physical sciences, recent scholarship has highlighted Kant's interest in and contributions to the life sciences. Scholars are now investigating the extent to which Kant appealed to and incorporated insights from the life sciences and considering the ways he may have contributed to a new conception of living beings. The scholarship remains, however, divided in its interest: historians of science are concerned with the content of Kant's claims, and the ways in which they may or may not have contributed to the emerging science of life, while historians of philosophy focus on the systematic justifications for Kant's claims, e.g., the methodological and theoretical underpinnings of Kant's statement that living beings are mechanically inexplicable. My aim in this paper is to bring together these two strands of scholarship into dialogue by showing how Kant's methodological concerns (specifically, his notion of reflective judgment) contributed to his conception of living beings and to the ontological concern with life as a distinctive object of study. I argue that although Kant's explicit statement was that biology could not be a science, his implicit and more fundamental claim was that the study of living beings necessitates a distinctive mode of thought, a mode that is essentially analogical. I consider the implications of this view, and argue that it is by developing a new methodology for grasping organized beings that Kant makes his most important contribution to the new science of life.     

Mathematical method and Newtonian science in the philosophy of Christian Wolff

This paper aims to illuminate Christian Wolff’s view of mathematical reasoning, and its use in metaphysics, by comparing his and Leibniz’s responses to Newton’s work. Both Wolff and Leibniz object that Newton’s metaphysics is based on ideas of sense and imagination that are suitable only for mathematics. Yet Wolff expresses more regard (than Leibniz) for Newton’s scientific achievement. Wolff’s approval of the use of imaginative ideas in Newtonian mathematical science seems to commit him to an inconsistent triad. For he rejects their use in metaphysics, and also holds that every scientific discipline must follow mathematics’ method. A facile resolution would be to suppose Wolff identifies the method of mathematics with the order in which propositions are deduced, or with “analysis” that reveals the structure of concepts. This would be to assimilate Wolff’s view to Leibniz’s (on which all mathematical propositions are ultimately derived from definitions, and definitions are justified by conceptual analysis). On this construal, mathematical reasoning involves only the understanding. But Wolff conceives mathematics’ method more broadly, to include processes of concept-formation which involve perception and imagination. Thus my way of resolving the tension is to find roles for perception and imagination in the formation of metaphysical concepts.     

Natural history and the formation of the human being: Kant on active forces

In his 1785-review of the Ideen zur Philosophie der Geschichte der Menschheit, Kant objects to Herder's conception of nature as being imbued with active forces. This attack is usually evaluated against the background of Kant's critical project and his epistemological concern to caution against the “metaphysical excess” of attributing immanent properties to matter. In this paper I explore a slightly different reading by investigating Kant's pre-critical account of creation and generation. The aim of this is to show that Kant's struggle with the forces of matter has a long history and revolves around one central problem: that of how to distinguish between the non-purposive forces of nature and the intentional powers of the mind. Given this history, the epistemic stricture that Kant's critical project imposes on him no longer appears to be the primary reason for his attack on Herder. It merely aggravates a problem that Kant has been battling with since his earliest writings.     

Kant's universal conception of natural history

Scholars often draw attention to the remarkably individual and progressive character of Kant's Universal Natural History and Theory of the Heavens (1755). What is less often noted, however, is that Kant's project builds on several transformations that occurred in natural science during the seventeenth and eighteenth centuries. Without contextualising Kant's argument within these transformations, the full sense of Kant's achievement remains unseen. This paper situates Kant's essay within the analogical form of Newtonianism developed by a diverse range of naturalists including Georges Buffon, Albrecht von Haller and Thomas Wright. It argues that Kant's universal conception of natural history can be viewed within the free-thinking and anti-clerical movement associated with Buffon. This does not mean, however, that it breaks from the methodological rules of Newtonianism. The claim of this paper is that Kant's essay contributes to the transformation of natural history from a logical system of classification to an explanation for the physical diversity of natural products according to laws.     

The intelligibility of motion and construction: Descartes’ early mathematics and metaphysics, 1619-1637

I argue for an interpretation of the connection between Descartes’ early mathematics and metaphysics that centers on the standard of geometrical intelligibility that characterizes Descartes’ mathematical work during the period 1619 to 1637. This approach remains sensitive to the innovations of Descartes’ system of geometry and, I claim, sheds important light on the relationship between his landmark Geometry (1637) and his first metaphysics of nature, which is presented in Le monde (1633). In particular, I argue that the same standard of clear and distinct motions for construction that allows Descartes to distinguish ‘geometric’ from ‘imaginary’ curves in the domain of mathematics is adopted in Le monde as Descartes details God’s construction of nature. I also show how, on this interpretation, the metaphysics of Le monde can fruitfully be brought to bear on Descartes’ attempted solution to the Pappus problem, which he presents in Book I of the Geometry. My general goal is to show that attention to the standard of intelligibility Descartes invokes in these different areas of inquiry grants us a richer view of the connection between his early mathematics and philosophy than an approach that assumes a common method is what binds his work in these domains together.     

The metaphor of epigenesis: Kant,Blumenbach and Herder

Over the last few decades, the meaning of the scientific theory of epigenesis and its significance for Kant's critical philosophy have become increasingly central questions. Most recently, scholars have argued that epigenesis is a key factor in the development of Kant's understanding of reason as self-grounding and self-generating. Building on this work, our claim is that Kant appealed to not just any epigenetic theory, but specifically Johann Friedrich Blumenbach's account of generation, and that this appeal must be understood not only in terms of self-organization, but also in terms of the demarcation of a specific domain of inquiry: for Blumenbach, the study of life; for Kant, the study of reason. We argue that Kant adopted this specific epigenetic model as a result of his dispute with Herder regarding the independence of reason from nature. Blumenbach's conception of epigenesis and his separation of a domain of the living from the non-living lent Kant the tools to demarcate metaphysics, and to guard reason against Herder's attempts to naturalize it.     

Structuralism and the conformity of mathematics and nature

Structuralists typically appeal to some variant of the widely popular ‘mapping’ account of mathematical representation to suggest that mathematics is applied in modern science to represent the world’s physical structure. However, in this paper, I argue that this realist interpretation of the ‘mapping’ account presupposes that physical systems possess an ‘assumed structure’ that is at odds with modern physical theory. Through two detailed case studies concerning the use of the differential and variational calculus in modern dynamics, I show that the formal structure that we need to assume in order to apply the mapping account is inconsistent with the way in which mathematics is applied in modern physics. The problem is that a realist interpretation of the ‘mapping’ account imposes too severe of a constraint on the conformity that must exist between mathematics and nature in order for mathematics to represent the structure of a physical system.     

Defending eliminative structuralism and a whole lot more (or less)

Ontic structural realism argues that structure is all there is. In (French, 2014) I argued for an ‘eliminativist’ version of this view, according to which the world should be conceived, metaphysically, as structure, and objects, at both the fundamental and ‘everyday’ levels, should be eliminated. This paper is a response to a number of profound concerns that have been raised, such as how we might distinguish between the kind of structure invoked by this view and mathematical structure in general, how we should choose between eliminativist ontic structural realism and alternative metaphysical accounts such as dispositionalism, and how we should capture, in metaphysical terms, the relationship between structures and particles. In developing my response I shall touch on a number of broad issues, including the applicability of mathematics, the nature of representation and the relationship between metaphysics and science in general.     

Radical mathematical Thomism: beings of reason and divine decrees in Torricelli’s philosophy of mathematics

Evangelista Torricelli (1608-1647) is perhaps best known for being the most gifted of Galileo’s pupils, and for his works based on indivisibles, especially his stunning cubature of an infinite hyperboloid. Scattered among Torricelli’s writings, we find numerous traces of the philosophy of mathematics underlying his mathematical practice. Though virtually neglected by historians and philosophers alike, these traces reveal that Torricelli’s mathematical practice was informed by an original philosophy of mathematics. The latter was dashed with strains of Thomistic metaphysics and theology. Torricelli’s philosophy of mathematics emphasized mathematical constructs as human-made beings of reason, yet mathematical truths as divine decrees, which upon being discovered by the mathematician ‘appropriate eternity’. In this paper, I reconstruct Torricelli’s philosophy of mathematics—which I label radical mathematical Thomism—placing it in the context of Thomistic patterns of thought.     

Hilbert's axiomatic method and Carnap's general axiomatics

This paper compares the axiomatic method of David Hilbert and his school with Rudolf Carnap's general axiomatics that was developed in the late 1920s, and that influenced his understanding of logic of science throughout the 1930s, when his logical pluralism developed. The distinct perspectives become visible most clearly in how Richard Baldus, along the lines of Hilbert, and Carnap and Friedrich Bachmann analyzed the axiom system of Hilbert's Foundations of Geometry—the paradigmatic example for the axiomatization of science. Whereas Hilbert's axiomatic method started from a local analysis of individual axiom systems in which the foundations of mathematics as a whole entered only when establishing the system's consistency, Carnap and his Vienna Circle colleague Hans Hahn instead advocated a global analysis of axiom systems in general. A primary goal was to evade, or formalize ex post, mathematicians' ‘material’ talk about axiom systems for such talk was held to be error-prone and susceptible to metaphysics.     

The abundant world: Paul Feyerabend's metaphysics of science

The goal of this paper is to provide an interpretation of Feyerabend's metaphysics of science as found in late works like Conquest of Abundance and Tyranny of Science. Feyerabend's late metaphysics consists of an attempt to criticize and provide a systematic alternative to traditional scientific realism, a package of views he sometimes referred to as “scientific materialism.” Scientific materialism is objectionable not only on metaphysical grounds, nor because it provides a poor ground for understanding science, but because it implies problematic claims about the epistemic and cultural authority of science, claims incompatible with situating science properly in democratic societies. I show how Feyerabend's metaphysical view, which I call “the abundant world” or “abundant realism,” constitute a sophisticated and challenging form of ontological pluralism that makes interesting connections with contemporary philosophy of science and issues of the political and policy role of science in a democratic society.     

Rehabilitating the regulative use of reason: Kant on empirical and chemical laws

In his Kritik der reinen Vernunft, Kant asserts that laws of nature “carry with them an expression of necessity” (A159/B198). There is, however, widespread interpretive disagreement regarding the nature and source of the necessity of empirical laws of natural sciences in Kant's system. It is especially unclear how chemistry—a science without a clear, straightforward connection to the a priori principles of the understanding—could contain such genuine, empirical laws. Existing accounts of the necessity of causal laws unfortunately fail to illuminate the possibility of non-physical laws. In this paper, I develop an alternative, ‘ideational’ account of natural laws, according to which ideas of reason necessitate the laws of some non-physical sciences. Chemical laws, for instance, are grounded on ideas of the elements, and the chemist aims to reduce her phenomena to these elements via experimentation. Although such ideas are beyond the possibility of experience, their postulation is necessary for the achievement of reason's theoretical ends: the unification and explanation of the cognitions of science.     

Into the ‘regions of physical and metaphysical chaos’: Maxwell’s scientific metaphysics and natural philosophy of action (agency,determinacy and necessity from theology,moral philosophy and history to mathematics,theory and experiment)

Maxwell’s writings exhibit an enduring preoccupation with the role of metaphysics in the advancement of science, especially the progress of physics. I examine the question of the distinction and the proper relation between physics and metaphysics and the way in which the question relies on key notions that bring together much of Maxwell’s natural philosophy, theoretical and experimental. Previous discussions of his attention to metaphysics have been confined to specific issues and polemics such as conceptions of matter and the problem of free will. I suggest a unifying pattern based on a generalized philosophical perspective and varying expressions, although never a systematic or articulated philosophical doctrine, but at least a theme of action and active powers, natural and human, intellectual and material, with sources and grounds in theology, moral philosophy and historical argument. While science was developing in the direction of professional specialization and alongside the rise of materialism, Maxwell held on to conservative intellectual outlook, but one that included a rich scientific life and held science as part of a rich intellectual, cultural and material life. His philosophical outlook integrated his science with and captured the new Victorian culture of construction and work, political, economic, artistic and engineering.     

Confirmational holism and its mathematical (w)holes

I critically examine confirmational holism as it pertains to the indispensability arguments for mathematical Platonism. I employ a distinction between pure and applied mathematics that grows out of the often overlooked symbiotic relationship between mathematics and science. I argue that this distinction undercuts the notion that (pure) mathematical theories fall under the holistic scope of the confirmation of our scientific theories.     

Between Kepler and Newton: Hooke’s ‘principles of congruity and incongruity’ and the naturalization of mathematics

ABSTRACT

Robert Hooke’s development of the theory of matter-as-vibration provides coherence to a career in natural philosophy which is commonly perceived as scattered and haphazard. It also highlights aspects of his work for which he is rarely credited: besides the creative speculative imagination and practical-instrumental ingenuity for which he is known, it displays lucid and consistent theoretical thought and mathematical skills. Most generally and importantly, however, Hooke’s ‘Principles?…?of Congruity and Incongruity of bodies’ represent a uniquely powerful approach to the most pressing challenge of the New Science: legitimizing the application of mathematics to the study of nature. This challenge required reshaping the mathematical practices and procedures; an epistemological framework supporting these practices; and a metaphysics which could make sense of this epistemology. Hooke’s ‘Uniform Geometrical or Mechanical Method’ was a bold attempt to answer the three challenges together, by interweaving mathematics through physics into metaphysics and epistemology. Mathematics, in his rendition, was neither an abstract and ideal structure (as it was for Kepler), nor a wholly-flexible, artificial human tool (as it was for Newton). It drew its power from being contingent on the particularities of the material world.