Kuhn and coherentist epistemology

The paper challenges a recent attempt by Jouni-Matti Kuukkanen to show that since Thomas Kuhn’s philosophical standpoint can be incorporated into coherentist epistemology, it does not necessarily lead to: (Thesis 1) an abandonment of rationality and rational interparadigm theory comparison, nor to (Thesis 2) an abandonment of convergent realism. Leaving aside the interpretation of Kuhn as a coherentist, we will show that Kuukkanen’s first thesis is not sufficiently explicated, while the second one entirely fails. With regard to Thesis 1, we argue that Kuhn’s view on inter-paradigm theory comparison allows only for (what we shall dub as) ‘the weak notion of rationality’, and that Kuukkanen’s argument is thus acceptable only in view of such a notion. With regard to Thesis 2, we show that even if we interpret Kuhn as a coherentist, his philosophical standpoint cannot be seen as compatible with convergent realism since Kuhn’s argument against it is not ‘ultimately empirical’, as Kuukkanen takes it to be.     

Superposition: on Cavalieri’s practice of mathematics

Bonaventura Cavalieri has been the subject of numerous scholarly publications. Recent students of Cavalieri have placed his geometry of indivisibles in the context of early modern mathematics, emphasizing the role of new geometrical objects, such as, for example, linear and plane indivisibles. In this paper, I will complement this recent trend by focusing on how Cavalieri manipulates geometrical objects. In particular, I will investigate one fundamental activity, namely, superposition of geometrical objects. In Cavalieri’s practice, superposition is a means of both manipulating geometrical objects and drawing inferences. Finally, I will suggest that an integrated approach, namely, one which strives to understand both objects and activities, can illuminate the history of mathematics.     

Grasping the spirit in nature: Anschauung in Ørsted's epistemology of science and beauty

The intersection between art, poetry, philosophy and science was the leitmotif which guided the lives and careers of romantic natural philosophers including that of the Danish natural philosopher, H. C. Ørsted. A simple model of Ørsted’s career would be one in which it was framed by two periods of philosophical speculation: the youth’s curious and idealistic interest in new attractive thoughts and the experienced man’s mature reflections at the end of his life. We suggest that a closer look at the epistemological aspects of his works on the theory of beauty reveals a connection between this late work and his early philosophical work including experimental philosophy, but also with the work in teaching and textbook writing, that lies in between. The latter includes Ørsted’s view on the application of mathematics in natural philosophy as well as his failed attempt at a genetic presentation of elementary geometry.     

Maimon's criticism of Kant's doctrine of mathematical cognition and the possibility of metaphysics as a science

The aim of this paper is to discuss Maimon's criticism of Kant's doctrine of mathematical cognition. In particular, we will focus on the consequences of this criticism for the problem of the possibility of metaphysics as a science. Maimon criticizes Kant's explanation of the synthetic a priori character of mathematics and develops a philosophical interpretation of differential calculus according to which mathematics and metaphysics become deeply interwoven. Maimon establishes a parallelism between two relationships: on the one hand, the mathematical relationship between the integral and the differential and on the other, the metaphysical relationship between the sensible and the supersensible. Such a parallelism will be the clue to the Maimonian solution to the Kantian problem of the possibility of metaphysics as a science.     

Holographic space and time: Emergent in what sense?

This paper proposes a metaphysics for holographic duality. In addition to the AdS/CFT correspondence I also consider the dS/CFT conjecture of duality. Both involve non-perturbative string theory and both are exact dualities. But while the AdS/CFT keeps time at the margins of the story, the dS/CFT conjecture gives to time the “space” it deserves by presenting an interesting holographic model of it. My goals in this paper can be summarized in the following way. First, I argue that the formal structure and physical content of the duality do not support the standard philosophical reading of the relation in terms of grounding. Second, I put forward a philosophical scheme mainly extrapolated from the double aspect monism theory. I read holographic duality in this framework as it seems to fit the mathematical and physical structure of the duality smoothly. Inside this framework I propose a notion of spacetime emergence alternative to those ones commonly debated in the AdS/CFT physics and philosophy circles.     

Abduction, tomography, and other inverse problems

Charles S. Peirce introduced in the late 19^th century the notion of abduction as inference from effects to causes, or from observational data to explanatory theories. Abductive reasoning has become a major theme in contemporary logic, philosophy of science, and artificial intelligence. This paper argues that the new growing branch of applied mathematics called inverse problems deals successfully with various kinds of abductive inference within a variety of scientific disciplines. The fundamental theorem about the inverse reconstruction of plane functions from their line integrals was proved by Johann Radon already in 1917. The practical applications of Radon’s theorem and its generalizations include computerized tomography which became a routine imaging technique of diagnostic medicine in the 1970s.     

Augustus De Morgan and the propagation of moral mathematics

In the early nineteenth century, Henry Brougham endeavored to improve the moral character of England through the publication of educational texts. Soon after, Brougham helped form the Society for the Diffusion of Useful Knowledge to carry his plan of moral improvement to the people. Despite its goal of improving the nation’s moral character, the Society refused to publish any treatises on explicitly moral or religious topics. Brougham instead turned to a mathematician, Augustus De Morgan, to promote mathematics as a rational subject that could provide the link between the secular and religious worlds. Using specific examples gleaned from the treatises of the Society, this article explores both how mathematics was intended to promote the development of reason and morality and how mathematical content was shaped to fit this particular view of the usefulness of mathematics. In the course of these treatises De Morgan proposed a fundamentally new pedagogical approach, one which focused on the student and the role mathematics could play in moral education.     

Philosophy,history and sociology of science: Interdisciplinary relations and complex social identities

Sociology and philosophy of science have an uneasy relationship, while the marriage of history and philosophy of science has—on the surface at least—been more successful. I will take a sociological look at the history of the relationships between philosophy and history as well as philosophy and sociology of science. Interdisciplinary relations between these disciplines will be analysed through social identity complexity theory in order to draw out some conclusions on how the disciplines interact and how they might develop. I will use the relationships between the disciplines as a pointer for a more general social theory of interdisciplinarity which will then be used to sound a caution on how interdisciplinary relations between the three disciplines might be managed.     

Prelude to dimension theory: The geometrical investigations of Bernard Bolzano

This paper treats Bernard Bolzano's (1781–1848) investigations into a fundamental problem of geometry: the problem of adequately defining the concepts of line (or curve), surface, solid, and continuum. Bolzano's interest in this problem spanned most of his creative lifetime. In this paper a full discussion is given of the philosophical and mathematical motivation of Bolzano's problem as well as his two solutions to the problem. Bolzano's work on this part of geometry is relevant to the history of modern mathematics, because it forms a prelude to the more recent development of topological dimension theory.     

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.     

Why did John Herschel fail to understand polarization? The differences between object and event concepts

This paper offers a solution to a problem in Herschel studies by drawing on the dynamic frame model for concept representation offered by cognitive psychology. Applying the frame model to represent the conceptual frameworks of the particle and wave theories, this paper shows that discontinuity between the particle and wave frameworks consists mainly in the transition from a particle notion ‘side’ to a wave notion ‘phase difference’. By illustrating intraconceptual relations within concepts, the frame representations reveal the ontological differences between these two concepts. ‘Side’ is an object concept built on spatial relations, but ‘phase difference’ is an event concept built on temporal relations. The conceptual analyses display a possible cognitive source of Herschel’s misconception of polarization. Limited by his experimental works and his philosophical beliefs, Herschel comprehended polarization solely in terms of spatial relations, which prevented him from replacing the object concept ‘side’ with the event concept ‘phase difference’, and eventually resulted in his failure to understand the wave account of polarization.     

The ascendancy of the Laplace transform and how it came about

The modern Laplace transform is relatively recent. It was first used by Bateman in 1910, explored and codified by Doetsch in the 1920s and was first the subject of a textbook as late as 1937. In the 1920s and 1930s it was seen as a topic of front-line research; the applications that call upon it today were then treated by an older technique — the Heaviside operational calculus. This, however, was rapidly displaced by the Laplace transform and by 1950 the exchange was virtually complete. No other recent development in mathematics has achieved such ready popularisation and acceptance among the users of mathematics and the designers of undergraduate curricula. Communicated by C. Truesdell     

Über den Auslesewert chromosomaler Strukturtypen vonDrosophila subobscura unter experimentellen Bedingungen

Summary In artificial populations ofDrosophila subobscura, arising from the mating of strains of different origin and different chromosomal structure, it was demonstrated that also in the X-chromosome an adaptive equilibrium is reached between two different structural types, although heterosis only acts in the females. There is a significant excess of females in the hybrid-generation in both reciprocal matings, especially at low temperature. The sexratio of the pure strains is 1:1. The excess may be caused by a selective advantage of the structurally heterozygous females in competition with the hemizygous males.     

Probability in 1919/20: the von Mises-Pólya-Controversy

The correspondence between Richard von Mises and George Pólya of 1919/20 contains reflections on two well-known articles by von Mises on the foundations of probability in the Mathematische Zeitschrift of 1919, and one paper from the Physikalische Zeitschrift of 1918. The topics touched on in the correspondence are: the proof of the central limit theorem of probability theory, von Mises' notion of randomness, and a statistical criterion for integer-valuedness of physical data. The investigation will hint at both the fruitfulness and the limits of several of von Mises' notions such as ``collective', ``distribution' and ``complex adjuncts' (characteristic functions) for further developments in probability theory and in ``directional statistics'. By pointing to the selectiveness of Pólya's criticism, the historical analysis shows the differing expectations of the two men with respect to the further development of the theory of probability and its applications. The paper thus gives a glimpse of the provisional state of the theory around 1920, before others such as P. Lévy (1886–1971) and A. N. Kolmogorov (1903–1987) stepped in and created a new paradigm for probability theory.     

Is complexity a scientific concept?

Complexity science has proliferated across academic domains in recent years. A question arises as to whether any useful sense of ‘generalized complexity’ can be abstracted from the various versions of complexity to be found in the literature, and whether it could prove fruitful in a scientific sense. Most attempts at defining complexity center around two kinds of notions: Structural, and temporal or dynamic. Neither of these is able to provide a foundation for the intuitive or generalized notion when taken separately; structure is often a derivative notion, dependent on prior notions of complexity, and dynamic notions such as entropy are often indefinable. The philosophical notion of process may throw light on the tensions and contradictions within complexity. Robustness, for instance, a key quality of complexity, is quite naturally understood within a process-theoretical framework. Understanding complexity as process also helps one align complexity science with holistically oriented predecessors such as General System Theory, while allowing for the reductionist perspective of complexity. These results, however, have the further implication that it may be futile to search for general laws of complexity, or to hope that investigations of complex objects in one domain may throw light on complexity in unrelated domains.     

Negotiating history: Contingency,canonicity, and case studies

Objections to the use of historical case studies for philosophical ends fall into two categories. Methodological objections claim that historical accounts and their uses by philosophers are subject to various biases. We argue that these challenges are not special; they also apply to other epistemic practices. Metaphysical objections, on the other hand, claim that historical case studies are intrinsically unsuited to serve as evidence for philosophical claims, even when carefully constructed and used, and so constitute a distinct class of challenge. We show that attention to what makes for a canonical case can address these problems. A case study is canonical with respect to a particular philosophical aim when the features relevant to that aim provide a reasonably complete causal account of the results of the historical process under investigation. We show how to establish canonicity by evaluating relevant contingencies using two prominent examples from the history of science: Eddington’s confirmation of Einstein’s theory of general relativity using his data from the 1919 eclipse and Watson and Crick’s determination of the structure of DNA.     

The Russell Archives: Some new light on Russell's logicism

This paper describes the materials in the Russell Archives relevant to Russell's work on logic and the foundations of mathematics, and suggests the kinds of information that may and may not be drawn about the historical development of his ideas. By way of illustration, a couple of episodes are described. The first (arts. 2–4) concerns a logical system closely related to his theory of denoting, which preceeds the system used in Principia mathematics, while the second (arts. 5) describes a delay in publishing the second volume of that work due to the discovery by Whitehead of a conceptual error.     

No understanding without explanation

Scientific understanding, this paper argues, can be analyzed entirely in terms of a mental act of “grasping” and a notion of explanation. To understand why a phenomenon occurs is to grasp a correct explanation of the phenomenon. To understand a scientific theory is to be able to construct, or at least to grasp, a range of potential explanations in which that theory accounts for other phenomena. There is no route to scientific understanding, then, that does not go by way of scientific explanation.