In an extension of speculations that physical space–time is a fractal which might itself be embedded in a high-dimensional continuum, it is hypothesized to “compensate” for local variations of the fractal dimension by instead varying the metric in such as way that the intrinsic dimensionality remains an integer. Thereby, an extrinsic fractal continuum is intrinsically perceived as a classical continuum. Conversely, it is suggested that any variation of the metric from its Euclidean form can be “shifted” to nontrivial fractal topology. (...) Thereby “holes” or “gaps” in spacetime could give rise to curvature. (shrink)

I trace the development of arguments for the consistency of non-Euclidean geometries and for the independence of the parallel postulate, showing how the arguments become more rigorous as a formal conception of geometry is introduced. I analyze the kinds of arguments offered by Jules Hoüel in 1860-1870 for the unprovability of the parallel postulate and for the existence of non-Euclidean geometries, especially his reaction to the publication of Beltrami’s seminal papers, showing that Beltrami was much more concerned with the existence (...) of non-Euclidean objects than he was with the formal consistency of non-Euclidean geometries. The final step towards rigorous consistency proofs is taken in the 1880s by Henri Poincaré. It is the formal conception of geometry, stripping the geometric primitive terms of their usual meanings, that allows the introduction of a modern fully rigorous consistency proof. (shrink)

The purpose of this article is to integrate two outstanding problems within the philosophy of science. The first concerns what role aesthetics plays in scientific thinking. The second is the problem of how logically testable ideas are generated (the so-called "psychology of research" versus "logic of (dis)proof" problem). I argue that aesthetic sensibility is the basis for what scientists often call intuition, and that intuition in turn embodies (in a literal physiological sense) ways of thinking that have their own meta-logic. (...) Thus, aesthetics is a form of cognition. Scientists think not in equations or words or other logical abstractions, but emotionally and sensually, using visual and aural images, kinesthetic and other proprioceptive feelings, sensations, patterns, and analogies. These aesthetic forms of thinking have their own logics that I call "synosia", from the root words synaesthesia (a combining of senses) and gnosis, "to know". Synosia denotes understanding that integrates feeling that one knows with feeling what one knows. Eminent scientists universally describe an explicitly secondary process in which such personal knowledge must be "translated" into a formal language, such as words or equations, in order to be communicated to other people. Many of the unsolved problems that philosophers of science (as well as psychologists and artificial intelligence researchers) have had in making sense of scientific thinking have arisen from confusing the form and content of the final translations with the hidden means by which scientific insights are actually achieved. (shrink)

Professor Rahn takes the approach to the analysis of Western art music developed recently by theorists such as Benjamin Boretz and extends it to address non-Western forms. In the process, he rejects recent ethnomusicological formulations based on mentalism, cultural determinism, and the psychology of perception as potentially fruitful bases for analysing music in general. Instead he stresses the desirability of formulating a theory to deal with all music, rather than merely Western forms, and emphasizes the need to evaluate an analysis (...) and compare it with other interpretations, and demonstrates how this may be done. The theoretical concepts which form the basis of Rahn's approach are discussed and applied: first to individual pieces of non-Western music which have enjoyed a fairly high profile in ethnomusicological literature, and second to repertoires or groups of pieces. The author also discusses the fields of anthropology and psychology, showing how his approach serves as a starting point for studies of perception and the concepts, norms, and values found in specific music cultures. In conclusion, he lists what he considers to be musical universals and takes up the more controversial issues implicit in his discussion. (shrink)

often insisted existence in mathematics means logical consistency, and formal logic is the sole guarantor of rigor. The paper joins this to his view of intuition and his own mathematics. It looks at predicativity and the infinite, Poincaré's early endorsement of the axiom of choice, and Cantor's set theory versus Zermelo's axioms. Poincaré discussed constructivism sympathetically only once, a few months before his death, and conspicuously avoided committing himself. We end with Poincaré on Couturat, Russell, and Hilbert.

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, (...) Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity. (shrink)

A modern assessment of the classical Boltzmann-Schuetz argument for large-scale entropy fluctuations as the origin of our observable cosmological domain is given.The emphasis is put on the central implication of this picture which flatly contradicts the weak anthropic principle as an epistemological statement about the universe. Therefore, to associate this picture with the anthropic principle as it is usually done is unwarranted. In particular, Feynman's criticism of theanthropic principle based on the entropy-fluctuation picture is a product of this semantic confusion.

Dance anthropologists and ethnomusicologists are trained to treat the labels ‘dance’ and ‘music’ with caution, because the terms carry preconceptions that may mask significant aspects of the structured movement/sound systems they study. Yet many talk about ‘the body’ – the medium through which these systems come into being – as something given and ‘true’, without investigating its emic conceptualizations or looking into the implications these may have in terms of how music and dance are experienced. The article investigates the dancing (...) bodies of western ballet dancers and of Tiwi dancers of northern Australia. The latter have an extensive vocabulary for body parts, but no word for ‘body’ as a bounded entity, comparable to the ‘body’ conceptualized by ballet dancers. The article argues that for the Tiwi landscape and human bodies are unavoidably tied together and that the dancers incarnate both land and cosmology. While specific parts have a concrete practical existence, the whole body expands into the landscape so that the landscape becomes flesh. Through the aesthetic transformation of dancing, dancers and land become one, and life is sustained and the land revitalized. (shrink)

(2000). CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM: THE PHILOSOPHY OF THE EXACT SCIENCES IN THE BACKGROUND TO EARLY LOGICAL EMPIRICISM. British Journal for the History of Philosophy: Vol. 8, No. 1, pp. 71-106.

This piece is a commentary on a precis of Maggie Boden's book "The creative mind" published in Behavioral and Brain Sciences.

In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians who interacted with Luitzen Egbertus Jan Brouwer in order to compare their conceptions of intuition. We will see how to the same word “intuition” very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty of considering concepts (...) that habitually occur in our thinking separately. Furthermore, we will discover that these different meanings had a philosophical, very relevant counterside: they passed from a racial characterization of mathematics to a pluralistic view of it. (shrink)

Discussions of the metaphysical status of spacetime assume that a spacetime theory offers a causal explanation of phenomena of relative motion, and that the fundamental philosophical question is whether the inference to that explanation is warranted. I argue that those assumptions are mistaken, because they ignore the essential character of spacetime theory as a kind of physical geometry. As such, a spacetime theory does notcausally explain phenomena of motion, but uses them to construct physicaldefinitions of basic geometrical structures by coordinating (...) them with dynamical laws. I suggest that this view of spacetime theories leads to a clearer view of the philosophical foundations of general relativity and its place in the historical evolution of spacetime theory. I also argue that this view provides a much clearer and more defensible account of what is entailed by realism concerning spacetime. (shrink)

Conventionalism in sport philosophy has been rejected as unable to provide a theory of normativity and as collapsing in ethical relativism, but this criticism is rather imprecise about its target, which invites doubt about the legitimacy of the concept of conventionalism described by its critics. Instead, a more charitable and legitimate account of conventionalism is proposed, one that draws inspiration from conventionalism in axiomatic geometry and is able to avoid the counterarguments directed against conventionalism. This new model allows for a (...) number of non-conventional elements of sport, namely the definition of sport and certain central moral norms, while at the same time arguing that normativity in sport is not exhausted by them, which leaves athletic communities with authority over a broad range of norms. (shrink)

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

This work is an analysis of F. P. Ramsey's philosophy of science. Twentieth-century philosophy of science was marked by attempts to consider the relation between scientific theories and our knowledge of the empirical world through considerations of abstract mathematical structure. Such considerations led Bertrand Russell to an account of the relation between our theoretical picture of the world and its real nature as a relation of structural similarity. Subsequently, Max Newman gave what has become a well-known logico-mathematical objection to this (...) account. William Demopoulos recently showed that Newman's problem applied not only to Russell's realist account, but also to a variety of otherwise disparate accounts of theoretical knowledge. The common element underlying these accounts is a conception of theories as abstract formal structures. Many such accounts have incorporated key elements of Ramsey's views, most notably the Ramsey-sentence. Moreover, Demopoulos has interpreted Ramsey's own view of theories as sharing the essential features of those abstract views, and therefore their common problem. My analysis aims to show that this abstract conception of theories does not adequately characterize Ramsey's view. Namely, his account of theories was not an attempt to do the epistemology of science in the fashion of Russell or Eddington, or of subsequent structuralist views that have adopted the Ramsey-sentence. I show this by a broader exposition of Ramsey's work on the nature of theories, comparing his seminal paper with his many other remarks on the nature and purpose of theories. I begin by discussing the historical context of Newman's objection, and a generalization of it that shows its broad applicability to abstract characterizations of theoretical knowledge. I then reconstruct Ramsey's view of theories, to show how far it extends beyond the Ramsey-sentence picture. Finally, I discuss the relevance of this view to contemporary debates concerning realism and instrumentalism. I characterize Ramsey's view as focused not on grounding our theoretical knowledge in abstract structure, but instead on demystifying the role of theoretical language and concepts in a theory's application to the world. (shrink)

The main purpose of this dissertation is to examine critically and discuss the role of imagination in science and religion, with particular emphasis on its possible epistemic, creative, and meaning-making functions. In order to answer my research questions, I apply theories and concepts from contemporary philosophy of mind on scientific and religious practices. This framework allows me to explore the mental state of imagination, not as an isolated phenomenon but, rather, as one of many mental states that co-exist and interplay (...) in our cogntive architecture. Based on the philosophical discourse of philosophy of mind, four types of imagination are indentified and conceptualized: sensory, propositional, experiential, and creative imagination. These categories are then employed on five phenomena that can be found in scientific and religious environments: metaphors, models, thought experiments, aspect perception, and - in the religious case - rituals. In relation to the concept of religious "seeing" I consider how imaginings may influence visionary experiences and visualization, and compare these phenomena with cases of scientific visualization and eureka experiences. In regard to scientific and religious models, a distinction is made between, on the one hand, two notions of truth and, on the other hand, truth-independent meaning-making. In light of these categories, I differentiate between, and critically discuss, the use of imagination in doxastic, non-doxastic and fictionalist accounts. In light of this investigation, I formulate and defend the position of interactivism, which acknowledges a constant interplay between different attitudes and mental states. In my examination of rituals and scientific and religious thought experiments, special attention is given to the mental capacity to recreate the experiences that are entailed in an imagined scenario. At the end of the investigation, I consider the possible impact that my study might have on how we view science and religion as well as the dialogue bewteen these two fields. (shrink)