In this paper, I present the case of the discovery of complex numbers by Girolamo Cardano. Cardano acquires the concepts of (specific) complex numbers, complex addition, and complex multiplication. His understanding of these concepts is incomplete. I show that his acquisition of these concepts cannot be explained on the basis of Christopher Peacocke’s Conceptual Role Theory of concept possession. I argue that Strong Conceptual Role Theories that are committed to specifying a set of transitions that is both necessary and sufficient (...) for possession of mathematical concepts will always face counterexamples of the kind illustrated by Cardano. I close by suggesting that we should rely more heavily on resources of Anti-Individualism as a framework for understanding the acquisition and possession of concepts of abstract subject matters. (shrink)

Recently Carrie S. Jenkins formulated an epistemology of mathematics, or rather arithmetic, respecting apriorism, empiricism, and realism. Central is an idea of concept grounding. The adequacy of this idea has been questioned e.g. concerning the grounding of the mathematically central concept of set (or class), and of composite concepts. In this paper we present a view of concept formation in mathematics, based on ideas from Carnap, leading to modifications of Jenkins’s epistemology that may solve some problematic issues with her ideas. (...) But we also present some further problems with her view, concerning the role of proof for mathematical knowledge. (shrink)

The view that contradictions cannot be true has been part of accepted philosophical theory since at least the time of Aristotle. In this regard, it is almost unique in the history of philosophy. Only in the last forty years has the view been systematically challenged with the advent of dialetheism. Since Graham Priest introduced dialetheism as a solution to certain self-referential paradoxes, the possibility of true contradictions has been a live issue in the philosophy of logic. Yet, despite the arguments (...) advanced by dialetheists, many logicians and philosophers still hold the opinion that contradictions cannot be true. -/- Rather than advocating the truth of certain contradictions, this thesis offers a different challenge to the classical logician. By showing that it can be philosophically coherent to propose that true contradictions are metaphysically possible, the thesis suggests that the classical logician must do more than she currently has to justify her confidence in the impossibility of true contradictions. Simply fighting off the dialetheist’s putative examples of true contradictions at the actual world isn’t enough to justify the classical logician’s conclusion that true contradictions are impossible. -/- To aid the thesis dialectically, we introduce a new position, absolutism, which hypothesises that it’s metaphysically possible for at least one contradiction to be true, contrasting with the dialetheic hypothesis that some contradictions are true in the actual world. We demonstrate that absolutism can be given a philosophically coherent interpretation, an appropriate logic, and that certain criticisms are completely toothless against absolutism. The challenge put to the classical logician is then: On what logical or philosophical grounds can we rule out the metaphysical possibility of true contradictions? (shrink)

Semantic analysis in early analytic philosophy belongs to a long tradition of adopting geometrical methodologies to the solution of philosophical problems. In particular, it adapts Descartes’ development of formalization as a mechanism of analytic representation, for its application in natural language semantics. This article aims to trace the mathematical roots of Frege, Russel and Carnap’s analytic method. Special attention is paid to the formal character of modern analysis introduced by Descartes. The goal is to identify the particular conception of “form” (...) developed by the analytic tradition, from Descartes to early analytic philosophy, and to determine its relation to similar notions, like ‘function’ and ‘syntax’. Finally, I focus on how Frege, Russell and Carnap’s methods of semantic analysis fit the general characterization of formal analysis previously developed. (shrink)

The role of conventions in the formulation of Thomas Reid’s theory of the geometry of vision, which he calls the “geometry of visibles”, is the subject of this investigation. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reid’s “geometry of visibles” and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject to a choice of conventions regarding the (...) construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reid’s direct realist theory of perception with his geometry of visibles. (shrink)

In 1813, J.-V. Poncelet discovered that if there exists a polygon of n-sides, which is inscribed in a given conic and circumscribed about another conic, then infinitely many such polygons exist. This theorem became known as Poncelet’s porism, and the related polygons were called Poncelet’s polygons. In this article, we trace the history of the research about the existence of such polygons, from the “prehistorical” work of W. Chapple, of the middle of the eighteenth century, to the modern approach of (...) P. Griffiths in the late 1970s, and beyond. For reasons of space, the article has been divided into two parts, the second of which will appear in the next issue of this journal. (shrink)

This is an attempt to explain Kepler’s invention of the first “non-cone-based” system of conics, and to put it into a historical perspective.

RésuméCet article est consacré à l’histoire de la théorie locale des courbes “à double courbure”. Initiée par Clairaut en 1731, cette théorie se développe en parallèle à la théorie des surfaces et trouve son achèvement avec les formules de Serret et Frenet et leur interprétation par Darboux, en 1887. Au delà de l’analyse des contributions de nombreux mathématiciens, parmi lesquels Monge bien sûr mais aussi Fourier, Lagrange et Cauchy, notre étude donne un regard particulier sur l’évolution conjointe de l’Analyse et (...) de la Géométrie, dans une longue période riche de nombreuses remises en cause théoriques. (shrink)

In this paper I assess the obstacles to a transfer of Lakatos's methodology of scientific research programmes to mathematics. I argue that, if we are to use something akin to this methodology to discuss modern mathematics with its interweaving theoretical development, we shall require a more intricate construction and we shall have to move still further away from seeing mathematical knowledge as a collection of statements. I also examine the notion of rivalry within mathematics and claim that this appears to (...) be significant only at a high level. In addition, ideas of ‘progress’ in mathematics are outlined. (shrink)

Originally published in 1904, Whittaker’s A Treatise on the Analytical Dynamics of Particles and Rigid Bodies soon became a classic of the subject and has remained in print for most of these 108 years. In this paper, we follow the book as it develops from a report that Whittaker wrote for the British Society for the Advancement of Science to its influence on Dirac’s version of quantum mechanics in the 1920s and beyond.

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...) involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

In this paper I present an argument for belief in inconsistent objects. The argument relies on a particular, plausible version of scientific realism, and the fact that often our best scientific theories are inconsistent. It is not clear what to make of this argument. Is it a reductio of the version of scientific realism under consideration? If it is, what are the alternatives? Should we just accept the conclusion? I will argue (rather tentatively and suitably qualified) for a positive answer (...) to the last question: there are times when it is legitimate to believe in inconsistent objects. (shrink)

Indispensability arguments for realism about mathematical entities have come under serious attack in recent years. To my mind the most profound attack has come from Penelope Maddy, who argues that scientific/mathematical practice doesn't support the key premise of the indispensability argument, that is, that we ought to have ontological commitment to those entities that are indispensable to our best scientific theories. In this paper I defend the Quine/Putnam indispensability argument against Maddy's objections.

We characterize abstraction in computer science by first comparing the fundamental nature of computer science with that of its cousin mathematics. We consider their primary products, use of formalism, and abstraction objectives, and find that the two disciplines are sharply distinguished. Mathematics, being primarily concerned with developing inference structures, has information neglect as its abstraction objective. Computer science, being primarily concerned with developing interaction patterns, has information hiding as its abstraction objective. We show that abstraction through information hiding is a (...) primary factor in computer science progress and success through an examination of the ubiquitous role of information hiding in programming languages, operating systems, network architecture, and design patterns. (shrink)

What are intuitions? Stereotypical examples may suggest that they are the results of common intellectual reflexes. But some intuitions defy the stereotype: there are hard-won intuitions that take d...

In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993 ) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One example is (...) how Björling interprets Cauchy’s definition of the logarithm function with respect to complex variables, which is investigated in the paper. Furthermore, in view of an article written by Björling (Kongl Vetens Akad Förh Stockholm 166–228, 1852 ) we consider Cauchy’s theorem on power series expansions of complex valued functions. We investigate Björling’s, Cauchy’s and the Belgian mathematician Lamarle’s different conditions for expanding a complex function of a complex variable in a power series. We argue that one reason why Cauchy’s theorem was controversial could be the ambiguities of fundamental concepts in analysis that existed during the mid-nineteenth century. This problem is demonstrated with examples from Björling, Cauchy and Lamarle. (shrink)

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such (...) as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presentation inviting the contextualization of pieces of mathematical reasoning within encompassing bodies of (...) explicit and implicit, formal and informal background knowledge. (shrink)

I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner's old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.

The paper discusses to what extent the conceptual issues involved in solving the simple harmonic oscillator model fit Wigner’s famous point that the applicability of mathematics borders on the miraculous. We argue that although there is ultimately nothing mysterious here, as is to be expected, a careful demonstration that this is so involves unexpected difficulties. Consequently, through the lens of this simple case we derive some insight into what is responsible for the appearance of mystery in more sophisticated examples of (...) the Wigner problem. (shrink)

Leibniz never tired of stressing the fundamental importance of the concept of continuity for philosophy, nor was he shy of attributing major importance to his own struggle through “the labyrinth of the continuum” for the subsequent development of his whole system of thought. Unfortunately, however, his own thought on the subject is something of a labyrinth itself, and from a modern point of view many of his pronouncements are apt to seem blatantly contradictory.Certain quotations seem to commit him unambiguously to (...) atomism. Thus to de Voider he writes: “Matter is not continuous, but discrete…. The same holds for changes, which are not truly continuous.” (To de Voider, 11th October 1705: G.II.279).2. (shrink)

This paper investigates the notion of semiotic scaffolding in relation to mathematics by considering its influence on mathematical activities, and on the evolution of mathematics as a research field. We will do this by analyzing the role different representational forms play in mathematical cognition, and more broadly on mathematical activities. In the main part of the paper, we will present and analyze three different cases. For the first case, we investigate the semiotic scaffolding involved in pencil and paper multiplication. For (...) the second case, we investigate how the development of new representational forms influenced the development of the theory of exponentiation. For the third case, we analyze the connection between the development of commutative diagrams and the development of both algebraic topology and category theory. Our main conclusions are that semiotic scaffolding indeed plays a role in both mathematical cognition and in the development of mathematics itself, but mathematical cognition cannot itself be reduced to the use of semiotic scaffolding. (shrink)

This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...

A new hypothesis on the basic features characterising the Foundations of Mathematics is suggested. By means of them the entire historical development of Mathematics before the 20th Century is summarised through a table. Also the several programs, launched around the year 1900, on the Foundations of Mathematics are characterised by a corresponding table. The major difficulty that these programs met was to recognize an alternative to the basic feature of the deductive organization of a theory - more precisely, to Hilbert’s (...) main tenet. Ironically, already half a century before the births of these programs the alternative organization has been substantially represented by Lobachevsky's theory on parallel lines. Moreover, although each program’s founder recognised the basic features in a partial way only, all together these programs represented just the four possible foundational approaches. (shrink)

I argue that we find the articulation of a problem concerning bodily agency in the early works of the Merleau-Ponty which he explicates as analogous to what he explicitly calls the problem of perception. The problem of perception is the problem of seeing how we can have the object given in person through it perspectival appearances. The problem concerning bodily agency is the problem of seeing how our bodily movements can be the direct manifestation of a person’s intentions in the (...) world. In both cases what, according to Merleau-Ponty, obscures a recognition of the phenomenon in question is a conception of our bodily capacities, i.e. our sensibility and our motility, which reduces these to the workings of mechanisms that are blind to meaning. I argue that both the problem concerning perception and the problem concerning bodily agency can be properly called transcendental. The problem of perception is transcendental because it concerns the very intelligibility of appearances and judgements with empirical content. The problem of bodily agency is transcendental in the sense that it concerns the very intelligibility of our bodily capacity to carry out intentions and by implication the intelligibility of our intentions as such. (shrink)

Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician (...) Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed. (shrink)

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by (...) exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge. (shrink)

This book, presented in two parts, offers a slow introduction to mathematical logic, and several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions. Its first part, Logic Sets, and Numbers, shows how mathematical logic is used to develop the number structures of classical mathematics. The exposition does not assume any prerequisites; it is rigorous, but as informal as possible. All necessary concepts are introduced exactly as they would be in a course in mathematical logic; (...) but are accompanied by more extensive introductory remarks and examples to motivate formal developments. The second part, Relations, Structures, Geometry, introduces several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions, and shows how they are used to study and classify mathematical structures. Although more advanced, this second part is accessible to the reader who is either already familiar with basic mathematical logic, or has carefully read the first part of the book. Classical developments in model theory, including the Compactness Theorem and its uses, are discussed. Other topics include tameness, minimality, and order minimality of structures. The book can be used as an introduction to model theory, but unlike standard texts, it does not require familiarity with abstract algebra. This book will also be of interest to mathematicians who know the technical aspects of the subject, but are not familiar with its history and philosophical background. (shrink)

The meaning of the wave function has been a hot topic of debate since the early days of quantum mechanics. Recent years have witnessed a growing interest in this long-standing question. Is the wave function ontic, directly representing a state of reality, or epistemic, merely representing a state of knowledge, or something else? If the wave function is not ontic, then what, if any, is the underlying state of reality? If the wave function is indeed ontic, then exactly what physical (...) state does it represent? In this book, I aim to make sense of the wave function in quantum mechanics and find the ontological content of the theory. The book can be divided into three parts. The first part addresses the question of the nature of the wave function. After giving a comprehensive and critical review of the competing views of the wave function, I present a new argument for the ontic view in terms of protective measurements. In addition, I also analyze the origin of the wave function by deriving the free Schroedinger equation. The second part analyzes the ontological meaning of the wave function. I propose a new ontological interpretation of the wave function in terms of random discontinuous motion of particles, and give two main arguments supporting this interpretation. The third part investigates whether the suggested quantum ontology is complete in accounting for our definite experience and whether it needs to be revised in the relativistic domain. (shrink)

Kirchhoff’s 1882 theory of optical diffraction forms the centerpiece in the long-term development of wave optics, one that commenced in the 1820s when Fresnel produced an empirically successful theory based on a reinterpretation of Huygens’ principle, but without working from a wave equation. Then, in 1856, Stokes demonstrated that the principle was derivable from such an equation albeit without consideration of boundary conditions. Kirchhoff’s work a quarter century later marked a crucial, and widely influential, point for he produced Fresnel’s results (...) by means of Green’s theorem and function under specific boundary conditions. In the late 1880s, Poincaré uncovered an inconsistency between Kirchhoff’s conditions and his solution, one that seemed to imply that waves should not exist at all. Researchers nevertheless continued to use Kirchhoff’s theory—even though Rayleigh, and much later Sommerfeld, developed a different and mathematically consistent formulation that, however, did not match experimental data better than Kirchhoff’s theory. After all, Kirchhoff’s formula worked quite well in a specific approximation regime. Finally, in 1964, Marchand and Wolf employed the transformation of Kirchhoff’s surface integral that had been developed by Maggi and Rubinowicz for other purposes. The result yielded a consistent boundary condition that, while introducing a species of discontinuity, nevertheless rescued the essential structure of Kirchhoff’s original formulation from Poincaré’s paradox. (shrink)

This paper explores some of the constructive dimensions and specifics of human theoretic cognition, combining perspectives from (Husserlian) genetic phenomenology and distributed cognition approaches. I further consult recent psychological research concerning spatial and numerical cognition. The focus is on the nexus between the theoretic development of abstract, idealized geometrical and mathematical notions of space and the development and effective use of environmental cognitive support systems. In my discussion, I show that the evolution of the theoretic cognition of space apparently follows (...) two opposing, but in truth, intrinsically aligned trajectories. On the epistemic plane, which is the main focus of Husserl’s genetic phenomenological investigations, theoretic conceptions of space are progressively constituted by way of an idealizing emancipation of spatial cognition from the concrete, embodied intentionality underlying the human organism’s perception of space. As a result of this emancipation, it ultimately becomes possible for the human mind to theoretically conceive of and posit space as an ideal entity that is universally geometrical and mathematical. At the same time, by synthesizing a range of literature on spatial and mathematical cognition, I illustrate that for the theoretic mind to undertake precisely this emancipating process successfully, and further, for an ideal and objective notion of geometrical and mathematical space to first of all become fully scientifically operative, the cognitive support provided by a range of specific symbolic technologies is central. These include lettered diagrams, notation systems, and more generally, the technique of formalization and require for their functioning various cognitively efficacious types of embodiment. Ultimately, this paper endeavors to understand the specific symbolic-technological dimensions that have been instrumental to major shifts in the development of idealized, scientific conceptions of space. The epistemic characteristics of these shifts have been previously discussed in genetic phenomenology, but without devoting sufficient attention to the constructive role of symbolic technologies. At the same time, this paper identifies some of the irreducible phenomenological and epistemic dimensions that characterize the functioning of the historically situated, embodied and distributed theoretic mind. (shrink)

The ArgumentIn the minds of many, the Bourbakist trend in mathematics was characterized by pursuit of rigor to the detriment of concern for applications or didactic concessions to the nonmathematician, which would seem to render the concept of a Bourbakist incursion into a field of applied mathematices an oxymoron. We argue that such a conjuncture did in fact happen in postwar mathematical economics, and describe the career of Gérard Debreu to illustrate how it happened. Using the work of Leo Corry (...) on the fate of the Bourbakist program in mathematics, we demonstrate that many of the same problems of the search for a formalstructurewith which to ground mathematical practice also happened in the case of Debreu. We view this case study as an alternative exemplar to conventional discussions concerning the “unreasonable effectiveness” of mathematics in science. (shrink)

Classic of Changes is a Chinese cultural classic born more than 3000 years ago. Its profound philosophical thoughts and the use of divination have brought Classic of Changes to a strong oriental mysticism. The view of the heaven and man of yin and yang and the five elements states of Classic of Changes are completely different from the Western elemental theory of ancient Greece. The latter gave birth to classical and modern scientific theories, and the yin and yang and the (...) eight trigrams symbol has become synonymous with oriental mysticism. In fact, the cosmology of the Holism of Classic of Changes is a precious scientific heritage of mankind. The axiomatization of the symbolic formal system of Classic of Changes aims to unveil the veil of oriental mysticism and provide oriental wisdom for the development of modern science. Transforming the symbolic system of Classic of Changes into a formal axiom system is the crystallization of the fusion of wisdom between the East and the West. The axiomatization of the symbolic formal system of Classic of Changes shows that the oriental and the western scientific tradition harmony but not sameness and there is no conflict. Classic of Changes can also be interpreted by the axiomatic system like Euclid’s Elements. The main contribution of this paper is that the author skillfully uses mathematical language to formulate the system of Classic of Changes, reconstructs the ideological system of Classic of Changes with the axiomatic method and realizes the scientificalization of Chinese classical natural philosophy. The formal axiom system of Classic of Changes may give us such a revelation: the gap between the oriental and western scientific traditions is mainly in the axiomatization, but the lack of oriental scientific tradition may be the bottleneck of the development of modern science. (shrink)

Deleuze's theory of time set out in Difference and Repetition is a complex structure of three different syntheses of time – the passive synthesis of the living present, the passive synthesis of the pure past and the static synthesis of the future. This article focuses on Deleuze's third synthesis of time, which seems to be the most obscure part of his tripartite theory, as Deleuze mixes different theoretical concepts drawn from philosophy, Greek drama theory and mathematics. Of central importance is (...) the notion of the cut, which is constitutive of the third synthesis of time defined as an a priori ordered temporal series separated unequally into a before and an after. This article argues that Deleuze develops his ordinal definition of time with recourse to Kant's definition of time as pure and empty form, Hölderlin's notion of ‘caesura’ drawn from his ‘Remarks on Oedipus’ (1803) and Dedekind's method of cuts as developed in his pioneering essay ‘Continuity and Irrational Numbers’ (1872). Deleuze then ties together the conceptions of the Kantian empty form of time and the Nietzschean eternal return, both of which are essentially related to a fractured I or dissolved self. This article aims to assemble the different heterogeneous elements that Deleuze picks up on and to show how the third synthesis of time emerges from this differential multiplicity. (shrink)