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)

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)

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)

This historiographical study aims at introducing the category of “situated mathematics” to the case of Ancient Egypt. However, unlike Situated Learning Theory, which is based on ethnographic relativity, in this paper, the goal is to analyze a mathematical craft knowledge based on concrete particulars and case studies, which is ubiquitous in all human activity, and which even covers, as a specific case, the Hellenistic style, where theoretical constructs do not stand apart from practice, but instead remain grounded in it.The historiographic (...) interpretation that we will give of situated mathematics is inscribed in a characterization of mathematical styles that focuses on the role of mathematical practice. This categorization describes three types of mathematization, where, on the one hand, type I represents the classical and dominant Hellenocentric approach, which seeks to generate a body of principles that could then be applied in other fields. On the other hand, types II and III represent two kinds of situated mathematics, a parametrized and a concrete one. Type II proceeds in the opposite direction from Type I describing an application of a previously obtained theory. That is, given a practice in any domain, it seeks to build a mathematical systematization a posteriori to explain said practice. Finally, type III starts from a concrete practice and develops another similar practice that explains analogically the relationship.Based on the typology adopted, we seek to describe a case study within ancient Egyptian mathematics, which reveals how it is possible to subsume it in the two types of situated mathematization II and III. The foregoing will allow to bridge the gap between theory and practice. (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)

The development of rigorous foundations of differential calculus in the course of the nineteenth century led to concerns among physicists about its applicability in physics. Through this development, differential calculus was made independent of empirical and intuitive notions of continuity, and based instead on strictly mathematical conditions of continuity. However, for Boltzmann and Poincaré, the applicability of mathematics in physics depended on whether there is a basis in physics, intuition or experience for the fundamental axioms of mathematics—and this meant that (...) to determine the status of differential equations in physics, they had to consider whether there was a justification for these mathematical continuity conditions in physics. For this reason, their ideas about continuity and discreteness in nature were entangled with epistemology and philosophy of mathematics. They reached opposite conclusions: Poincaré argued that physicists must work with a continuous representation of nature, and thus with differential equations, while Boltzmann argued that physicists must ultimately take nature to be discrete. (shrink)

Frege's docent's dissertation Rechnungsmethoden, die sich auf eine Erweiterung des Grössenbegriffes gründen(1874) contains indications of a bold attempt to extend arithmetic. According to it, arithmetic means the science of magnitude, and magnitude must be understood structurally without intuitive support. The main thing is insight into the formal structure of the operation of ?addition?. It turns out that a general ?magnitude domain? coincides with a (commutative) group. This is an interesting connection with simultaneous developments in abstract algebra. As his main application, (...) Frege studies iterations of functions. He does not yet pose the question of existence proofs. Measurement of magnitudes is also connected to numbers, but the discussion is here ambiguous in a way which calls for the systematic account of numbers in Grundgesetze. (shrink)

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice (...) mathematics teachers. (shrink)

When it comes to the relation of modern mathematics and philosophy, most people tend to think of the three major schools of thought—i.e. logicism, formalism, and intuitionism—that emerged as profound researches on the foundations and nature of mathematics in the beginning of the 20th century and have shaped the dominant discourse of an autonomous discipline of analytic philosophy, generally known under the rubric of “philosophy of mathematics” since then. What has been completely disregarded by these philosophical attitudes, these foundational researches (...) which seek to provide pure mathematics with a philosophically plausible justification by founding it on firm logico-philosophical bases, is that the genuine self-foundation of pure mathematics had been done before, namely during the 19th century, when it was developing into an entirely new and independent discipline as a concomitant of the continuous dissociation of mathematics from the physical world. This self-foundation of the 19th-century pure mathematics, however, was more akin to the German-idealist interpretations of Kant’s transcendental philosophy, than the post-factum, retrospective 20th-century researches on the foundations of mathematics. This article aims to demonstrate this neglected historical fact via delving into the philosophical inclinations of the three major founders of the 19th-century pure mathematics, Riemann, Dedekind and Cantor. Consequently, pure mathematics, with respect to its idealist origins, proves to be a formalization and idealization of certain activities specific to a self-conscious transcendental subjectivity. (shrink)

Diophantos' solutions to the problems of Arithmetica have been the object of extensive reading and interpretation in modern times, especially from the point of view of identifying ``hidden steps'' or ``general methods''. In this paper, after examining the relevance of various interpretations given for the famous problem II 8 in the context of modern algebra or geometry, we focus on a close reading of the ancient text of some problems of Arithmetica in order to investigate Diophantos' solving practices. This inquiry (...) reveals certain pointers, which enable us to create a framework for defining the generality of Diophantos' methods. (shrink)

This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable.

This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and geometry and (...) the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer. (shrink)

Standard histories of mathematics and of analytic philosophy contend that work on the foundations of mathematics was motivated by a crisis such as the discovery of paradoxes in set theory or the discovery of non-Euclidean geometries. Recent scholarship, however, casts doubt on the standard histories, opening the way for consideration of an alternative motive for the study of the foundations of mathematics—unification. Work on foundations has shown that diverse mathematical practices could be integrated into a single framework of axiomatic systems (...) and that much of mathematics could be expressed in a single language. The new framework was the product of an interdisciplinary coalition whose ideas resemble those later adopted by the Vienna Circle and logical empiricists. (shrink)

The recent discovery of an indeterministic system in classical mechanics, the Norton dome, has shown that answering the question whether classical mechanics is deterministic can be a complicated matter. In this paper I show that indeterministic systems similar to the Norton dome were already known in the nineteenth century: I discuss four nineteenth century authors who wrote about such systems, namely Poisson, Duhamel, Boussinesq and Bertrand. However, I argue that their discussion of such systems was very different from the contemporary (...) discussion about the Norton dome, because physicists in the nineteenth century conceived of determinism in essentially different ways: whereas in the contemporary literature on determinism in classical physics, determinism is usually taken to be a property of the equations of physics, in the nineteenth century determinism was primarily taken to be a presupposition of theories in physics, and as such it was not necessarily affected by the possible existence of systems such as the Norton dome. (shrink)

We consider the informal concept of "computability" or "effective calculability" and two of the formalisms commonly used to define it, "(Turing) computability" and "(general) recursiveness". We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful (...) historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of "computability" and "recursion." Specifically we recommend that: the term "recursive" should no longer carry the additional meaning of "computable" or "decidable;" functions defined using Turing machines, register machines, or their variants should be called "computable" rather than "recursive;" we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be "Computability Theory" or simply Computability rather than "Recursive Function Theory.". (shrink)

Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defects concerns the traditional treatment of time derivatives in Eulerian hydrodynamic description. The other one resides in the conventional demonstration of the so-called Convection Theorem. Both approaches are thought to be necessary for cross-verification of the standard (...) differential form of continuity equation. Any revision of these fundamental results might have important implications for all classical field theories. Rigorous reconsideration of time derivatives in Eulerian description shows that it evokes Minkowski metric for any flow field domain without any previous postulation. Mathematical approach is developed within the framework of congruences for general four-dimensional differentiable manifold and the final result is formulated in form of a theorem. A modified version of the Convection Theorem provides a necessary cross-verification for a reconsidered differential form of continuity equation. Although the approach is developed for one-component (scalar) flow field, it can be easily generalized to any tensor field. Some possible implications for classical electrodynamics are also explored. (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)