Treating mathematical science as a distinct cultural entity subject to environmental factors which influence its evolution, the author examines the creation and development of its major concepts since early times.

Can concepts be acquired by testing hypotheses about these concepts? Fodor famously argued that this is not possible. Testing the correct hypothesis would require already possessing the concept. I argue that this does not generally hold for mathematical concepts. I discuss specific, empirically motivated, hypotheses for number concepts that can be tested without needing to possess the relevant number concepts. I also argue that one can test hypotheses about the identity conditions of other (...) class='Hi'>mathematical concepts, and then fix the application conditions based on those hypotheses—under the assumption that the neo‐logicist view on abstraction principles is correct. (shrink)

When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: (...) poorly conceptualised intuitions are further conceptualised while poorly intuited concepts are further intuited. In this paper I study this latter process in mathematics during the twentieth century and, more specifically, show the role of set theory and category theory in this process. I use this material for defending the following claims: (1) mathematical intuitions are subject to historical development just like mathematical concepts; (2) mathematical intuitions continue to play their traditional role in today's mathematics and will plausibly do so in the foreseeable future. This second claim implies that the popular view, according to which modern mathematical concepts, unlike their more traditional predecessors, cannot be directly intuited, is not justified. (shrink)

ABSTRACTIn this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make (...) the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory. (shrink)

In this paper I investigate two notions of concepts that have played a dominant role in 20th century philosophy of mathematics. According to the first, concepts are definite and fixed; in contrast, according to the second notion concepts are open and subject to modifications. The motivations behind these two incompatible notions and how they can be used to account for conceptual change are presented and discussed. On the basis of historical developments in mathematics I argue that both (...) notions of concepts capture important aspects of mathematical and scientific reasoning, and, consequently, that a pluralistic approach that allows for representing both of these aspects is most useful for an adequate account of investigative practices. (shrink)

Dedekind's major work on the foundations of arithmetic employs several techniques that have left him open to charges of psychologism, and through this, to worries about the objectivity of the natural-number concept he defines. While I accept that Dedekind takes the foundation for arithmetic to lie in certain mental powers, I will also argue that, given an appropriate philosophical background, this need not make numbers into subjective mental objects. Even though Dedekind himself did not provide that background, one can nevertheless (...) be found in the work of the neo-Kantian Ernst Cassirer on mathematical concept formation. (shrink)

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and (...) neuropsychology support the nativist hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge? (shrink)

In this paper, I have attempted to make the role of mathematical thinking clear in Husserl’s theory of sciences. Husserl believed that phenomenology could afford to provide a safe foundation for individual sciences. Hence, the first task of the project was reorganizing the system of sciences and to show the possibility of apodictic knowledge regarding the world. Husserl was inspired by the progress of mathematics at that time because mathematics is the most logical discipline and deals with abstract objects. (...) It was the most suitable model for Husserl’s project. In fact, we can find structural similarities between his project and F. Klein’s Erlangen Program; further, the procedure of the essence intuition can be explained by a mathematical induction. Mathematics is certainly a new path for understanding Husserl’s phenomenology. In order to clarify the relation between Husserl’s theory of sciences and mathematics, this study focused on the problem of classification. Lastly, another implication of Husserl’s phenomenology as a theory of sciences is that his work is still meaningful for today’s dynamic reality of sciences. (shrink)

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and (...) neuropsychology support the nativist hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge? (shrink)

Treating mathematical science as a distinct cultural entity subject to environmental factors which influence its evolution, the author examines the creation and development of its major concepts since early times.

RésuméSelon Avicenne, certains objets des mathématiques existent et d'autres non. Chaque objet mathématique existant est un attribut connotationnel non sensible d'un objet physique et peut être perçu par la faculté d'estimation. Les objets mathématiques non existants peuvent être représentés et perçus par la faculté d'imagination en séparant et en combinant des parties d'images d'objets mathématiques existants qui sont précédemment perçues par estimation. Dans tous les cas, même les objets mathématiques non existants doivent être considérés comme des propriétés d'entités matérielles. Ils (...) ne peuvent jamais être saisis comme des entités totalement immatérielles. Avicenne pense que nous ne pouvons saisir aucun concept mathématique à moins d'avoir au préalable des expériences perceptives spécifiques. Ce n'est que par l'opération non éliminable et irremplaçable des facultés d'estimation et d'imagination sur certaines données sensibles que nous pouvons saisir les concepts mathématiques. Cela montre qu'Avicenne approuve une sorte d'empirisme conceptuel sur les mathématiques. (shrink)

Scientific and mathematical concepts are significantly different from everyday concepts and are notoriously difficult to learn. It is shown that particular instances of such concepts can be identified or generated by different possible modes of concept interpretation. Some of these modes use formally explicit knowledge and thought processes; others rely on less formal case‐based knowledge and more automatic recognition processes. The various modes differ in attainable precision, likely errors, and ease of use. A combination of such (...) modes can be used to formulate an “ideal” model for interpreting scientific concepts both reliably and efficiently. Comparisons are made with the actual concept interpretations of expert scientists and novice students. The discussion elucidates some cognitive and metacognitive reasons why the learning of scientific or mathematical concepts is difficult. It also suggests instructional guidelines for teaching such concepts more effectively. (shrink)

Mathematics often seems incomprehensible, a melee of strange symbols thrown down on a page. But while formulae, theorems, and proofs can involve highly complex concepts, the math becomes transparent when viewed as part of a bigger picture. What Is a Number? provides that picture. Robert Tubbs examines how mathematical concepts like number, geometric truth, infinity, and proof have been employed by artists, theologians, philosophers, writers, and cosmologists from ancient times to the modern era. Looking at a broad (...) range of topics -- from Pythagoras's exploration of the connection between harmonious sounds and mathematical ratios to the understanding of time in both Western and pre-Columbian thought -- Tubbs ties together seemingly disparate ideas to demonstrate the relationship between the sometimes elusive thought of artists and philosophers and the concrete logic of mathematicians. He complements his textual arguments with diagrams and illustrations. This historic and thematic study refutes the received wisdom that mathematical concepts are esoteric and divorced from other intellectual pursuits -- revealing them instead as dynamic and intrinsic to almost every human endeavor. (shrink)

Responding to widespread interest within cultural studies and social inquiry, this book addresses the question 'what is a mathematical concept?' using a variety of vanguard theories in the humanities and posthumanities. Tapping historical, philosophical, sociological and psychological perspectives, each chapter explores the question of how mathematics comes to matter. Of interest to scholars across the usual disciplinary divides, this book tracks mathematics as a cultural activity, drawing connections with empirical practice. Unlike other books in this area, it is highly (...) interdisciplinary, devoted to exploring the ontology of mathematics as it plays out in different contexts. This book will appeal to scholars who are interested in particular mathematical habits - creative diagramming, structural mappings, material agency, interdisciplinary coverings - that shed light on both mathematics and other disciplines. Chapters are also relevant to social sciences and humanities scholars, as each offers philosophical insight into mathematics and how we might live mathematically. (shrink)

This paper explores the nature of the language used when teaching mathematics to young children. It proposes that an important part of the teaching of a mathematical concept is the introduction of specific terminology. Children may need to be taught new meanings for already familiar words. The timing of these introductions to new words or meanings is critical to their understanding of the concepts being taught. It will be argued that there are two aspects of the children's learning (...) that need to be considered. First, their understanding of the concept being introduced, and secondly, their learning the appropriate word to describe that concept. By assessing children's understanding of new mathematical concepts through their own use of the terminology, the teacher can then negotiate new meanings with them through practical experiences, introducing new word meanings only when the concepts have been understood. (shrink)

Read in December 1905 at the Royal Society of London, Alfred North Whitehead’s memoir “On mathematical concepts of the material world” is not only, according to Whitehead’s own retrospective assessment, “the most original thing that he had done”; it also provides very interesting clues to understand Whitehead’s contemporary collaboration with Bertrand Russell, as well as his later philosophical – epistemological and metaphysical – work.

When science of the 20th century is spoken of in opposition to that of the 19th century, a particularly characteristic attribute is often cited: namely, that since the time of Galileo and Newton the task of science has been to explain everything mechanistically. By analogy the world was to be conceived as a great machine. But the theories of the 20th century, above all the relativity and quantum theories, caused a revolution in science. It is seen today that nature can (...) be described and understood not ‘mechanistically’ but only through abstract mathematical formulas. The world is no longer a machine but a mathematical formula. (shrink)

Dans cet article je propose d'exprimer et de défendre une conception des pratiques et du domaine de discours mathématiques qui soit sensible, d'une part, au pluralisme des relations entre pratiques inférentielles et intérêts, et d'autre part, à la structure objective et déterminante des concepts mathématiques. J'ébauche tout d'abord une caractérisation générale des pratiques, pour ensuite préciser certains phénomènes propres aux pratiques mathématiques. Suit un recensement des idées qui se dégagent des arguments pluralistes, et de celles qui sont à retenir. (...) Mais je défends par la suite la nécessité d'une forme de réalisme mathématique, qui toutefois ne peut être le réalisme d'objets que prônent les partisans de l'argument d'indispensabilité. Je défends plutôt un réalisme des concepts, soutenu par ce que je baptise l'« argument de la contrainte ». (shrink)

Dans cet article je propose d'exprimer et de défendre une conception des pratiques et du domaine de discours mathématiques qui soit sensible, d'une part, au pluralisme des relations entre pratiques inférentielles et intérêts, et d'autre part, à la structure objective et déterminante des concepts mathématiques. J'ébauche tout d'abord une caractérisation générale des pratiques, pour ensuite préciser certains phénomènes propres aux pratiques mathématiques. Suit un recensement des idées qui se dégagent des arguments pluralistes, et de celles qui sont à retenir. (...) Mais je défends par la suite la nécessité d'une forme de réalisme mathématique, qui toutefois ne peut être le réalisme d'objets que prônent les partisans de l'argument d'indispensabilité. Je défends plutôt un réalisme des concepts, soutenu par ce que je baptise l'« argument de la contrainte ». (shrink)

This book examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem leads to the (...) exclusion of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today. (shrink)

In the chapter of the Critique of Pure Reason entitled ‘The Discipline of Pure Reason in Dogmatic Use’, Kant contrasts mathematical and philosophical knowledge in order to show that pure reason does not (and, indeed, cannot) pursue philosophical truth according to the same method that it uses to pursue and attain the apodictically certain truths of mathematics. In the process of this comparison, Kant gives the most explicit statement of his critical philosophy of mathematics; accordingly, scholars have typically focused (...) their interpretations and criticisms of Kant’s conception of mathematics on this small section of the Critique. (shrink)

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example, they actually reveal the behavior of a real function not “close to” a point but “in” the point. We are interested in our research in (...) the diagrams which play an optical role –microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role, and an unveiling role. In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations. (shrink)

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)

Purpose: The paper contributes to the naturalization of epistemology. It suggests tentative itineraries for the progression from elementary experiential situations to the abstraction of the concepts of unit, plurality, number, point, line, and plane. It also provides a discussion of the question of certainty in logical deduction and arithmetic. Approach: Whitehead's description of three processes involved in criticizing mathematical thinking (1956) is used to show discrepancies between a traditional epistemological stance and the constructivist approach to knowing and communication. (...) Practical implications: Reducing basic abstract terms to experiential situations should make them easier to conceive for students. (shrink)

Hartry Field has recently examined the question whether our logical and mathematical concepts are referentially indeterminate. In his view, (1) certain logical notions, such as second-order quantification, are indeterminate, but (2) important mathematical notions, such as the notion of finiteness, are not (they are determinate). In this paper, I assess Field's analysis, and argue that claims (1) and (2) turn out to be inconsistent. After all, given that the notion of finiteness can only be adequately characterized in (...) pure secondorder logic, if Field is right in claiming that second-order quantification is indeterminate (see (1)), it follows that finiteness is also indeterminate (contrary to (2)). After arguing that Field is committed to these claims, I provide a diagnosis of why this inconsistency emerged, and I suggest an alternative, consistent picture of the relationship between logical and mathematical indeterminacy. (shrink)

Purpose: The paper contributes to the naturalization of epistemology. It suggests tentative itineraries for the progression from elementary experiential situations to the abstraction of the concepts of unit, plurality, number, point, line, and plane. It also provides a discussion of the question of certainty in logical deduction and arithmetic. Approach: Whitehead’s description of three processes involved in criticizing mathematical thinking (1956) is used to show discrepancies between a traditional epistemological stance and the constructivist approach to knowing and communication. (...) Practical implications: Reducing basic abstract terms to experiential situations should make them easier to conceive for students. (shrink)

New technologies, such as functional magnetic resonance imaging and transcranial magnetic stimulation, are currently touted as, not only giving us a better picture of the structure of the brain, but also a better understanding of our thinking. As Alan Snyder demonstrates when he claims his aim is to understand the ‘architecture of thought’ by investigating the brain. Against this backdrop, I will argue that new technologies present a worrying extension of mathematical natural science into the domain of human affairs. (...) Extrapolating upon Heidegger, I will put forward that neuroscientific experiments force thinking to conform to the mathematical conception of nature, rather than reveal something about the ‘true’ nature of our thinking. In a time when the expansion of mathematical natural science threatens to reduce every domain to that which is quantifiable, I will conclude by suggesting that the responsibility of the philosopher is to question the presuppositions of modern science and psychology.Nuevas tecnologías como la imagen de resonancia magnética funcional y la estimulación magnética transcranial se consideran presumiblemente capaces de darnos no solo una mejor imagen de nuestro cerebro, sino también una mejor comprensión de nuestro pensamiento, tal como demuestra Alan Snyder cuando afirma que su objetivo es entender la “arquitectura del pensamiento” mediante la investigación del cerebro. Contra este marco general, argumentaré que las nuevas tecnologías nos presentan una preocupante extensión de la ciencia natural matemática al dominio de los asuntos humanos. Extrapolando a partir de Heidegger, propondré que los experimentos neurocientíficos fuerzan al pensamiento a conformarse a la concepción matemática de la naturaleza en vez de revelar algo sobre la verdadera naturaleza de nuestro pensamiento. Nos encontramos en un momento en que o la expansión de la ciencia natural matemática amenaza con reducir todos los dominios a lo que es cuantificable. Concluiré sugiriendo que la responsabilidad del filósofo es cuestionar los presupuestos de la ciencia moderna y de la psicología. (shrink)

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the (...) foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition. (shrink)

Es indudable la importancia de la noción de función para la matemática y la lógica actuales y es sabido que es G. W. Leibniz quien utiliza por vez primera el término función en un sentido matemático, un término que, además, es introducido en el marco de su cálculo infinitesimal. Puesto que el pensador alemán es, junto con I. Newton, uno de los descubri­dores del cálculo, suele pensarse que también debemos a él el concepto de función. Sin embargo, poco se ha (...) escrito sobre la manera específica en la que Leibniz utilizó el término función, evadiendo con frecuencia la pregunta de si acaso el término y el concepto han correspondido siempre. En el presente trabajo exploramos el significado del término en el contexto de su surgimiento: el manuscrito de 1673 De functionibus plagulae quattuor. Con tal objetivo, hacemos, en primer lugar, una breve reconstrucción de la historia del instinto de funcionalidad que sirva de marco para comprender el significado del término función. Este significado es obtenido, en segundo lugar, atendiendo a los usos que el autor hace del término en dicho manuscrito. (shrink)