Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects.

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and (...) Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel’s sense) structures exist, which alleviates the cosmological measure problem and may help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems. (shrink)

We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept (...) to highlight a potential danger of intellectual enculturation. (shrink)

This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to cover the (...) field, it lays down a much-needed marker of progress to date and provides a platform for informed and coherent future analysis and research of the subject. -/- The publication comes at a time of heightened worldwide concern over the standard of science and mathematics education, attended by fierce debate over how best to reform curricula and enliven student engagement in the subjects There is a growing recognition among educators and policy makers that the learning of science must dovetail with learning about science; this handbook is uniquely positioned as a locus for the discussion. -/- The handbook features sections on pedagogical, theoretical, national, and biographical research, setting the literature of each tradition in its historical context. Each chapter engages in an assessment of the strengths and weakness of the research addressed, and suggests potentially fruitful avenues of future research. A key element of the handbook’s broader analytical framework is its identification and examination of unnoticed philosophical assumptions in science and mathematics research. It reminds readers at a crucial juncture that there has been a long and rich tradition of historical and philosophical engagements with science and mathematics teaching, and that lessons can be learnt from these engagements for the resolution of current theoretical, curricular and pedagogical questions that face teachers and administrators. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

The core idea of social constructivism in mathematics is that mathematical entities are social constructs that exist in virtue of social practices, similar to more familiar social entities like institutions and money. Julian C. Cole has presented an institutional version of social constructivism about mathematics based on John Searle’s theory of the construction of the social reality. In this paper, I consider what merits social constructivism has and examine how well Cole’s institutional account meets the challenge of accounting for the (...) characteristic features of mathematics, especially objectivity and applicability. I propose that in general social constructivism shows promise as an ontology of mathematics, because the view can agree with mathematical practice and it offers a way of understanding how mathematical entities can be real without conflicting with a scientific picture of reality. However, I argue that Cole’s specific theory does not provide an adequate social constructivist account of mathematics. His institutional account fails to sufficiently explain the objectivity and applicability of mathematics, because the explanations are weakened and limited by the three-level theoretical model underlying Cole’s account of the construction of mathematical reality and by the use of the Searlean institutional framework. The shortcomings of Cole’s theory give reason to suspect that the Searlean framework is not an optimal way to defend the view that mathematical reality is socially constructed. (shrink)

We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct speech acts within mathematics, and, as we intend to show, distinct illocutionary force indicators as well. Even mathematics in its most formalized version cannot do without some such indicators. This goes against a certain orthodoxy both in contemporary philosophy of mathematics and in speech act theory. As we will comment, the recognition of distinct illocutionary acts within logic and mathematics and the incorporation (...) of illocutionary force indicators in the formal language for both goes back to Frege’s conception of these topics. We are, therefore, going back to a Fregean perspective. This paper is part of a larger project of applying contemporary speech act theory to the scientific language of mathematics in order to uncover the varieties and regular combinations of illocutionary acts present in it. For reasons of space, we here concentrate only on assertive and declarative acts within mathematics, leaving the investigation of other kinds of acts for a future occasion. (shrink)

Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two forms, and examine the (...) only available argument for it. I argue that this argument, the so-called Failure Argument, itself fails. My overall conclusion is that although there is no self-refutation argument against reinterpretation naturalism, there are as yet no good reasons to accept it. Naturalism in mathematics The consistency of mathematical naturalism The failure argument Objections to the failure argument Philosophy as the default. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the (...) effectiveness of mathematics in natural science. (shrink)

This article derives from a project attempting to show that Western formal logic, from Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. In the present article, I will first discuss, in light of Frege’s honorary role as founder of the philosophy of mathematics, Reuben Hersh’s What is Mathematics, Really? Second, I will explore how the infamous section of Frege’s 1924 diary (specifically the entries from March 10 to April 9) supports (...) Hersh's claim regarding the link between political conservatism and the (historically and currently) dominant school of the philosophy of mathematics (to which Frege undeniably belongs). Third, I will examine Frege’s attempt at a more reader-friendly introduction to his philosophy of mathematics, The Foundations of Arithmetic. And finally, I will briefly analyze Frege’s Begriffsschrift to see how questions of race arise even at the heights of his logical abstraction. (shrink)

ABSTRACT This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and (...) assessing the relation between cognitive processes and certain properties of the stimuli at which they are directed. (shrink)

In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the (...) role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role. (shrink)

A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be developed on the basis of these and other studies, (...) locating the substance of mathematics in the various informal argument methods used by mathematicians. (shrink)

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...) of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)

A traditional problem of ethics in mathematics is the denial of social responsibility. Pure mathematics is viewed as neutral and value free, and therefore free of ethical responsibility. Applications of mathematics are seen as employing a neutral set of tools which, of themselves, are free from social responsibility. However, mathematicians are convinced they know what constitutes good mathematics. Furthermore many pure mathematicians are committed to purism, the ideology that values purity above applications in mathematics, and some historical reasons for this (...) are discussed. MacIntyre’s virtue ethics accommodates both the good mathematician and the ethics of the social practice of mathematics. It demonstrates that purism is compatible with acknowledging the social responsibility of mathematics. Four aspects of this responsibility are mentioned, two concerning the impact of mathematics via education, and two concerning explicit and implicit applications of mathematics. The last of these opens up the performativity of mathematical and measurement applications in society, which change the very processes they are supposed to measure. Although these applications are not explored in detail, they illustrate the importance of considering the ethics and social responsibility of mathematics in society. MacIntyre’s virtue theory opens a broad approach to the controversial topic of the ethics of mathematics encompassing purism, and absolutist and social constructivist philosophies of mathematics, but still enabling ethical critiques of the impact of mathematics on society. (shrink)

The object of this paper is to study the analogy, drawn both positively and negatively, between mathematics and fiction. The analogy is more subtle and interesting than fictionalism, which was discussed in part I. Because analogy is not common coin among philosophers, this particular analogy has been discussed or mentioned for the most part just in terms of specific similarities that writers have noticed and thought worth mentioning without much attention's being paid to the larger picture. I intend with this (...) analogy (looking at others' comparisons) to shed a little light on what is going on in mathematics, how one can understand it a bit other than experientially. This intention is philosophical and the way that I am attempting to accomplish it is also philosophical. I shall conclude my attempt to explain how it is possible and even natural for mathematics and fiction to have the analogy they have, taking it for granted, as argued in part I, that they are not to be identified. To this end I shall discuss philosophers' comparisons, mainly those of Hodes, Resnik, Tharp, and Wagner, who are the writers that seem to me to have written most thoughtfully and sufficiently extensively about fiction in making the comparison and of Körner, whose comparison is different. Whether either of these comparisons or the more general analogy is of permanent philosophical interest will have to be decided by philosophers now that they have had a fuller examination. I shall mention some other writers' reference to fiction, but not all; indeed, I am sure that I have not even found all the comparisons that there are. (shrink)

The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which are essential (...) to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic-neutral reasoning. These in turn are classified under seven headings: retroduction, citation, intuition, meta-argument, closure, generalization, and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed. (shrink)

The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some (...) general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach. (shrink)

We document the influence on programming language semantics of the Platonism/formalism divide in the philosophy of mathematics.

In this article, modern standards of early years mathematics education are criticized and a proposal for change is presented. Today's early years mathematics education standards rest on a view of mathematics that became obsolete already at the end of the 19th century while the spirit of children's mathematics is precisely the spirit of modern mathematics. The proposal for change is not a return to the “new mathematics” movement, but something different.

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There are “verifications” of (...) hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed. (shrink)

This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. a matter (...) of recognizing trutlis wliich had previoasly been unrecognized, but of extending the domain of what is true. (shrink)

Recently, certain philosophers of mathematics (Fallis [1997]; Womack and Farach [(1997]) have argued that there are no epistemic considerations that should stop mathematicians from using probabilistic methods to establish that mathematical propositions are true. However, mathematicians clearly should not use methods that are unreliable. Unfortunately, due to the fact that randomized algorithms are not really random in practice, there is reason to doubt their reliability. In this paper, I analyze the prospects for establishing that randomized algorithms are reliable. I end (...) by arguing that it would be inconsistent for mathematicians to suspend judgement on the truth of mathematical propositions on the basis of worries about the reliability of randomized algorithms. (shrink)

This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and assessing (...) the relation between cognitive processes and certain properties of the stimuli at which they are directed. (shrink)

Using Carnap’s concept explication, we propose a theory of concept formation in mathematics. This theory is then applied to the problem of how to understand the relation between the concepts formal proof and informal, mathematical proof.

This contribution defends two claims. The first is about why thought experiments are so relevant and powerful in mathematics. Heuristics and proof are not strictly and, therefore, the relevance of thought experiments is not contained to heuristics. The main argument is based on a semiotic analysis of how mathematics works with signs. Seen in this way, formal symbols do not eliminate thought experiments (replacing them by something rigorous), but rather provide a new stage for them. The formal world resembles the (...) empirical world in that it calls for exploration and offers surprises. This presents a major reason why thought experiments occur both in empirical sciences and in mathematics. The second claim is about a looming aporia that signals the limitation of thought experiments. This aporia arises when mathematical arguments cease to be fully accessible, thus violating a precondition for experimenting in thought. The contribution focuses on the work of Vladimir Voevodsky (1966–2017, Fields medalist in 2002) who argued that even very pure branches of mathematics cannot avoid inaccessibility of proof. Furthermore, he suggested that computer verification is a feasible path forward, but only if proof is not modeled in terms of formal logic. (shrink)

The sum of all objects of a science, the objects’ features and their mutual relations compose the reality described by that sense. The reality described by mathematics consists of objects such as sets, functions, algebraic structures, etc. Generally speaking, the use of terms reality and existence, in relation to describing various objects’ characteristics, usually implies an employment of physical and perceptible attributes. This is not the case in mathematics. Its reality and the existence of its objects, leaving aside its application, (...) are completely virtual and yet clearly organized. This organization can be recognized in the creation of axioms or in the arrival at new theorems and definitions. It results either from a formalization of an intuitive idea or from a combinatorics that has not been guided by intuition. In all four possible cases—therefore, also in the two in which there is no intuitive “lead”, we can plausibly talk about a discovery of mathematical facts and thus support the Platonist view. (shrink)

This paper discerns two types of mathematization, a foundational and an explorative one. The foundational perspective is well-established, but we argue that the explorative type is essential when approaching the problem of applicability and how it influences our conception of mathematics. The first part of the paper argues that a philosophical transformation made explorative mathematization possible. This transformation took place in early modernity when sense acquired partial independence from reference. The second part of the paper discusses a series of examples (...) from the history of mathematics that highlight the complementary nature of the foundational and exploratory types of mathematization. (shrink)

open journal of Philosophical Investigations (PI) is an international journal dedicated to the latest advancements in philosophy. The goal of this journal is to provide a platform for academicians all over the world to promote, share, and discuss various new issues and developments in different areas of philosophy. -/- All manuscripts to be prepared in English or Persian and are subject to a rigorous and fair peer-review process. Generally, accepted papers will appear online. The journal publishes papers including the following (...) fields: -/- · Analytic Philosophy · Ancient Greek and Roman Philosophy · Art Aesthetics · Comparative Philosophy West and Chinese · Eastern Philosophy . Islamic philosophy · Epistemology · Ethics · Hermeneutics · History of Religious Thought · History of Western Philosophy · Language Logic . (shrink)

This study applied a method of assisted introspection to investigate the phenomenology of mathematical intuition arousal. The aim was to propose an essential structure for the intuitive experience of mathematics. To achieve an intersubjective comparison of different experiences, several contemporary mathematicians were interviewed in accordance with the elicitation interview method in order to collect pinpoint experiential descriptions. Data collection and analysis was then performed using steps similar to those outlined in the descriptive phenomenological method that led to a generic structure (...) that accounts for the intuition surge in the experience of mathematics which was found to have four irreducible structural moments. The interdependence of these moments shows that a perceptualist view of intuition in mathematics, as defended by Chudnoff, is relevant to the characterization of mathematical intuition. The philosophical consequences of this generic structure and its essential features are discussed in accordance with Husserl’s philosophy of ideal objects and theory of intuition. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

Mathematics textbooks teach logical reasoning by example, a practice started by Euclid; while logic textbooks treat logic as a subject in its own right without practical application to mathematics. Stuck in the middle are students seeking mathematical proficiency and educators seeking to provide it. To assist them, the article explains in practical detail how to teach logic-based skills such as: making mathematical reasoning fully explicit; moving from step to step in a mathematical proof in logically correct ways; and checking to (...) make sure inferences are logically correct. The methodology can easily be extended beyond the four examples analyzed. (shrink)

The paper shows how to use the Husserlian phenomenological method in contemporary philosophical approaches to mathematical practice and mathematical ontology. First, the paper develops the phenomenological approach based on Husserl's writings to obtain a method for understanding mathematical practice. Then, to put forward a full-fledged ontology of mathematics, the phenomenological approach is complemented with social ontological considerations. The proposed ontological account sees mathematical objects as social constructions in the sense that they are products of culturally shared and historically developed practices. (...) At the same time the view endorses the sense that mathematical reality is given to mathematicians with a sense of independence. As mathematical social constructions are products of highly constrained, intersubjective practices and accord with the phenomenologically clarified experience of mathematicians, positing them is phenomenologically justified. The social ontological approach offers a way to build mathematical ontology out of the practice with no metaphysical magic. (shrink)

Understanding is a valued trait in any epistemic practice, scientific or not. Yet, when it comes to characterizing its nature, the notion has not received the philosophical attention it deserves. We have set ourselves three tasks in this paper. First, we defend the importance of this endeavor. Second, we consider and criticize a number of proposals to this effect. Third, we defend an alternative account, focusing on abilities as the proper mark of understanding.

The collection Virtual Mathematics: the logic of difference brings together a range of new philosophical engagements with mathematics, using the work of French philosopher Gilles Deleuze as its focus. Deleuze’s engagements with mathematics rely upon the construction of alternative lineages in the history of mathematics in order to reconfigure particular philosophical problems and to develop new concepts. These alternative conceptual histories also challenge some of the self-imposed limits of the discipline of mathematics, and suggest the possibility of forging new connections (...) between philosophy and more recent developments in mathematics. This component of Deleuze’s work has, to date, been rather neglected by commentators working in the field of Deleuze studies. One of the aims of this collection is to address this critical deficit by providing a philosophical presentation of Deleuze’s relation to mathematics; one that is adequate to his project of constructing a philosophy of difference and to the exploration of some of its applications. This project developed as a result of encounters with an increasing number of researchers who have been working on the mathematical aspects of Deleuze’s work and the key role that these play in his philosophy. By bringing this work together for the first time, this collection makes an important contribution to the emerging body of work that endeavours to explore the broad range of Deleuze’s philosophy. (shrink)

This essay will be divided into three parts. In the first part, we discuss the case of infintesimals seen as a bridge between the discrete and the continuous. This leads in the second part to a discussion of the nature of numbers. In the last part, we follow up with some observations on the obvious applicability of mathematics.

RésuméDans leur livre récent, George Lakoff et Rafael Núñez se livrent à une critique naturaliste soutenue du platonisme traditionnel concernant les entités mathématiques. Ils affirment que des résultats récents en sciences cognitives démontrent qu'il est faux. En particulier, ils estiment que la découverte que la cognition mathématique s'appuie pour une large part sur les métaphores conceptuelles est incompatible avec le platonisme. Nous montrons ici que tel n'est pas le cas. Nous examinons et rejetons également quelques arguments philosophiques que formulent Lakoff (...) et Núñez contre le platonisme. Enfin, nous lions leur hostilité au platonisme à certaines opinions sur les ramifications politiques de celui-ci. Pour conclure, nous faisons valoir que ces opinions sont sans fondements et présentent une image distordue des dimensions politiques du platonisme. (shrink)

Certain mathematical objects bear the name “pathological”. They either occur as unexpected and unwilling in mathematical research practice, or are constructed deliberately, for instance in order to delimit the scope of application of a theorem. I discuss examples of mathematical pathologies and the circumstances of their emergence. I focus my attention on the creative role of pathologies in the development of mathematics. Finally, I propose a few reflections concerning the degree of cognitive accessibility of mathematical objects. I believe that the (...) problems discussed in the paper may attract the attention of philosophers interested in concept formation and the development of mathematical ideas. (shrink)

Gender greatly impacts access to opportunities, potential, and success in corporate leadership roles. We begin with a general presentation of why such discussion is necessary for basic considerations of justice and fairness in gender equality and how the issues we raise must impact any ethical perspective on gender in the corporate workplace. We continue with a breakdown of the central categories affecting the success of women in corporate leadership roles. The first of these includes gender-influenced behavioral factors, such as the (...) requirements and expectations of gendered verbal and nonverbal communication styles as well as appearance. We move on to address the impact of family on corporate leadership opportunities and success, discussing the asymmetrical evaluation of an individual’s potential, authority, and competence based on gender stereotypes of familial obligations and expectations. Finally, we address how gender impacts access to networking and sponsorship opportunities and the long-term effects of systematic limitations on women’s inclusion in the upper echelons of corporate leadership. We conclude with a summary of the questions and issues raised by our discussion and direct individuals to consider how different ethical systems and moral requirements might influence their interpretations of gender and leadership in the corporate workplace. (shrink)

In this article we contend that attempts to foster democratic education in the United States' public schools rarely include mathematics class in meaningful ways. We begin with Dewey's conception of democracy and then argue that current ways of thinking about mathematics do not provide adequate foundations for democratic mathematics education. Our reconceptualization of mathematics draws on Dewey's uniquely humanistic philosophy of mathematics. We conclude with some implications of democratic mathematics education for school and society. Thus, this project seeks to blur (...) the theory-practice dualism, developing a theoretical argument which draws sustenance from and seeks to contribute back to educational practice. (shrink)

The initial motivating question for this thesis is what the standard of rigour in modern mathematics amounts to: what makes a proof rigorous, or fail to be rigorous? How is this judged? A new account of rigour is put forward, aiming to go some way to answering these questions. Some benefits of the norm of rigour on this account are discussed. The account is contrasted with other remarks that have been made about mathematical proof and its workings, and is tested (...) and illustrated by considering a case study discussed in the literature. On the view put forward here one can obtain a manner of informal, rigorous mathematics founded on any of a variety of proof systems. The latter part of the thesis is concerned with the question of how we should decide which of these competing proof systems we should base our mathematics on: i.e., the question of which proof system we should take as a foundation for our mathematics. A novel answer to this question is proposed, in which the key property we should require of a proof system is that for as many different kinds of structures as possible, when the proof system allows a generalization about that kind of structure to be proved, the generalization actually holds of all real examples of that kind of structure which exist. This is the requirement of soundness of the proof system. It is argued that the best way to establish the soundness of a proof system may be by giving an interpretation of its axioms on which they are established as true. As preparation for this discussion, the thesis first investigates the logical and conceptual basis of various arithmetic concepts, with the results obtained used in the final discussion of soundness. (shrink)

Concepts of infinity have been subjects of dispute since antiquity. The main problems of this paper are: is the mind able to acquire a concept of infinity? and: how are concepts of infinity acquired? The aim of this paper is neither to say what the meanings of the word “infinity” are nor what infinity is and whether it exists. However, those questions will be mentioned, but only in necessary extent.