This project explored what constitutes “ethical practice of mathematics”. Thematic analysis of ethical practice standards from mathematics-adjacent disciplines (statistics and computing), were combined with two organizational codes of conduct and community input resulting in over 100 items. These analyses identified 29 of the 52 items in the 2018 American Statistical Association Ethical Guidelines for Statistical Practice, and 15 of the 24 additional (unique) items from the 2018 Association of Computing Machinery Code of Ethics for inclusion. Three of the 29 items (...) synthesized from the 2019 American Mathematical Society Code of Ethics, and zero of the Mathematical Association of America Code of Ethics, were identified as reflective of “ethical mathematical practice” beyond items already identified from the other two codes. The community contributed six unique items. Item stems were standardized to, “The ethical mathematics practitioner…”. Invitations to complete the 30-min online survey were shared nationally (US) via Mathematics organization listservs and other widespread emails and announcements. We received 142 individual responses to the national survey, 75% of whom endorsed 41/52 items, with 90–100% endorsing 20/52 items on the survey. Items from different sources were endorsed at both high and low rates. A final thematic analysis yielded 44 items, grouped into “General” (12 items), “Profession” (10 items) and “Scholarship” (11 items). Moreover, for the practitioner in a leader/mentor/supervisor/instructor role, there are an additional 11 items (4 General/7 Professional). These results suggest that the community perceives a much wider range of behaviors by mathematicians to be subject to ethical practice standards than had been previously included in professional organization codes. The results provide evidence against the argument that mathematics practitioners engaged in “pure” or “theoretical” work have minimal, small, or no ethical obligations. (shrink)

In 1892, Eliakim Hastings Moore accepted the task of building a mathematics department at the University of Chicago. Working in close conjuction with the other original department members, Oskar Bolza and Heinrich Maschke, Moore established a stimulating mathematical environment not only at the University of Chicago, but also in the Midwest region and in the United States in general. In 1893, he helped organize an international congress of mathematicians. He followed this in 1896 with the organization of the Midwest (...) Section of the New York City-based American Mathematical Society. He became the first editor-in-chief of the Society's Transactions in 1899, and rose to the presidency of the Society in 1901. (shrink)

Abstract This paper explores the role of symbolic languages within and between positivist disciplines. Symbolic languages, of which mathematics is the most important example, consist of tautologically true statements, such as 2 + 2 = 4. These must be operationalized before being useful for positivist research agendas (i.e. two apples and two oranges make four fruit). Disciplines may borrow either the symbolic languages of another discipline or the symbolic language and the accompanying operationalizations. The choice has important theoretical effects, and (...) affects the kind of interdisciplinary community created. The game theory community is an example of a community based on the interdisciplinary exchange of symbolic language only, while the ? political economy? community exchanges both symbolic languages and operationalizations. (shrink)

The focus on discourse and communication in the recent sociology of scientific knowledge offers new perspectives for an integration of qualitative and quantitative approaches in science studies. The common point of interest is the question of how reflexive communication systems communicate. The elaboration of the mathematical theory of communication into a theory of potentially self-organizing entropical systems enables us to distinguish the various layers of communication, and to specify the dynamic changes in these configurations over (...) time. For example, a paradigmatic discourse can be considered as a virtual communication system at the supra-individual level. Communication systems, however, cannot be directly observed. One observes only their instantaneous operations. The reflexive analyst is able to attribute the observed uncertainty to hypothesized systems that interact in the events. The implications of this perspective for various programmes in the sociology of scientific knowledge are discussed. (shrink)

In 1935, a group of mathematical statisticians in the United States announced the founding of a new professional organization, the Institute of Mathematical Statistics. In the coming years, this institution became a significant source of support for the growing community of mathematical statisticians. Thus it would seem that the formation of the Institute marked the beginning of the process of the creation of that community. This paper, however, argues that those who initially participated in the Institute constituted (...) a small, loosely organized, but wellconnected group within the broader scientific world even before they joined together formally to promote their discipline. We will examine some aspects of their education, employment, and interests in order to determine how American mathematical statisticians set themselves apart from other scientists and helped to define their discipline before the community acquired its official status. (shrink)

The author analyzes proof and argumentation as a form of appeal to a scientific community with deep ethical meaning. He presents proof primarily as an effort to persuade a scientific community rather than a search for true knowledge, as an instrument by which responsibility is taken for the correctness of the thesis being proved, which usually originates in a sudden flash of insight.

This paper argues that the shared intersubjective accessibility of mathematical objects has its roots in a stratum of experience prior to language or any other form of concrete social interaction. On the basis of Husserl’s phenomenology, I demonstrate that intersubjectivity is an essential stratum of the objects of mathematical experience, i.e., an integral part of the peculiar sense of a mathematical object is its common accessibility to any consciousness whatsoever. For Husserl, any experience of an objective nature (...) has as its correlate a “we,” which he terms the “community of monads”. Thus, even before mathematical objects gain expression, formalization, and axiomatization through natural and scientific language, from a phenomenological viewpoint their objectivity has its roots in raw pre-linguistic though intersubjective experience. Accordingly, I demonstrate the different senses in which the experience of mathematical objects is permeated by intersubjectivity, suggesting a picture of mathematical intersubjectivity as pre-linguistic common experience based on Husserl’s idea of a “community of monads”. (shrink)

What understanding of mathematical objectivity is promoted by Peirce’s pragmatism? Can Peirce’s theory help us to further comprehend the role of intersubjectivity in mathematics? This paper aims to answer such questions, with special reference to recent debates on mathematical practice, where Peirce is often quoted, although without a detailed scrutiny of his theses. In particular, the paper investigates the role of intersubjectivity in the constitution of mathematical objects according to Peirce. Generally speaking, this represents one of the (...) key issues for the philosophy of mathematical practice, whereas – with regard to Peirce – these two aspects (intersubjectivity and objectivity) have been studied for a long time, but mostly as unrelated topics. To reconstruct the connection between Peirce’s reflection on intersubjectivity and the objectivity of mathematical theories, the paper is divided into three parts: (1) the first part introduces Peirce’s view of mathematics; (2) the second analyzes his peculiar “pragmatist realism,” which overcomes common dichotomies in the philosophy of mathematics; (3) the third illustrates the role that Peirce attributes to intersubjectivity in science, and investigates to what extent the intersubjective dimension is also essential in mathematics. (shrink)

This research aims at establishing the impact of information and communication technologies (ICT) on e-commerce development of industrial enterprises by means of economic and mathematical modelling. The goal was achieved using the following methods: theoretical generalization, analysis and synthesis (to critically analyse the scientific approaches of scientists regarding the expediency of using mathematical models in the context of enterprises’ e-commerce development), target, comparison and grouping (to reveal innovative methodological approach to assessing ICT impact on e-commerce development of (...) industrial enterprises), tabular, analytical and integral method (for summarizing the analysis results of enterprises readiness to implement ICT, ICT use in the activities of industrial enterprises of Ukraine and the analysis of e-commerce development), mathematical modelling (to build a regression model determining impact of changes in ICT use on the market share occupied by industrial enterprises), generalization (to determine promising directions of e-commerce developing of industrial enterprises). The implementation of a comprehensive approach to understanding the importance of ICT influence on e-commerce development of industrial enterprises will ensure acceleration of the digitalization of business processes, will contribute to the speed increase of enterprises response to customer requests, and increase the market share occupied by enterprises. A new vision of directions for developing e-commerce of industrial enterprises is suggested, which are determined by the need for enterprise rebranding, the development of e-commerce tools and technologies, the importance of outsourcing service automation and promotion of subscription trade. ICT is considered as integration factor that determines prospects for e-commerce development of industrial enterprises and contributes to increasing efficiency of online business management. Research results demonstrate that the use of economic and mathematical modelling is an important tool for assessing ICT impact, and its absence can negatively affect the accuracy and validity of online business management. (shrink)

Examine how community college faculty in science, technology, engineering, and mathematics partnered with K–12 teachers to mentor community college science and math students and open their minds to pursuing a career in K–12 teaching. This report outlines the lessons learned by the community colleges that participated in AACC’s Teaching Scholar Partnerships, an initiative supported by the National Science Foundation.

An inclusive vision of mathematics-- its beauty, its humanity, and its power to build virtues that help us all flourish. For mathematician Francis Su, a society without mathematical affection is like a city without museums. To miss out on mathematics is to live without experiencing some of humanity's most beautiful ideas. In this profound book, written for a diverse audience but especially for those disenchanted by their past experiences, an award-winning mathematician and educator weaves personal reflections, puzzles, and stories (...) to show how mathematics meets basic human desires and cultivates virtues essential for human flourishing. Readers will explore mathematical concepts-- and see how mathematical thinking can even fulfill such longings as for love, play, freedom, justice, and community. Some lessons come from those who have struggled, including philosopher Simone Weil, whose own mathematical contributions were overshadowed by her brother's, and Christopher Jackson, who discovered mathematics as an inmate in a federal prison. Christopher Jackson's letters to the author appear throughout the book and show how this intellectual pursuit can-- and must-- be open to all. (shrink)

Open peer commentary on the target article “Who Conceives of Society?” by Ernst von Glasersfeld. Excerpt: My goal in this commentary is to say enough to suggest that the meanings children impute to the language and actions of other children are based on their current conceptual schemes and that, if the schemes are at different levels of the constructive process, it is no easy feat for children to use their schemes in interactive mathematical communication.

The emergence of mathematical probability has something to do with dice games: all the early discussions (Cardano, Galileo, Pascal) suggest as much. Although this has long been recognized, the problem is that gambling at dice has been a popular pastime since antiquity. Why, then, did gamblers wait until the sixteenth century ce to calculate the math of dicing? Many theories have been offerred, but there may be a simple solution: early-modern gamblers played different sorts of dice games than in (...) antiquity. While ancients diced at communal risk, early-moderns diced at individualized risk. These individual wager-games, for example, the game “hazard”, incentivized solving questions like “how likely is it to roll a triple-six?”. Before the early modern period there was no gambling game to which working out mathematical probability would have brought an advantage. (shrink)

Mathematicians suggest that some proofs are valued for their explanatory value. This has led to a philosophical debate about the distinction between explanatory and non-explanatory proofs. In this paper, we explore whether contrasting views about the explanatory value of proof are possible and how to understand these diverging assessments. By considering an epistemic and contextual conception of explanation, we can make sense of disagreements about explanatoriness in mathematics by identifying differences in the background knowledge, skill corpus, or epistemic aims of (...) mathematicians or mathematical communities. We focus on the relation between explanation, epistemic aims and diverging explanatory assessments by looking at cases from mathematical practice. (shrink)

This paper seeks to utilize mathematical methods to formally define and analyze the metaethical theory that is ethical reductionism. In contemporary metaethics, realist-antirealist debates center on the ontology of moral properties. Our research reflects an innovative methodology using methods from Graph Theory to clarify a debated position of Meta-Ethics, previously encumbered by intrinsic vagueness and ambiguity. We employ rigorous mathematical formalism to symbolize, parse, and thus disambiguate, particular philosophical questions regarding ethical ontological materialism of the reductionist variety. In (...) this paper, we seek to revisit the once vexed question regarding the multiple-realizability of moral properties by employing the mathematical machinery of hypergraphs and category theory. The utilization of Mathematical formalism offers explanatory flexibility specifically to the hypothesis that there are potentially an infinite number of unrelated subvening facts which may constitute supervening moral properties. We by no means have attempted to settle the argument on the side of naturalism, but have only identified an obstacle to rigorous argumentation, and pioneered a method to eliminate it. We hope our research will be of interest to both the Analytic Philosophy and the Applied Mathematics communities, and will help facilitate discussions of metaethics, particularly pertaining to ethical naturalism. We believe there is much research yet to be done. (shrink)

Though some studies describe attempts to integrate Facebook in education, little is known how to use it in mathematics education. This article describes an attempt to populate Facebook with mathematicians from the past, as well as strategies to involve friends with the mathematics of the mathematicians. The experiment shows that Facebook can attract friends to content knowledge, beginning with social talk, and transiting gradually and smoothly to mathematics content knowledge through cultural discourse. The experiment implies that Facebook, representing social networks (...) not intended from the beginning for education, can be adopted successfully for mathematics education. (shrink)

Le rôle des mathématiques dans la formation des officiers de l’armée néerlandaise, a profondément changé pendant le premier xixe siècle avec la fondation de l’Académie militaire en 1828. Les mathématiques étaient au centre de la formation. L’Académie était un des lieux les plus importants de diffusion des connaissances mathématiques aux Pays-Bas pendant la première moitié du xixe siècle, mais elle a perdu ce rôle pendant les années 1860-1870. Dans cet article, je me propose d’examiner à la fois les programmes de (...) l’Académie et le rôle central des enseignants de mathématiques de l’Académie dans la Société Mathématique néerlandaise et dans l’Académie royale des Sciences. Par ailleurs, l’évolution des mathématiques, visible dans ces deux institutions dans les années 1850, n’a pas entraîné une adaptation des programmes au sein de l’Académie. Ainsi, à partir de 1870, l’Académie a presque complètement disparu de la vie mathématique des Pays-Bas. The role of mathematics in the education of Dutch army officers changed dramatically in the early 19th century. This shift became visible with the founding of the Dutch Royal Military Academy in 1828, which shows the role of mathematics in military education shifting toward a purely pedagogical function. This change in approach, which primarily occurred in 1860’s, affected the standing and perception of an academy once highly respected in the field of mathematics. In the present work, I examine both the curricula at the Academy and the crucial role of the Academy’s mathematics instructors in the Dutch Mathematical Society and the Royal Academy of Sciences. Importantly, these two institutions managed to successfully adapt to the changing character of mathematical sciences that occurred around 185 Mathematicians at the Royal Military Academy, however, appeared to cling to an outdated view of mathematics. Failing to update their approach ultimately caused the Academy to lose its once prominent place in the Dutch mathematical community. (shrink)

The primary aim of this article is to bring the work of Deleuze and Guattari to bear on the question ofcommunication in the classroom. I focus on the mathematics classroom, where agency and subjectivity are highly regulated by the rituals of the discipline, and where neoliberal psychological frameworks continue to dominate theories of teaching and learning. Moreover, the nature ofcommunication in mathematics classrooms remains highlyelusive and problematic, due in part to the distinct relationship the discipline has with verbal language and (...) thought. I first discuss current attempts to better address the embodied nature ofcommunication in mathematics classrooms, and argue that these remain overly logo-centric and language-centric in their conception of thinking. I then show how the work of Deleuze and Guattari on thought as a radical disruptive event can be used effectively to critique current pedagogical practices that privilege a narrow conception of communication in the classroom. I examine a set of exemplary classroom videos used in mathematics teacher education to argue that the current approach fails to honor the highly creative and disruptive nature ofthinking. (shrink)

Wittgenstein's role was vital in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, the three phases of Wittgenstein's reflections on mathematics are viewed as a progressive whole, rather than as separate entities. Frascolla builds up a systematic construction of Wittgenstein's representation of the role of arithmetic in the theory of logical operations. He also presents a new interpretation of Wittgenstein's rule-following considerations - the `community view of internal relations'.

This work explores the later Wittgenstein’s philosophy of mathematics in relation to Lakatos’ philosophy of mathematics and the philosophy of mathematical practice. I argue that, while the philosophy of mathematical practice typically identifies Lakatos as its earliest of predecessors, the later Wittgenstein already developed key ideas for this community a few decades before. However, for a variety of reasons, most of this work on philosophy of mathematics has gone relatively unnoticed. Some of these ideas and their significance as (...) precursors for the philosophy of mathematical practice will be presented here, including a brief reconstruction of Lakatos’ considerations on Euler’s conjecture for polyhedra from the lens of late Wittgensteinian philosophy. Overall, this article aims to challenge the received view of the history of the philosophy of mathematical practice and inspire further work in this community drawing from Wittgenstein’s late philosophy. (shrink)

This essay argues that the diversity of the history of mathematics community in the United Kingdom has influenced the development of the subject and is a significant factor behind the different concerns often evident in work on the history of mathematics when compared with that of historians of science. The heterogeneous nature of the community, which includes many who are not specialist historians, and the limited opportunities for academic careers open to practitioners have had a profound effect on the discipline, (...) leading to a focus on elite mathematics and great mathematicians. More recently, reflecting earlier developments in the history of science, an increased interest in the context and culture of the practice of mathematics has become evident. (shrink)

Widespread interest in Frege's general philosophical writings is, relatively speaking, a fairly recent phenomenon. But it is only very recently that his philosophy of mathematics has begun to attract the attention it now enjoys. This interest has been elicited by the discovery of the remarkable mathematical properties of Frege's contextual definition of number and of the unique character of his proposals for a theory of the real numbers. This collection of essays addresses three main developments in recent work on (...) Frege's philosophy of mathematics: the emerging interest in the intellectual background to his logicism; the rediscovery of Frege's theorem; and the reevaluation of the mathematical content of The Basic Laws of Arithmetic. Each essay attempts a sympathetic, if not uncritical, reconstruction, evaluation, or extension of a facet of Frege's theory of arithmetic. Together they form an accessible and authoritative introduction to aspects of Frege's thought that have, until now, been largely missed by the philosophical community. (shrink)

This paper attempts to look at the contemporary mathematical practice through the lenses of the distributed cognition approach. The ubiquitous use of personal computers and the internet as a key attribute of the digital society is interpreted here as a means to achieve a more effective distribution of the human cognitive activity. The major challenge that determines the transformation of mathematical practice is identified as ‘the problem of complexity’. The computer-assisted complete formalization of mathematical proofs as a (...) current tendency is viewed as one of the strands along which the mathematical community responds to the challenge. It is shown that this tendency gives live to the project calling to revisit and rebuild the very foundations of mathematics to secure more effective communication and thus guarantee the reliability of contemporary mathematics. (shrink)

Inferential communities are communities using specific substantial argumentative schemes. The religious or scientific communities are examples. I discuss the status of the mathematical community as it appears through the position held by the French mathematician Henri Poincaré during his famous ar-guments with Russell, Hilbert, Peano and Cantor. The paper focuses on the status of complete induction and how logic and psychology shape the community of mathematicians and the teaching of mathematics.

One of the criticisms of standard teaching practices is that they support merely ‘ritual’ as opposed to ‘principled’ knowledge, that is, knowledge which is procedural rather than being founded on principled explanation. This paper addresses issues and assumptions in current debate concerning the nature of mathematical knowledge, focusing on the ritual/principle distinction. Taking a discussion of centralism in logic and mathematics as its start-point, it seeks to resolve these issues through an examination of mathematics as a community of practice (...) and the teacher's role as epistemological authority in inducting pupils into such practices. (shrink)

This book presents a comprehensive treatment of basic mathematical logic. The author's aim is to make exact the vague, intuitive notions of natural number, preciseness, and correctness, and to invent a method whereby these notions can be communicated to others and stored in the memory. He adopts a symbolic language in which ideas about natural numbers can be stated precisely and meaningfully, and then investigates the properties and limitations of this language. The treatment of mathematical concepts in the (...) main body of the text is rigorous, but, a section of 'historical remarks' traces the evolution of the ideas presented in each chapter. Sources of the original accounts of these developments are listed in the bibliography. (shrink)

The OMDoc (Open Mathematical Documents) format is a content markup scheme for (collections of) mathematical documents including articles, textbooks, interactive books, and courses. OMDoc also serves as the content language for agent communication of mathematical services on a mathematical software bus.

It is demonstrated that Maxwellian electrodynamics was created as a result of the old pre-Maxwellian programmes’s reconciliation: the electrodynamics of Ampère–Weber, the wave theory of Young–Fresnel and Faraday’s programme. Maxwell’s programme finally superseded the Ampère–Weber one because it assimilated the ideas of the Ampère–Weber programme, as well as the presuppositions of the programmes of Young–Fresnel and Faraday. Maxwell’s victory became possible because the core of Maxwell’s unification strategy was formed by Kantian epistemology. Maxwell put forward as a basic synthetic principle (...) an idea that radically differed from that of rival approaches by its open, flexible and contra-ontological character. “Action at a distance”, “incompressible fluid”, “molecular vortices” were contrived analogies for Maxwell, capable only of directing the researcher to the “right” mathematical relations Kantian epistemology subsequently enabled Helmholtz and Hertz to arrive at a version of Maxwell’s theory that served as a heuristical basis for the discovery of radio waves. Finally, though neither of Einstein’s relativistic ideas came directly from Kant, they were made possible by the Kantian worldview that had permeated Einstein’s thinking. The Maxwellian revolution can be described in terms of Habermas’s communicative rationality encouraging the establishment of mutual understanding between the various scientific communities. Maxwell’s programme constituted a progressive step in respect to its rivals because it constituted a basis of communication and interpenetration between the main paradigms of 19th century physics. (shrink)

Alan Baker argues that mathematical objects play an indispensable explanatory role in science. There are several examples cited in the literature as solid candidates for such a role. We discuss two such examples and show that they are very different in their strength and (im)perfection, although both are recognized by the scientific community as examples of the best scientific explanations of particular phenomena. More specifically, it will be shown that the explanation of the cicada case has serious shortcomings compared (...) with the explanation of the case of Königsberg’s bridges. We will argue that the latter is a perfectly reliable scientific explanation that employs mathematical reasoning whereas the former is not. (shrink)

This volume honours the life and work of Solomon Feferman, one of the most prominent mathematical logicians of the latter half of the 20th century. In the collection of essays presented here, researchers examine Feferman’s work on mathematical as well as specific methodological and philosophical issues that tie into mathematics. Feferman’s work was largely based in mathematical logic, but also branched out into methodological and philosophical issues, making it well known beyond the borders of the mathematics community. (...) With regard to methodological issues, Feferman supported concrete projects. On the one hand, these projects calibrate the proof theoretic strength of subsystems of analysis and set theory and provide ways of overcoming the limitations imposed by Gödel’s incompleteness theorems through appropriate conceptual expansions. On the other, they seek to identify novel axiomatic foundations for mathematical practice, truth theories, and category theory. In his philosophical research, Feferman explored questions such as “What is logic?” and proposed particular positions regarding the foundations of mathematics including, for example, his “conceptual structuralism.” The contributing authors of the volume examine all of the above issues. Their papers are accompanied by an autobiography presented by Feferman that reflects on the evolution and intellectual contexts of his work. The contributing authors critically examine Feferman’s work and, in part, actively expand on his concrete mathematical projects. The volume illuminates Feferman’s distinctive work and, in the process, provides an enlightening perspective on the foundations of mathematics and logic. (shrink)

Mathematics is a highly specialised arena of human endeavour, one in which complex notations are invented and are subjected to complex and involved manipulations in the course of everyday work. What part do these writing practices play in mathematical communication, and how can we understand their use in the mathematical world in relation to theories of communication and cognition? To answer this, I examine in detail an excerpt from a research meeting in which communicative board-writing practices (...) can be observed, and attempt to explain the observed exchanges with reference to relevance theory (a cognitivist theory of pragmatics), in combination with the situated cognition paradigm. Successful communication in the excerpt appears well described by the notion of metacognitive acquaintance, as described inSperber & Wilson (2015), especially when a situated view of cognition is adopted. Though this example is taken from a relatively informal face-to-face communicative situation, characteristics of such situations extend even into formal written mathematics, since even the technical terminology developed in the course of the meeting includes details that may provide readers with a kind of access to the mathematicians’ cognitive processes. (shrink)

Mathematical images occur in lectures, books, notes and posters, and on the internet. We extend Kennedy's proposal for classifying these images. In doing so we distinguish three uses of images in mathematics: iconic images; incidental images; and integral images. An iconic image is one that so captures the essence of a concept or proof that it serves for a community of mathematicians as a motto or a meme for an area or a result. A system such as Euclid's can (...) combine such apprehensions with other forms of logical inference and an image that is built into a system of exposition is called an integral image. An incidental image is an image used by a mathematician to reason with a particular concept. In addition to this thematic characterization, we also explore one concept, infinity, in some depth by comparing representations of the infinite by mathematicians and artists. (shrink)