Today’s students are accustomed to a world where information is available on-demand, anywhere and anytime. They bring this expectation to their academic world where they want to work cooperatively and flexibly, using the modern information processing tools and access with which they are familiar. New hardware platforms such as e-Readers and tablet computers have made substantial inroads in the consumer market. E-Readers are becoming more prevalent in universities – replacing the need for physical textbooks, lecturing notes and other academic documents. (...) Many universities are now running pilot programs, while some are already using e-Readers. Tablet computers, apart from their ability to read, edit or create various types of information documents, also offer additional features such as collaborative and social networking services. This empirical research project investigated student perspectives on the educational use of e-Readers and tablets within the University of Cape Town. A questionnaire was distributed to a significant portion of the student population and stratified according to seniority and faculty. Quantitative data was gathered and analysed on themes ranging from awareness to preferences to facilitating conditions around e-Readers within the academic environment. The students sent a clear signal: that they are keen to embrace these new technologies and the advantages they offer. E-Books can be updated automatically and downloaded to e-Readers instantly from almost anywhere in the world. Students feel they would benefit greatly through the use of e-Readers and many believe they are set to replace books completely. Having access to a portable library of information in their backpacks would benefit both students and the environment. However, there are a number of barriers to the comprehensive introduction of these technologies, including the cost of the devices; cost, rights and file format of e-books; the need for a critical mass of textbooks available in digital format and device characteristics such as battery life, and ubiquitous Wi-Fi access. (shrink)

The Polymath Project is an online collaborative enterprise that was initiated in 2009, when Timothy Gowers asked whether and how groups could work together to solve mathematical problems that “do not naturally split up into a vast number of subtasks.” Gowers proposed to answer this question himself by actually trying to set up such a collaboration, based on interactions taking place in the comment-threads of a series of posts on a WordPress blog. Hence, the first project officially started in early (...) 2009, to be proclaimed successful only 6 weeks later (Gowers and Nielsen 2009). From its inception until April 2018, 15 more Polymath problems (and a handful of smaller or related ones) have been launched. These projects have attracted attention from different scholarly communities, including the philosophy of mathematical practices, from the perspective of which the Polymath Project can be seen as a vast repository of mathematics in action. This chapter continues previous work by its authors on the topic in question and topics related. More specifically, for the purposes of this volume, it is our aim to both summarize and expand upon these earlier contributions. The starting point is the above observation that in the past decade, the issue of “massively collaborative mathematics” (to be qualified below) has drawn quite some attention. However, does it also warrant intensive philosophical attention in particular? For, as exciting as these developments might be from mathematical and other points of view, the enhanced possibilities that have come with it do not per se give rise to any philosophical import. In Van Bendegem (2011), one of us has indeed tentatively argued for the philosophical relevance of these new dynamics of proof construction flowing from the wide availability and efficiency of Internet technology. We shall below briefly rehearse and update the argument given there. (shrink)

In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice and epistemology.

The paper gives an impression of the multi-dimensionality of mathematics as a human activity. This 'phenomenological' exercise is performed within an analytic framework that is both an expansion and a refinement of the one proposed by Kitcher. Such a particular tool enables one to retain an integrated picture while nevertheless welcoming an ample diversity of perspectives on mathematical practices, that is, from different disciplines, with different scopes, and at different levels. Its functioning is clarified by fitting in illustrations based on (...) actual research. (shrink)

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

Philosophy of mathematics today has transformed into a very complex network of diverse ideas, viewpoints, and theories. Sometimes the emphasis is on the "classical" foundational work (often connected with the use of formal logical methods), sometimes on the sociological dimension of the mathematical research community and the "products" it produces, then again on the education of future mathematicians and the problem of how knowledge is or should be transmitted from one generation to the next. The editors of this book felt (...) the urge, first of all, to bring together the widest variety of authors from these different domains and, secondly, to show that this diversity does not exclude a sufficient number of common elements to be present. In the eyes of the editors, this book will be considered a success if it can convince its readers of the following: that it is warranted to dream of a realistic and full-fledged theory of mathematical practices, in the plural. If such a theory is possible, it would mean that a number of presently existing fierce oppositions between philosophers, sociologists, educators, and other parties involved, are in fact illusory. (shrink)

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic...

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic...

This is not "another collection of contributions on a traditional subject." Even more than we dared to expect during the preparatory stages, the papers in this volume prove that our thinking about science has taken a new turn and has reached a new stage. The progressive destruction of the received view has been a fascinating and healthy experience. At present, the period of destruction is over. A richer and more equilibrated analysis of a number of problems is possible and is (...) being cru'ried out. In this sense, this book comes right on time. We owe a lot to the scholars of the Kuhnian period. They not only did away with obstacles, but in several respects instigated a shift in attention that changed history and philosophy of science in a irreversible way. A c1earcut example - we borrow it from the paper by Risto Hilpinen - concerns the study of science as a process, Rnd not only as a result. Moreover, they apparently reached several lasting results, e.g., concerning the tremendous impact of theoretical conceptions on empirical data. Apart from baffling people for several decades, this insight rules out an other return to simple-minded empiricism in the future. (shrink)

An international group of distinguished scholars brings a variety of resources to bear on the major issues in the study and teaching of mathematics, and on the problem of understanding mathematics as a cultural and social phenomenon. All are guided by the notion that our understanding of mathematical knowledge must be grounded in and reflect the realities of mathematical practice. Chapters on the philosophy of mathematics illustrate the growing influence of a pragmatic view in a field traditionally dominated by platonic (...) perspectives. In a section on mathematics, politics, and pedagogy, the emphasis is on politics and values in mathematics education. Issues addressed include gender and mathematics, applied mathematics and social concerns, and the reflective and dialogical nature of mathematical knowledge. The concluding section deals with the history and sociology of mathematics, and with mathematics and social change. Contributors include Philip J. Davis, Helga Jungwirth, Nel Noddings, Yehuda Rav, Michael D. Resnik, Ole Skovsmose, and Thomas Tymoczko. (shrink)

For the first time in history, scholars working on language and culture from within an evolutionary epistemological framework, and thereby emphasizing complementary or deviating theories of the Modern Synthesis, were brought together. Of course there have been excellent conferences on Evolutionary Epistemology in the past, as well as numerous conferences on the topics of Language and Culture. However, until now these disciplines had not been brought together into one all-encompassing conference. Moreover, previously there never had been such stress on alternative (...) and complementary theories of the Modern Synthesis. Today we know that natural selection and evolution are far from synonymous and that they do not explain isomorphic phenomena in the world. ‘Taking Darwin seriously’ is the way to go, but today the time has come to take alternative and complementary theories that developed after the Modern Synthesis, equally seriously, and, furthermore, to examine how language and culture can merit from these diverse disciplines. As this volume will make clear, a specific inter- and transdisciplinary approach is one of the next crucial steps that needs to be taken, if we ever want to unravel the secrets of phenomena such as language and culture. (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)

Paraconsistent logic, logic in which inconsistent information does not deliver arbitrary conclusions, is one of the fastest growing areas of logic, with roots in profound philosophical issues, and applications in information processing and philosophy of science. This book contains selected papers presented at the First World Congress on Paraconsistency, held in Ghent in 1997. It contains papers on various aspects of the subject. As such, it should be of interest to all who want to learn what the subject is, and (...) where it is going. (shrink)

In this paper several arguments are discussed and evaluated concerning the possibility of discrete space and time.

A solution of the zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by hermann weyl, The so-Called tile argument. This note shows that, Given a set of reasonable assumptions for a discrete geometry, The weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The pythagorean theorem is shown to hold for arbitrary right triangles.

The heuristics and strategies presented in Lakatos' Proofs and Refutations are well-known. However they hardly present the whole story as many authors have shown. In this paper a recent, rather spectacular, event in the history of mathematics is examined to gather evidence for two new strategies. The first heuristic concerns the expectations mathematicians have that a statement will be proved using given methods. The second heuristic tries to make sense of the mathematicians' notion of the quality of a proof.

It is a generally accepted idea that strict finitism is a rather marginal view within the community of philosophers of mathematics. If one therefore wants to defend such a position (as the present author does), then it is useful to search for as many different arguments as possible in support of strict finitism. Sometimes, as will be the case in this paper, the argument consists of, what one might call, a “rearrangement” of known materials. The novelty lies precisely in the (...) rearrangement, hence on the formal-axiomatic level most of the results presented here are not new. In fact, the basic results are inspired by and based on Mycielski (1981). (shrink)

Most philosophers still tend to believe that mathematics is basically about producing formal proofs. A consequence of this view is that some aspects of mathematical practice are entirely lost from view. My contention is that it is precisely in those aspects that similarities can be found between practices in the exact sciences and in mathematics. Hence, if we are looking for a (more) unified treatment of science and mathematics it is necessary to incorporate these elements into our view of what (...) mathematics is about. As a helpful tool I introduce the notion of a mathematical argument as a more liberalized version of the notion of mathematical proof. (shrink)

The first part of this paper presents asympathetic and critical examination of the approachof Shahid Rahman and Walter Carnielli, as presented intheir paper “The Dialogical Approach toParaconsistency”. In the second part, possibleextensions are presented and evaluated: (a) top-downanalysis of a dialogue situation versus bottom-up, (b)the specific role of ambiguities and how to deal withthem, and (c) the problem of common knowledge andbackground knowledge in dialogues. In the third part,I claim that dialogue logic is the best-suitedinstrument to analyse paradoxes of the (...) Sorites type.All these considerations lead to philosophicallyrelevant observations concerning principles of charityon the one hand, and compactness on the other. (shrink)

The complex story of mathematics in the Tractatus In this paper some thoughts are presented about the treatment of mathematics in the Tractatus Logico-Philosophicus of Ludwig Wittgenstein. After introducing a metaphor for the mathematical ‘building’, we look at the scattered ideas about mathematics in the Tractatus itself. Although the general consensus is that Wittgenstein rejects the entire ‘building’, there are recent insights that suggest that a more coherent view of ‘Tractarian’ mathematics can be presented, if we are willing to leave (...) behind a foundational form of thinking. What this means will be outlined in some detail. The concluding general assessment is that the final word on the status of mathematics in the Tractatus is still pending. (shrink)

In previous papers (see Van Bendegem [1993], [1996], [1998], [2000], [2004], [2005], and jointly with Van Kerkhove [2005]) we have proposed the idea that, if we look at what mathematicians do in their daily work, one will find that conceiving and writing down proofs does not fully capture their activity. In other words, it is of course true that mathematicians spend lots of time proving theorems, but at the same time they also spend lots of time preparing the ground, if (...) you like, to construct a proof. (shrink)

Kurt Gödel’s incompleteness theorems and the limits of knowledge In this paper a presentation is given of Kurt Gödel’s pathbreaking results on the incompleteness of formal arithmetic. Some biographical details are provided but the main focus is on the analysis of the theorems themselves. An intermediate level between informal and formal has been sought that allows the reader to get a sufficient taste of the technicalities involved and not lose sight of the philosophical importance of the results. Connections are established (...) with the work of Alan Turing and Hao Wang to show the present-day relevance of Gödel’s research and how it relates to the limitations of human knowledge, mathematical knowledge in particular. (shrink)

Philosophy as an academic discipline has grown into something highly specific. This raises the question whether alternatives are available within the academic world itself – what I call the Lutheran view – and outside of academia – what I call the Calvinist view. Since I defend the thesis that such alternatives partially exist and as yet non-existent possibilities could in principle be realised, the main question thus becomes what prevents us from acting appropriately. In honour of Paul Smeyers, the (...) fitting metaphor has to be the Wittgensteinian fly-bottle. (shrink)

Is alternative mathematics possible? More specifically,is it possible to imagine that mathematics could havedeveloped in any other than the actual direction? Theanswer defended in this paper is yes, and the proofconsists of a direct demonstration. An alternativemathematics that uses vague concepts and predicatesis outlined, leading up to theorems such as ``Smallnumbers have few prime factors''.

In this chapter, the possibility of experiments in mathematics is examined. A general scheme is proposed as a tool to handle the different forms of experiments that are being used in mathematical practices: computations, “experimental mathematics” as a new research domain in mathematics and computer science, real-world experiments, and thought experiments. In a final section, extensions of the scheme are proposed that further support the conclusion that mathematical experiments are indeed facts and the future.