Jean Paul Van Bendegem

Vrije Universiteit Brussel | VUB · Department of Philosophy

As knowledge can be condensed in different non-verbal ways of representation, the integration of graphic and visual representations and design in research output helps to expand insight and understanding. Layers of visual charts, maps, diagrams not only aim at synergizing the complexity of a topic with visual simplicity, but also to guide a persona...

How is science performative and how do the arts make science perform? Performance scholar Bleeker and mathematician and philosopher of science Van Bendegem explore various examples of scientific processes that expose performative mechanisms resulting from the entanglement of technological tools and human agency. A dynamic that characterizes scienti...

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 so...

This paper investigates the impact of publication pressure on the ethics and the scientific integrity in the domain of mathematics and of the arts. Both research fields are specific in their methodology, being that they do not start from a classical hypothesis and researchers in these areas are not knowing what the outcome will be. The research des...

1. WHAT TO EXPECT (OR NOT) The study of the combination of mathematics and the arts is not a novel one, witness the simple fact that an elementary search on Google Scholar, using the keywords ‘mathematics’ and ‘art’, produced more than three million hits. Rather striking is the fact that the heterogeneity of this sample is quite high. There are boo...

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...

We invite readers for discussion. Suggestions and questions are very welcome.

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 (with or without support from the inside) – what I call the Calvinist view. Since I defend the thesis that such alternat...

In the first part an outline is presented of the emergent new field of the study and the philosophy of mathematical practices, including (the philosophy of) mathematics education. In the second part the focus is on particular themes within this field that correspond more or less to my personal contributions over a thirty-year period. As the title o...

The development of the philosophy of science in the twentieth century has created a framework where issues concerning funding dynamics can be easily accommodated. It combines the historical-philosophical approach of Thomas Kuhn (The structure of scientific revolutions (2nd ed., enlarged). The University of Chicago Press, Chicago, [1962] (1970)) wit...

This chapter looks at the impact of recent societal approaches of knowledge and science from the perspectives of two rather distant educational domains, mathematics and music. Science’s attempt at ‘self-understanding’ has led to a set of control mechanisms, either generating ‘closure’—the scientists’ non-involvement in society—or ‘economisation’, p...

This article focuses on the writings of Hardy, Poincaré, Birkhoff, and Whitehead, in order to substantiate the claim that mathematicians experience a mathematical proof as beautiful when it offers a maximum of insight while demanding a minimum of effort. In other words, it claims that the study of the aesthetic success of theorem-proofs can benefit...

This survey provides a brief and selective overview of research in the philosophy of mathematics education. It asks what makes up the philosophy of mathematics education, what it means, what questions it asks and answers, and what is its overall importance and use? It provides overviews of critical mathematics education, and the most relevant moder...

To understand present-day research in logic and argumentation theory in Belgium, it is necessary to highlight the unique contribution of Leo Apostel. The first part of the paper deals with his main intellectual influences, namely Chaïm Perelman, Rudolf Carnap, and Jean Piaget. In the second part the Signific Movement and the Erlangen School are dis...

This article discusses the concept of mathematics discourses by analysing two main questions. The first question is about the plurality of mathematics and the possibility of the simultaneous existence of culturally different mathematics. The second question is about the respective value of the different mathematics and its means of power in terms o...

The core thesis of this contribution is that, if we wish to construct formal-logical models of mathematical practices, taking into account the maximum of detail, then it is a wise strategy to see mathematics as a heterogeneous entity. This thesis is supported by two case studies: the first one concerns a mathematical puzzle, the second one concerns...

This survey provides a brief and selective overview of research in the philosophy of mathematics education. It asks what makes up the philosophy of mathematics education, what it means, what questions it asks and answers, and what is its overall importance and use? It provides overviews of critical mathematics education, and the most relevant moder...

In this chapter we explore how mathematics education is caught by a meritocratic sense of the useful and how it could benefit from a more creative and experiential approach. The notion of olympification in mathematics education comes to the fore in the analysis of the differences between the measurements of PISA and TIMSS, further detailed by an ex...

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient mo...

This paper is a plea for a less monolithic interpretation of the (mathematical) style concept, in order for it to serve well as a methodological tool in the historiography of mathematics. Drawing inspiration from Le Lionnais, the Bourbaki movement and the French literary-mathematical OuLiPo movement, we introduce an approach along the path of a ‘pr...

In our rapidly globalizing world, continuous readjustment of the scientific basis of sustainable development (SD) is a prerequisite for sustainability. We shed light on the shift in international discourse concerning culural diversity and SD. We analyse worldviews as a constitutive element of SD, proposing to re-interpret SD as a joint worldview-co...

It is a rather safe statement to claim that the social dimensions of the scientific process are accepted in a fair share of studies in the philosophy of science. It is a somewhat safe statement to claim that the social dimensions are now seen as an essential element in the understanding of what human cognition is and how it functions. But it would...

Because the conclusion of a correct proof follows by necessity from its premises, and is thus independent of the mathematician’s beliefs about that conclusion, understanding how different pieces of mathematical knowledge can be distributed within a larger community is rarely considered an issue in the epistemology of mathematical proofs. In the pre...

The chapter by Karen François, Kathleen Coessens and Jean Paul Van Bendegem turns to ‘The Spaces of Mathematics: Dynamic Encounters Between Local and Universal’ (Chap. 10). No doubt mathematics is the last place (or face?) to look for situatedness, that is, to show that mathematics too is linked to places, to people, to instruments and to practices...

Although Victoria Lady Welby did not write that much explicitly about mathematics, what she did write shows a very strong connection with Gerrit Mannoury's ideas about mathematics and mathematicians. These views, however, underwent dramatic changes in the hands of L. E. J. Brouwer. We claim that today traces of Welby's and Mannoury's legacy are to...

It is a rather safe statement to claim that the social dimensions of the scientific process are accepted in a fair share of studies in the philosophy of science. It is a somewhat safe statement to claim that the social dimensions are now seen as an essential element in the understanding of what human cognition is and how it functions. It would be a...

Context • As one of the major approaches within the philosophy of mathematics, constructivism is to be contrasted with realist approaches such as Platonism in that it takes human mental activity as the basis of mathematical content. 〉 Problem • Mathematical constructivism is mostly identified as one of the so-called foundationalist accounts interna...

〉 Context • Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. 〉 Problem • First, it is necessary to present a "decent" history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict f...

We prove an omitting types theorem and one direction of the related Ryll-Nardzewski theorem for semi-classical theories introduced in [2].

This chapter examines the possibility of discrete time. At first sight, the answer seems trivial, but actually, it raises a number of interesting questions, both philosophical and scientific. First, the chapter explains what interpretations of discrete time are not considered. Then, it addresses two key philosophical problems: if there are such thi...

‘Working’ mathematicians will claim without any hesitation that the above figure and formula represent the same thing, namely a normal distribution or Gauss bell-curve. The label ‘working’ is apposite here, as some more philosophically minded mathematicians will insist that the formula is the Gauss curve, whereas the drawing is a representation of...

In the paper it is shown that every physically sound Birkhoff --- von Neumann quantum logic, i.e., an orthomodular partially ordered set with an ordering set of probability measures can be treated as partial infinite-valued ¿ukasiewicz logic, which unifies ...

Both rhetoric and mathematics are ancient, elaborate and still active fields of study that cover a time span of more than two millennia. That much, they undisputedly have in common. However, in the domain of mathematics one will search in vain for traces, positive or negative, of rhetoric, and in the domain of rhetoric, although the relation betwee...

Research is as diverse as the different contextual, paradigmatic, social, and ideological perspectives that inform it. Opinions, expectations, and approaches are at the same time ‘cultivated’ and ‘bound’ by their embeddedness in specific traditions/ cultures and current economic, social, political, and educational developments (Crossley, 2005, p. 3...

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 in...

We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that als...

New developments are setting new challenges. Preceding societal movements, which include the ‘labour society’ or ‘work culture’, which were brought into being by the coterminous development of the economy and new technologies, have paved the way for further societal trends. Such trends have been named the ‘network society’, ‘information age’ or ‘kn...

Prelude.- Prelude.- Interlude.- The Untouchable and Frightening Status of Mathematics.- Interlude.- Philosophical Reflections in Mathematics Classrooms.- Interlude.- Integrating the Philosophy of Mathematics in Teacher Training Courses.- Interlude.- Learning Concepts Through the History of Mathematics.- Interlude.- The Meaning and Understanding of...

Without Abstract

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. H...

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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 ma...

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 (1984). Such a particular tool enables one to retain an integrated picture while nevertheless welcoming an...

It is a trivial remark that to discuss the philosophy of any topic, one must have at least a good understanding of the topic itself in order to raise philosophical problems about it. However, if the topic happens to be mathematics, this does not seem to be the case. Philosophers are not particularly interested in mathematical practice itself. Often...

The aim of the series Logic, Epistemology, and the Unity of Science, of which this is the first volume, is to take up anew the challenge of considering the scientific enterprise in its entirety in light of recent developments in logic and philosophy. Developments in logic are especially relevant to the current situation in philosophy of science. At...

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 th...

The dialogical approach to paraconsistency as developed by Rahman and Camielli ([1]), Rahman and Roetti ([2]) andRahman ([3], [4] and [5]) suggests a way ofstudying the dynamic process ofarguing with inconsistencies.In his paper on Paraconsistency and Dialogue Logic ([6]) Van Bendegem suggests that an adaptive version of paraconsistency is the natu...

This symposium is designed to stimulate discussions at the intersection of the sociology of mathematics, the philosophy of mathematics, and mathematics education. Our objective is to provide a forum that will focus the theory and research in sociology and philosophy of mathematics on practical issues facing classroom teachers of mathematics. If aud...

In this paper I will not confine myself exclusively to historical considerations. Both philosophical and technical matters will be raised, all with the purpose of trying to understand (better) what Newton, Leibniz and the many precursors (might have) meant when they talked about infinitesimals. The technical part will consist of an analysis why app...

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 rol...

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 s...

It may perhaps sound strange if not bizarre to suggest that metaphors and analogies could and should play a role in the practice of mathematics, let alone to claim that they are essential in present-day mathematics. Yet, that will be precisely the claim I will defend in this paper. I do insist that present-day mathematics is the domain of investiga...

How do scientists approach science? Scientists, sociologists and philosophers were asked to write on this intriguing problem and to display their results at the International Congress `Einstein Meets Magritte'. The outcome of their effort can be found in this rather unique book, presenting all kinds of different views on science. Quantum mechanics...