Jean Paul Van Bendegem

Philosophical Dimensions in Mathematics Education

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 Mathematics.- Interlude.- The Formalist Mathematical Tradition as an Obstacle to Stochastical Reasoning.- Interlude.- Logic and Intuition in Mathematics and Mathematical Education.- Interlude.- A Place for Education in the Contemporary Philosophy of Mathematics.- Interlude.- Ethnomathematics in Practice.- Postlude.

Pi on Earth, or Mathematics in the Real World

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 also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one’s means/ends ratio. Our story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like.

Mathematical Arguments and Distributed Knowledge

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 present chapter, we set out to question the received view expressed by the previous sentence. To that end, we study a prime example of collaborative mathematics, namely the Polymath Project, and propose a simple formal model based on epistemic logics to bring out some of the core features of this case-study.

Analogy and Metaphor as Essential Tools for the Working Mathematician

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 investigation I have in mind. From a historical perspective, and as an example, no one seems to doubt that mathematics one way or another must have arisen out of a specific set of practices, usually referred to as counting and measuring, and that mathematical concepts are metaphorically related to these practices. That view of the matter, however, does not necessarily say anything about the situation today and, indeed, in most cases it does not.

Fermat’s Last Theorem Seen as an Exercise in Evolutionary Epistemology

Out understanding of the mathematical process has been and still is (rightly so) associated with Lakatos’ Proofs and Refutations. But at first sight, the results of his research have little or nothing to do with evolutionary epistemology. The scheme he arrives at, is roughly this:

Olympification Versus Aesthetization: The Appeal of Mathematics Outside the Classroom

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 example of Flanders (Belgium). Besides the observation of the olympification we consider the possibility of another perspective on mathematics education, looking at a way of bringing classroom mathematics in interaction with the material grounding of mathematics and with other experiences in life. Based on the content analysis of eight international journals concerning mathematical education we demonstrate the extent in which teachers and researchers take care of outside classroom experiences as possible input for a mathematical curiosity and understanding. Focusing on the relation between mathematics and art we will shortly explore different examples of mathematics within the arts. Finally we bring an example of how a mathematician can creatively bring mathematics outside the classroom.

Another Look at Mathematical Style, as Inspired by Le Lionnais and the OuLiPo

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 ‘problem solving’ conception of mathematics, thus creating room for mathematical style to be significantly ‘more’ than a mere mode of presentation of immutable content. In our view, this very same approach opens us the possibility for a fruitful comparison between mathematical and literary styles.

Arguments and Proofs About Arguments and Proofs

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 between mathematics and rhetoric is often discussed, the standard claim is to deny that they are intimately related or intertwined. Moreover, things have hardly changed over two millennia. Let us present two examples, which I freely admit, present a slight bias. The first example is taken from what is commonly referred to as ‘old rhetoric’ and the second, from ‘new rhetoric’.

Epistemic Injustice in Mathematics

AbstractWe 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.

Math and Music: Slow and Not For Profit

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’, producing patents and other lucrative benefits. While scientometrics became the tool and the rule for measuring the economic impact of science, counter movements, like the slow science movement, citizen science, empowering music-art initiatives and other critical approaches focus on intrinsic and ethical questions of education and knowledge. Thinking about knowledge and research in terms of quantifiable products impacts heavily upon the domains of science and arts, while the complexity of knowledge acquisition forces society to consider also other parameters like equality, personal development and participatory processes.