Bart Van Kerkhove

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.

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.

Model-Based Reasoning in Mathematical Practice

The nature of mathematical reasoning has been the scope of many discussions in philosophy of mathematics. This chapter addresses how mathematicians engage in specific modeling practices. We show, by making only minor alterations to accounts of scientific modeling, that these are also suitable for analyzing mathematical reasoning. In order to defend such a claim, we take a closer look at three specific cases from diverse mathematical subdisciplines, namely Euclidean geometry, approximation theory, and category theory. These examples also display various levels of abstraction, which makes it possible to show that the use of models occurs at different points in mathematical reasoning. Next, we reflect on how certain steps in our model-based approach could be achieved, connecting it with other philosophical reflections on the nature of mathematical reasoning. In the final part, we discuss a number of specific purposes for which mathematical models can be used in this context. The goal of this chapter is, accordingly, to show that embracing modeling processes as an important part of mathematical practice enables us to gain new insights in the nature of mathematical reasoning.

The “Artificial Mathematician” Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.