Jean Paul Van Bendegem

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 from the analogy with the economic success of a business, which involves maximizing return on investment. On the other hand, the article also draws on Le Lionnais and Whitehead (again) in order to show that, whereas the kind of aesthetic delight offered by beautiful proofs is typical for well-established branches of mathematics, a romantic and adventurous spirit that goes beyond the search for classical aesthetic delights is needed when the exploration of new mathematics is at stake. The history of mathematics is not only a story of feelings of beauty invoked by perfect products, but also a survey of sublime periods of creative production. No account of mathematical beauty can be complete if it does not complement the classical product aesthetics with a romantic creation aesthetics. © 2017 Elsevier B.V., All rights reserved.

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 internal to mathematics. Other perspectives are possible, however. Results: The notion of “meaning finitism” is exploited to tie together internal and external directions within mathematical constructivism. The various contributions to this issue support our case in different ways. Constructivist content: Further contributions from a multitude of constructivist directions are needed for the puzzle of an integrative, overarching theory of mathematical practice to be solved.

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.

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 of this contribution indicates the relations and connections between the study of mathematical practices and ‘mainstream’ philosophy of mathematics need not be antagonistic but rather are to be seen as mutually beneficial.

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.

Suppose you attend a seminar where a mathematician presents a proof to some of his colleagues. Suppose further that what he is proving is an important mathematical statement Now the following happens: as the mathematician proceeds, his audience is amazed at first, then becomes angry and finally ends up disturbing the lecture (some walk out, some laugh, …). If in addition, you see that the proof he is presenting is formally speaking (nearly) correct, would you say you are witnessing an extraordinary event in urgent need of explanation? Surely, your answer would be yes. But do events of this type actually occur? The matter of the fact is, yes, they do. This paper presents the details of such an event, suggests a possible explanation and examines its implications for our understanding of mathematical practice.

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

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.