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.

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.

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 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.

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.