Proofs from THE BOOK, by Martin Aigner and Giinter M. Ziegler. Pp.199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag). I would like to begin with the authors' words from the Preface of this book: Paul Erd6s liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. ErdSs also said that 'you need not believe in God but, as a mathematician, you should believe in The Book'. This excellent book is dedicated to the memory of Paul ErdSs. Paul himself wrote several pages but he is not listed as a co-author because of his death in the summer of 1996. The idea of the book is to present beautiful proofs of some of the most exciting and historically important mathematical theorems. The book focuses on human endeavour to make the proofs more and more simple and striking.

Proofs from THE BOOK , by Martin Aigner and Günter M. Ziegler. Pp. 199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag).

Mathematical Practices Can Be Metaphysically Laden

https://dialnet.unirioja.es/descarga/articulo/4378585.pdf

"Iterated Cohen extensions and Souslin's problem", publicado en Annals of mathematics, de R. M. Solovay y S. Tennenbaum

In this sequence of philosophical essays about natural science, Nancy Cartwright argues that fundamental explanatory laws, the deepest and most admired successes of modern physics, do not in fact describe the regularities that exist in nature. Yet she is not 'anti-realist'. Rather, she draws a novel distinction, arguing that theoretical entities, and the complex and localized laws that describe them, can be interpreted realistically, but that the simple unifying laws of basic theory cannot.

How the Laws of Physics Lie.

In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections.

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Intellectual humility in mathematics

There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations. Our strategy is to frame current narratives of mathematics from a virtue-theoretic perspective. We identify the practice of mathematizing, put forward by Freudenthal’s ‘Realistic mathematics education’, as virtuous and use it to evaluate different narratives. We show that this can help to render the narratives more adequately, and to provide implications for societal organization.

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Mathematizing as a virtuous practice: different narratives and their consequences for mathematics education and society

/pdf/reflections-on-the-foundations-of-mathematics-univalent-xmtn4jvbao.pdf

Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts

The status of independent statements is the main problem in the philosophy of set theory. We address this problem by presenting the perspective of a practising set theorist. We thus give an authentic insight in the current state of thinking in set-theoretic practice, which is to a large extent determined by independence results. During several meetings, the second author asked the first author about the development of forcing, the use of new axioms and set-theoretic intuition on independence. Parts of these conversations are directly presented in this article. They are supplemented by important mathematical results as well as discussion sections. Finally, we present three hypotheses about set-theoretic practice: First, that most set theorists were surprised by the introduction of the forcing method. Second, that most set theorists think that forcing is a natural part of contemporary set theory. Third, that most set theorists prefer an answer to a problem with the help of a new axiom of lowest possible consistency strength, and that for most set theorists, a difference in consistency strength weighs much more than the difference between Forcing Axiom and Large Cardinal Axiom.

Deborah Kant

Papers

Mathematizing as a virtuous practice: different narratives and their consequences for mathematics education and society

Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts

Interview With a Set Theorist

Three Roles of Empirical Information in Philosophy: Intuitions on Mathematics do Not Come for Free