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Envisioning Transformations – The Practice of Topology

In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012-2014. Springer International Publishing. pp. 25-50 (2016)

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  1. The role of syntactic representations in set theory.Keith Weber - 2019 - Synthese 198 (Suppl 26):6393-6412.
    In this paper, we explore the role of syntactic representations in set theory. We highlight a common inferential scheme in set theory, which we call the Syntactic Representation Inferential Scheme, in which the set theorist infers information about a concept based on the way that concept can be represented syntactically. However, the actual syntactic representation is only indicated, not explicitly provided. We consider this phenomenon in relation to the derivation indicator position that asserts that the ordinary proofs given in mathematical (...)
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  • Instructions and constructions in set theory proofs.Keith Weber - 2023 - Synthese 202 (2):1-17.
    Traditional models of mathematical proof describe proofs as sequences of assertion where each assertion is a claim about mathematical objects. However, Tanswell observed that in practice, many proofs do not follow these models. Proofs often contain imperatives, and other instructions for the reader to perform mathematical actions. The purpose of this paper is to examine the role of instructions in proofs by systematically analyzing how instructions are used in Kunen’s Set theory: An introduction to independence proofs, a widely used graduate (...)
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  • Rigour and Intuition.Oliver Tatton-Brown - 2019 - Erkenntnis 86 (6):1757-1781.
    This paper sketches an account of the standard of acceptable proof in mathematics—rigour—arguing that the key requirement of rigour in mathematics is that nontrivial inferences be provable in greater detail. This account is contrasted with a recent perspective put forward by De Toffoli and Giardino, who base their claims on a case study of an argument from knot theory. I argue that De Toffoli and Giardino’s conclusions are not supported by the case study they present, which instead is a very (...)
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  • Mathematical Explanation in Practice.Ellen Lehet - 2021 - Axiomathes 31 (5):553-574.
    The connection between understanding and explanation has recently been of interest to philosophers. Inglis and Mejía-Ramos (Synthese, 2019) propose that within mathematics, we should accept a functional account of explanation that characterizes explanations as those things that produce understanding. In this paper, I start with the assumption that this view of mathematical explanation is correct and consider what we can consequently learn about mathematical explanation. I argue that this view of explanation suggests that we should shift the question of explanation (...)
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  • Why ‘scaffolding’ is the wrong metaphor: the cognitive usefulness of mathematical representations.Brendan Larvor - 2020 - Synthese 197 (9):3743-3756.
    The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the (...)
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  • From Euclidean geometry to knots and nets.Brendan Larvor - 2017 - Synthese:1-22.
    This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...)
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  • The material reasoning of folding paper.Michael Friedman & Colin Jakob Rittberg - 2021 - Synthese 198 (S26):6333-6367.
    This paper inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the nineteenth century, sufficiently stable to be considered as a practice. Moreover, we will argue (...)
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  • Induction and explanatory definitions in mathematics.Lehet Ellen - 2019 - Synthese 198 (2):1161-1175.
    In this paper, I argue that there are cases of explanatory induction in mathematics. To do so, I first introduce the notion of explanatory definition in the context of mathematical explanation. A large part of the paper is dedicated to introducing and analyzing this notion of explanatory definition and the role it plays in mathematics. After doing so, I discuss a particular inductive definition in advanced mathematics—CW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ CW}$$\end{document}-complexes—and argue that it is (...)
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  • Reconciling Rigor and Intuition.Silvia De Toffoli - 2020 - Erkenntnis 86 (6):1783-1802.
    Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this article, I (...)
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  • Reliability of mathematical inference.Jeremy Avigad - 2020 - Synthese 198 (8):7377-7399.
    Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of (...)
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  • Are Aesthetic Judgements Purely Aesthetic? Testing the Social Conformity Account.Matthew Inglis & Andrew Aberdein - 2020 - ZDM 52 (6):1127-1136.
    Many of the methods commonly used to research mathematical practice, such as analyses of historical episodes or individual cases, are particularly well-suited to generating causal hypotheses, but less well-suited to testing causal hypotheses. In this paper we reflect on the contribution that the so-called hypothetico-deductive method, with a particular focus on experimental studies, can make to our understanding of mathematical practice. By way of illustration, we report an experiment that investigated how mathematicians attribute aesthetic properties to mathematical proofs. We demonstrate (...)
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