Envisioning Transformations—The Practice of Topology

Toffoli, Silvia De; Giardino, Valeria

doi:10.1007/978-3-319-28582-5_3

Cited by 14 publications

(7 citation statements)

References 27 publications

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“…They and other authors have therefore begun to explore the broader epistemic benefits of informal proof [4,5,42,55,62]. It is even reasonable to maintain that philosophy of mathematics has suffered from excessive fixation on proof itself, and a number of philosophers of mathematics are turning their attention to aspects of mathematical understanding that extend well beyond justification and correctness [3,20,21,22,23,27,36,37,38,46,49,50,57,58,59,65].…”

Section: Discussionmentioning

confidence: 99%

“…Visualize. Examples of visual intuition in mathematics are compelling, and of all the strategies described here, this one has received the most philosophical attention in recent years [20,21,22,23,30,37,38,50,63,65]. Jill Larkin and Herbert Simon famously argued that diagrammatic representations leverage our spatial cognitive capacities by using features and relationships in the diagram to make relevant information salient [41].…”

Section: Specific Solutionsmentioning

confidence: 99%

“…Features absent from a diagram, or that can be eliminated by varying the diagram in a way that preserves the constraints, give rise to counterexamples. Various authors have explored ways that this plays out in mathematical reasoning, including geometry, topology, and algebra [7,20,21,22,23,30,47,50]. In the case of Euclidean geometry, Hamami, Mumma, and Amalric [38] provide experimental evidence to support such a cognitive model.…”

Section: Specific Solutionsmentioning

confidence: 99%

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Reliability of mathematical inference

Avigad

2020

Synthese

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Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. This is also a demand that is especially hard to fulfill, given the fragility and complexity of mathematical proof. This essay considers some of ways that mathematics supports reliable assessment, which is necessary to maintain the coherence and stability of the practice. * I am grateful to Silvia De Toffoli and Yacin Hamami for helpful comments and suggestions.

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Section: Discussionmentioning

confidence: 99%

Section: Specific Solutionsmentioning

confidence: 99%

Section: Specific Solutionsmentioning

confidence: 99%

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Reliability of mathematical inference

Avigad

2020

Synthese

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“…An indication of this diversity can be seen by studying the contents of Larvor's (2016) recent edited volume. The contributors to that collection drew conclusions about mathematical practice by studying historical episodes (e.g., Barany, 2016), by interviewing mathematicians (e.g., Johansen & Misfeldt, 2016), by analysing cultural artefacts (e.g., Pantsar, 2016) and by conducting detailed case studies of a particular mathematical notation (e.g., De Toffoli & Giardino, 2016). All these approaches are well-suited for generating insights about mathematical practice but are not always useful strategies if one wishes to test hypotheses that already exist, especially if those hypotheses involve causal claims.…”

Section: The Hypothetico-deductive Methods In the Study Of Mathematicamentioning

confidence: 99%

Are aesthetic judgements purely aesthetic? Testing the social conformity account

Inglis

Aberdein

2020

ZDM Mathematics Education

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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 that perceptions of the aesthetic properties of mathematical proofs are, in some cases at least, subject to social influence. Specifically, we show that mathematicians' aesthetic judgements tend to conform to the judgements made by others. Pedagogical implications are discussed.

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“…Movies are another option. Some of the illustrations in De Toffoli & Giardino (2016) are stills from an animation by Ester Dalvit.…”

Section: " (P 40) It Relies On What They Call a "Cognitive Capacitymentioning

confidence: 99%