An Inquiry into the Practice of Proving in Low-Dimensional Topology

Toffoli, Silvia De; Giardino, Valeria

doi:10.1007/978-3-319-10434-8_15

Cited by 24 publications

(11 citation statements)

References 16 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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“…One such example is the diagram accompanying the Snake lemma. There are also examples of results in contemporary mathematics that depend on diagrams as shown in, for example (De Toffoli and Giardino, 2015).…”

Section: Macbeth's Diagrammatic Reasoningmentioning

confidence: 99%

Exploring the fruitfulness of diagrams in mathematics

Carter

2017

Synthese

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The paper asks whether diagrams in mathematics are particularly fruitful compared to other types of representations. In order to respond to this question a number of examples of propositions and their proofs are considered. In addition I use part of Peirce's semiotics to characterise different types of signs used in mathematical reasoning, distinguishing between symbolic expressions and 2-dimensional diagrams. As a starting point I examine a proposal by Danielle Macbeth (2014). Macbeth explains how it can be that objects "pop up", e.g., as a consequence of the constructions made in the diagrams of Euclid, that is, why they are fruitful. It turns out, however, that diagrams are not exclusively fruitful in this sense. By analysing the proofs given in the paper I introduce the notion of a 'faithful representation'. A faithful representation represents as either an image (resembling what it stands for) or as a metaphor (sharing some underlying structure). Secondly it represents certain relevant relations (that is, as an iconic diagram in Peirce's terminology). Thirdly manipulations on the representations respect manipulations on the objects they represent, so that new relations may be found. The examples given in the paper illustrate how such representations can be fruitful. These examples include proofs based on both symbolic expressions as well * I especially thank Danielle Macbeth for comments on earlier versions of this paper. Furthermore I wish to thank the anonymous referees provided by Synthese for their helpful comments.1 as diagrams and so it seems diagrams are not special when it comes to fruitfulness. Having said this, I do present two features of diagrams that seem to be unique. One consists of the possibility of exhibiting the type of relation in a diagram -or simply showing that a relation exists -as a contrast to stating in words that it exists. The second is the spatial configurations possible when using diagrams, e.g., allowing to show multiple relations in a single diagram.

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“…We will expand on that and present the case in detail relating it to our general framework. Elsewhere, we have defended an analogous approach to diagrammatic reasoning in mathematics by offering other case studies such as knot theory (De Toffoli and Giardino 2014) and low-dimensional topology (De Toffoli and Giardino 2015).…”

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confidence: 99%