Epistemic roles of mathematical diagrams
Abstract/Contents
- Abstract
- Visual representations of various kinds are ubiquitous in pure and applied mathematics, in the natural and social sciences, as well as in many other human activities. By investigating these visualizations, many questions arise: What are they? How do they function? What are the conditions of their correct use? Why are they, at times, such effective aids to cognition? What type of knowledge can they promote? In this thesis, I focus on diagrams in mathematics, not exclusively in geometry, but in different mathematical domains as well. Despite the extreme variety of mathematical diagrams, it is possible to distill some of their characteristic properties. Diagrams are not usually static bookkeeping devices, but displays for advancing thought in a dynamic way: experts can manipulate diagrams in order to discover and prove mathematical results. In this thesis, I will first provide an epistemological framework for investigating mathematics in a way that appreciates how human agents actually practice it and in particular how they engage with concrete external artifacts. I will show that the normative dimension at play in mathematics is not monolithic, but multi-dimensional, with different objects of normative assessment and different appropriate sorts of evaluation. After this general discussion, I will investigate the importance of mathematical notations and then I will put diagrammatic notations into focus. I will propose a technical definition of mathematical diagrams based on their two-dimensionality and on their operative dimension, which is constituted by the supported manipulations that correspond to mathematical operations. This definition will allow me to unveil the features of diagrams that underwrite the possibility of them entering into the inferential structure of proofs. To exemplify and sustain my theses I will then present three case studies from contemporary mathematics; these examples are representative of different fields (knot theory, homological algebra, and category theory) and will allow me to exhibit the heterogeneity of diagrammatic notations in mathematics. I will show that in all these cases diagrams not only have important heuristic roles, but play a justificatory role as well.
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource. |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2018; ©2018 |
| Publication date | 2018; 2018 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | De Toffoli, Silvia |
|---|---|
| Degree supervisor | Ryckman, Thomas |
| Thesis advisor | Ryckman, Thomas |
| Thesis advisor | Friedman, Michael, 1947- |
| Thesis advisor | Giaquinto, M. (Marcus) |
| Thesis advisor | Lawlor, Krista |
| Degree committee member | Friedman, Michael, 1947- |
| Degree committee member | Giaquinto, M. (Marcus) |
| Degree committee member | Lawlor, Krista |
| Associated with | Stanford University, Department of Philosophy. |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Silvia De Toffoli. |
|---|---|
| Note | Submitted to the Department of Philosophy. |
| Thesis | Thesis Ph.D. Stanford University 2018. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2018 by Silvia De Toffoli
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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