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| Author(s): |
Tamires Dal Magro
Total Authors: 1
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| Document type: | Doctoral Thesis |
| Press: | Campinas, SP. |
| Institution: | Universidade Estadual de Campinas (UNICAMP). Instituto de Filosofia e Ciências Humanas |
| Defense date: | 2019-06-13 |
| Examining board members: |
Marco Ruffino;
José Manuel Ferreirós Domínguez;
María de Paz Amérigo;
Abel Lassalle Casanave;
Emiliano Boccardi
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| Advisor: | Marco Ruffino |
| Abstract | |
The main purpose of this thesis is investigating the role of diagrams in the achievement of the solid and lasting results of Euclidean geometry. For that purpose, the work is structured in the form of three academic papers. Each paper assesses different questions surrounding the main topic: (i) which epistemic roles diagrams can have in distinct mathematical practices; (ii) which abilities are employed in the regimented use of diagrams in Euclidean geometry; (iii) how particular diagrams can be employed in the justification of general propositions; (iv) how diagrams can be employed in proofs by reductio ad absurdum; (v) what is the nature of the kind of diagrammatic representation present in the Elements. The first paper that composes the thesis focuses on the analysis of the epistemic role of diagrams in distinct mathematical practices, using as case study the uses of diagrams as reasoning tools in two mathematical practices from antiquity: in the Greek treatise Elements and in the Chinese works Zhou Bi and Nine chapters of mathematical procedures. The second paper argues against the claim that the employment of diagrams in Euclidean geometry gives rise to gaps in the proofs. For this purpose, the paper is structured around three main steps: (1) arguing that it is misleading to evaluate the merits of the geometry presented in the Elements through the lenses of Hilbert¿s formal reconstruction; (2) elucidating the abilities employed in diagram-based inferences in the Elements and showing that, in this context, diagrams are mathematically reputable tools; (3) finally, reviewing recent experimental results purporting to show that, not only is the Euclidean diagram-based practice strictly regimented, it is rooted in cognitive abilities that are universally shared. The third paper is centered on a defense of a theory of the representational role of Euclidean diagrams according to which they are samples of co-exact properties (the `theory of samples¿). This theory is contrasted with two other conceptions ¿ the instantial conception and Macbeth¿s iconic view ¿ with respect to how well they accommodate three fundamental features of the role that diagrams play in the Euclidean mathematical practice: that they are used in proofs whose results are wholly general; that they exhibit the features that the geometer is allowed to infer from them; and that they are ever only used as a source of a specific type of (co-exact) information, something that clears up how they are employed in proofs by reductio. It is argued that the theory of samples is better suited to account for them in comparison with the two other competing views. The paper concludes with an illustration of the virtues of the theory of samples by means of an analysis of Saccheri¿s quadrilateral. The final section of the thesis relates the results achieved in the three papers and presents suggestions for future investigations (AU) | |
| FAPESP's process: | 14/23191-9 - Formal and diagrammatic reasoning in euclidian proofs by reduction ad absurdum |
| Grantee: | Tamires Dal Magro |
| Support Opportunities: | Scholarships in Brazil - Doctorate |