Villeneuve, Jean-Philippe (Author)

Université de Montréal (Canada)

Publication Date: 2008

Physical Details: 215 pp.

Language: French

Physical Details: 215 pp.

Language: French

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We offer a study on the process of generalization, not of statements but of mathematical notions. We will first consider research on that process by two mathematicians, George Pólya and Saunders Mac Lane, a philosopher, Imre Lakatos and a psychologist, Jean Piaget. That relatively small research corpus represents almost all of the research published on the subject, because little research has been done on generalization. Our analysis enables us to introduce two types of generalization. The first type is the logical generalization or the inductive inference. Initially this process only applies to statements and, because our considerations are mathematical notions, we will develop the extended logical generalization process which will be characterized by a fixed-variable relation between the initial notion and the new notion. The second type is what we will call the "notional" generalization process. This process can be illustrated as follows: the notion of feline is a notional generalization of the notion of cat, because all cats are felines but there are some felines that are not cats. In both processes, we will find that the extension of the initial notion is strictly included in the extension of the new notion. Following that analysis and also to illustrate it, we will produce an historical study of the mathematical notion of integral as developed in the 19th century by Cauchy, Dirichlet, Lipschitz, Riemann, Darboux, Jordan and Lebesgue. This study will enable us to link the generalization process with the notion of change process. To that effect, we will propose two types of change: the new interpretation, when the new notion is defined in the same way of the initial notion, and the reinterpretation, when the initial notion is redefined completely. In the first case, we will find a fix-variable relation between notions and, by the way, a strong link with the logical generalization process. This will not be the case for the second case, because it will be possible to find a notion that can be reinterpreted without being generalized. Note that we will also develop a variant of typed logic to formalize our results. Key Words: Philosophy, Philosophy of Mathematics, Theory of Mathematical Knowledge, Generalization, Integral, History of Mathematical Analysis.

...MoreDescription Cited in *ProQuest Diss. & Thes.* . ProQuest Doc. ID 304805396.

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