Ways of thinking in mathematics
Abstract
All mathematicians, whether they are engaged in teaching, in researching or in both at the same time, must master specific ways of thinking, of reasoning. These are ways of thinking that can be divided into three categories: reasoning and problem-solving; organizing already acquired knowledge; and characterizing, constructing or defining new elements and concepts. These are the modes that are presented in this essay, with a very brief indication of the historical moment in which they are constructed. The obligatory mastery of these modes of thinking, by itself, does not lead to mathematical creation; the imaginative and creative power of the mathematician is involved in this. Finally, we point out the appearance of an ontological problematic from the inversion that took place in the 19th century: are mathematical entities discovered or constructed? It is a problematic that has become a theme throughout the twentieth century about mathematical doing.
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