A Course in Arithmetic (Graduate Texts in Mathematics, by Jean Pierre Serre

By Jean Pierre Serre

Jean-Pierre Serre is Professor on the Collège de France. He has written a couple of books, together with "Algebraic teams and sophistication Fields", "Local Fields", "Complex Semisimple Lie Algebras", "Linear Representations of Finite Groups", amassed Papers (3 volumes), and "Trees" released by means of Springer-Verlag.

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Statements like Hardy’s “beauty is the first test: there is no permanent place in the world for ugly mathematics” [33, p. 85], show that there is a connection between judgements of beauty and ugliness. In general, the motivations we have to judge some things as beautiful are related to the motivations we have to judge some other things as ugly. However, although Rota sees beauty as related to enlightenment, he relates mathematical ugliness to an entirely different phenomenon, to non-definitiveness: The lack of beauty in a piece of mathematics is of frequent occurrence, and it is a strong motivation for further mathematical research.

The aesthetic induction induces a bias toward the properties of successful theories: By imagining the aesthetic induction in operation, we can infer how a community’s set of aesthetic preferences among theories will evolve in particular circumstances. A theory that achieves significant empirical success will cause its community’s aesthetic canon to be remodeled to a certain extent, in such a way, that the canon comes to attribute a greater weighting to that theory’s aesthetic properties. The canon will therefore acquire a bias in favor of any future theories that exhibit the aesthetic properties of current successful theories.

This type of inductive projection is particular in that the properties involved in past and future theories are not empirical properties, but aesthetic properties of theories. 5 Aesthetic Induction in Mathematics Although McAllister characterizes the aesthetic induction in terms of empirical adequacy, he argues that a variant of the aesthetic induction influences the development of mathematics [64]. The aesthetic induction operates in mathematics in a fashion similar to how it operates in the empirical sciences: [: : :] evidence that conceptions of mathematical beauty evolve under the influence of the aesthetic induction is provided by the gradual acceptance of new classes of numbers in mathematics, such as negative, irrational, and imaginary numbers.

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