26/09/2018

The crisis of the foundations of mathematics

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

At the beginning of the 20th century, mathematics was going through a serious crisis. In the previous century, the discipline had undergone extraordinary development, becoming more powerful and more abstract. It was necessary to organize and support this knowledge on solid foundations and ensure that it did not contain contradictions.

A good organizational model had already been invented in antiquity by the Hellenistic mathematician Euclid. In his treatise "Elements," he formulated five statements that he considered intuitively evident—the axioms—and showed how the other statements of plane geometry can be deduced from them through rigorous reasoning.

The work of Gauss, Bolyai, Lobachevsky, and Riemann, all in the 19th century, questioned the nature of Euclid's axioms and led to the discovery of non-Euclidean geometries.

But this did not call into question the usefulness of the axiomatic method; it only showed that axioms are not true or false in any physical sense, but merely convenient starting points for developing mathematical theory.

Still in the 19th century, Dedekind and Peano showed that the axiomatic method could also be applied to arithmetic. At the end of that century, Cantor developed set theory, which seemed to be the best foundation for all of mathematics.

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