From the Crisis of Foundations and the Loss of Certainties to a New Open Horizon

In the last decades of the nineteenth century, the aim of giving mathematics, starting from arithmetic, certain foundations inspired a formidable research effort. A hard-fought competition developed between the principal schools of foundamentalism: set theory, logicism, formalism and intuitionism. The great effort proved unsuccessful. The failure of attempts to provide a rigorous foundation for mathematical propositions caused the well-known crisis of foundations, which expanded to include philosophy, physics and in general the trust in the capacity of knowledge to achieve definitive certainties. Mathematicians were not too upset by the failure of the various schools. They rightly continued to dedicate themselves to mathematical practice. Once the storm had passed, the crisis of the foundations and the consequent loss of certainties turned into an extraordinary open horizon, first of all in the borderlands between mathematics, logic, physics and philosophy. We are still working in this new frontier.