Dynamics in foundations: what does it mean in the practice of mathematics?

The search for a synthesis between formalism and constructivism, and meditation on Gödel incompleteness, leads in a natural way to conceive mathematics as dynamic and plural, that is the result of a human achievement, rather than static and unique, that is given truth. This foundational attitude, called dynamic constructivism, has been adopted in the actual development of topology and revealed some deep structures that had remained hidden under other views. After motivations for and a brief introduction to dynamic constructivism, an overview is given of the changes it induces in the practice of mathematics and in its foundation, and of the new results it allows to obtain.