Why was the Scientific Revolution of the 17th century so successful for physics, but failed in biology? An easy answer is that the physical laws discovered by Kepler, Galilei and Newton could be expressed in the mathematical language of the calculus of Newton and Leibniz, but that biological structures are not readily amenable to a mathematical formalization. Mathematics became the guiding science of the Enlightenment, and its role was that of a tool for exploring the theoretical framework provided by physics. Other sciences then attempted to imitate that role model of physics, with varying success.
Today, the situation is different. Mathematics is typically no longer confronted with a clear theoretical framework provided by another science, but rather with data that may possess rather intricate structure, but that have been collected by advanced technical devices without such a theoretical framework within which they can be interpreted and understood. Such wild data collection is sometimes decried as agnostic or even unscientific. I shall take a more positive attitude and try to face the new challenges and opportunities for an abstract data science, which no longer is the formal tool of science, but becomes the science of formal tools. In particular, I shall try to identify some of the emerging new mathematical problems and concepts.
Examples will be taken not only from physics and biology, but also from many other fields, like the social sciences or linguistics. Therefore, this will be a very interdisciplinary talk.