As mathematics grows and changes, geometry changes. A more current example. Until the mid-twentieth century, the “derivative” or “slope” of a function at a point existed only if at that point the graph of the function was smooth—had a definite direction, and no jumps. Now mathematicians have adopted Laurent Schwartz’s generalized functions. Every function, no matter how rough, has a derivative.
**
The meaning of differentiation has changed. Newton and Leibniz’s differentiation operator has become something more general. Our generalized differentiation includes the old differentiation, and it’s much more powerful.
As mathematics grows and changes, functions and operators change.
A familiar cliché says that while other sciences throw away old theories, mathematics throws away nothing. But the old mathematics isn’t preserved intact. Mathematics is intensely interconnected and self-interactive. The new is vitally linked to the old. The old is revitalized, enriched, and complexified by interaction with the new.
