Vertical line and point symmetries of differentiable functions

Abstract

A heuristic account is given of the author's personal investigation of some aspects of the vertical line and point symmetries of differentiable functions. Starting from examples of third degree polynomials there is first generalized to polynomials in general, and later even more generally to differentiable functions. The role of quasi‐empirical testing is demonstrated throughout, not only in the gaining of confidence in conjectures, but also in improving them through the production of counter‐examples. In some examples the role of deductive proof is also clearly shown to be far less that of verification than that of explanation, systematization and/or discovery. Symmetry, as wide or as narrow as you may define it, is one idea by which man through the ages has tried to comprehend, and create order, beauty and perfection. (Hermann Weyl).