These papers resulted from the first workshop in a series exploring concepts that are of central importance to all of the sciences. The series will bring together the perspectives of natural and social sciences and the humanities in intimate, cross-disciplinary dialogue. The intention is to improve our understanding of the concepts discussed by emphasizing their different uses in the many different disciplines, while at the same time focusing on shared concerns and structural commonalities. And this is the peculiarity of the concepts to be examined here – they do not get defined in just one discipline, so that they only form part of the basic concepts of that discipline, as distinguished from the basic concepts of other disciplines. Rather, their unique character derives from the fact that they have different uses in different disciplines, and all of them in the foundations of the respective sciences. The concepts of natural law, causality, of complexity and truth, scientific truth, are of this kind, and so is the concept of symmetry, with which we will start this series.

As we all know, this concept has many different (scientific) meanings, while the basic meaning remains stable. In geometry, for instance, mirroring, rotation and periodicity prove to be symmetry-properties, that is, self-copies (automorphisms) of spaces, which leave the structure of figures and bodies unchanged (invariant). In natural philosophy, symmetry is the property of natural objects to remain invariant under certain operations, namely symmetry-operations. In logic, a two-place relation R is symmetric on a set S, if for any two elements x and y from that set, the following is the case: if the relation R holds between x and y, then it also holds between y and x.