Catastrophe and Transformations: «Geometries» of the Mind in the Bion's Transformational Perspective: Considerations in Terms of Bi-logical Epistemology

In 1872, when he was only twenty three, Felix Klein read a paper that was to be remembered in the history of mathematics as «The Erlangen Program».

Years of reflection and accumulated knowledge had finally brought the first systematization of geometric thought as an integrated whole. The term «geometry» thus came to mean the study of those properties of figures that are invariants with respect to a clearly determinated group of transformations(1). In other words on the one hand we have a set of elements: space S, and on the other a group of transformations of the set S. The geometric properties of the subsets of S are those properties that are invariants with respect to any given group of transformations. It follows that what we call the «property» of a figure is simply an invariant with respect to a set of transformations of the space in which the figure is immersed. The geometry thus varies with the transformation group. This simple idea makes it possible to construct different geometries corresponding to