2  SYMMETRY AS COGNITION: MATHEMATICS

Symmetries can be formulated in mathematical terms, and mathematics from its very beginning counted as a field of cognition that is expected to render a special kind of truth of exceptional certainty. Not only does this truth by far exceed the only „dependent“ truth of empirical knowledge, it even exceeds the limits of the - otherwise believed as unlimited - will power of God himself: Mathematic is believed to constitute a field of ideas independent from nature, mankind and God, a reigndom of pure and autonomous rationality that governs them all. The problem with this free and independent reason of course is its ontological status: is it an ordinary entity, or a transcendent idea, or identical with God, or a „free creation of the human mind“? - the diversity of possible answers, puristic, „fundamentalistic“, or synchretistic, constitute the history of philosophy and human selfunderstanding. We must confine ourselves to some aspects of the role symmetry plays in that global performance of mankind.

 Symmetry, when defined as „invariance under transformation“ (J.Rosen), holds an exceptional place as an intersection point of general laws of both, formal and empirical character:

-- Rosen demonstrated the identity of symmetry with „reproducibility“ and „predictability“, thus  with the basic, even defining cornerstones of scientific methodology at all;

-- Noether proved the equivalence of symmetries and conservational laws in physics;

-- Klein relates symmetry (as a formal equivalent of conservation) to the Kantian category of substance/accidence, the latter in relation to causality and interaction, thus to the general constituents of cognitive object formation;

-- von Weizsäcker postulates the identity of these forms of object formation with the basic formal laws of symmetries in contemporary (particle-) physics.