Birational Maps as Morphisms of Cognitive Structures

Abstract

Birational maps and their inverses are defined in terms of rational functions. They are very special in the sense that they map algebraic numbers in a given extension E of rationals to E itself. In the TGD framework, E defines a unique discretization of the space-time surface if the preferred coordinates of the allowed points belong to E. I refer to this discretization as cognitive representation. Birational maps map points in E to points in E so that they define what might be called cognitive morphism. M⁸-H duality duality (H = M⁴ x CP₂) relates the number vision of TGD to the geometric vision. M⁸-H duality maps  the 4-surfaces in M⁸-c to space-time surfaces in H: a natural condition is that in some sense it maps E to E and cognitive representations to cognitive representations. There are special surfaces in M⁸-c that allow cognitive explosion in the number-theoretically preferred coordinates. M⁴ and hyperbolic spaces H³ (mass shells), which contain 3-surfaces defining holographic data, are examples of these surfaces. Also the 3-D light-like partonic orbits defining holographic data. Possibly also string world sheets define holographic data. Does cognitive explosion happen also in these cases? In M⁸-c octonionic structure allows to identify natural preferred coordinates. In H, in particular M⁴, the preferred coordinates are not so unique but should be related by birational mappings. So called Hamilton-Jacobi structures define candidates for preferred coordinates: could different Hamilton-Jacobi structures relate to the each other by  birational maps? In this article these questions  are discussed.