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### 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.*M*duality duality (^{8}-H*H*=*M*x^{4}*CP*_{2}) relates the number vision of TGD to the geometric vision.*M*duality maps the 4-surfaces in^{8}-H*M*to space-time surfaces in^{8}-c*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*M*that allow cognitive explosion in the number-theoretically preferred coordinates.^{8}-c*M*and hyperbolic spaces^{4}*H*^{3}(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*M*octonionic structure allows to identify natural preferred coordinates. In^{8}-c*H*, in particular*M*, 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.^{4}