Ultrametric thinking and Freud's theory of unconscious mind.

One of the main reasons for extremely successful development of physics during the last three hundreds years was mathematical formalization of this science. This formalization was started with creating an adequate mathematical model of physical space, namely, {\it cartesian product} of real lines. Creation of this model took a few hundreds years and it was based on great contributions both from mathematics and physics. In the mathematical framework the highest point of development was constructing of the field of real numbers ${\bf R}$ and elaboration of the notion of the cartesian product $X_{\rm{phys}}= {\bf R}^3$ representing physical space. This mathematical object became the fruitful model of physical space. The notion of absolute physical space was introduced in physics through efforts of Galilei and Newton. Then through of efforts of Lobachevskii it became clear that the Euclidean geometry is not the unique possible geometry (at least from mathematical viewpoint). Then the great contribution was done by Riemann who introduced a notion of a manifold. The latter became the mathematical basis of Einstein's theory of general relativity. At the beginning of 20th century it became clear that it is impossible to incorporate so called quantum phenomena neither into the three dimensional ``physical space'' $X_{\rm{phys}}= {\bf R}^3$ nor into the four dimensional Minkovsky space-time. There was created a new model of space, {\it quantum Hilbert space} $X_{\rm{quantum}}.$ The main distinguishing mathematical feature of this space is its infinite dimension. It is very important for our further considerations to remark that, nevertheless, the space $X_{\rm{quantum}}$ geometrically does not differ so much from the space $X_{\rm{phys}}= {\bf R}^3.$ If we restrict geometry of $X_{\rm{quantum}}$ onto one of its finite dimensional subspaces we obtain again the Euclidean geometry (and in the relativistic framework the pseudo-Euclidean one). Thus even the transition from classical to quantum physics did not induce something new with respect to number structure of space and its geometry. One could say that during a few hundreds years physicists have been exploring more or less the same class of mathematical models of space. I believe that in cognitive sciences there should be used the same strategy of geometrization as in physics. And the starting point of such a geometric development of cognitive sciences should be creation of an adequate notion of {\it mental space} $X_{\rm{mental}}.$ First steps in realization of this program were done in \cite{KH1}--\cite{KH12}. The crucial point is that it seems that the real space model that was applied so successfully in physics is not adequate for mental phenomena. There should be found new mathematical models of space that would be more adequate for mental processes. In \cite{KH1}--\cite{KH12} we proposed to consider mental spaces having {\it treelike} structure. Such spaces have natural {\it ultrametric geometries} (which differs crucially from Euclidean or pseudo-Euclidean geometries used in physical theories). \medskip One of the main aims of this work is to give a detailed presentation of our program of {\it ultrametric geometrization of cognitive sciences.}