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).

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One of the main aims of this work is to give a detailed presentation

of our program of {\it ultrametric geometrization of cognitive

sciences.}