BioSystems 56 2000 95 – 120
Classical and quantum dynamics on p-adic trees of ideas
Andrei Khrennikov
Department of Mathematics, Statistics and Computer Sciences, Uni6ersity of Va¨xo¨, S-
35195
Va¨xo¨, Sweden Received 11 June 1999; received in revised form 27 December 1999; accepted 17 January 2000
Abstract
We propose mathematical models of information processes of unconscious and conscious thinking based on p-adic number representation of mental spaces. Unconscious thinking is described by classical cognitive mechanics which
generalizes Newton’s mechanics. Conscious thinking is described by quantum cognitive mechanics which generalizes the pilot wave model of quantum mechanics. The information state and motivation of a conscious cognitive system
evolve under the action of classical information forces and a new quantum information force, namely, conscious force. Our model might provide mathematical foundations for some cognitive and psychological phenomena:
collective conscious behavior, connection between physiological and mental processes in a biological organism, Freud’s psychoanalysis, hypnotism, homeopathy. It may be used as the basis of a model of conscious evolution of life.
© 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords
:
p-Adic number representation; Classical information force; Quantum information force; Conscious evolution of life www.elsevier.comlocatebiosystems
1. Introduction
It seems that the modern physics can in princi- ple explain or at least describe all phenomena
which are observed in reality: motion of classical and quantum systems, classical and quantum
fields, …, physiological processes in biological or- ganisms. This incredible power of physics induced
the common opinion that all biological processes could be reduced to some physical processes. This
concerns not only primary physiological processes in biological organisms such as, for example, the
functioning of the blood system, but even biologi- cal processes of the highest level of complexity,
namely, cognitive processes. The idea that by studying physiological processes in the brain we
could explain probably after many years of inten- sive research the functioning of the brain quickly
propagates throughout the biological community see, for example, Skinner, 1953; Lorenz, 1966;
Dawkins, 1976; Clark, 1980, for reductionist psy- chological theories. Hence it is widely supposed
that the phenomenon of the consciousness can be reduced to some probably still unknown physi-
cal phenomena. Such an idea seems natural and attractive at least at the present time. However,
I do not support this viewpoint. I think that the phenomenon of consciousness will be never re-
This investigation was supported by the grant ‘Strategical investigations’ of Va¨xo¨ University and visiting professor fel-
lowships at University of Clermont-Ferrand and Tokyo Sci- ence University.
0303-264700 - see front matter © 2000 Elsevier Science Ireland Ltd. All rights reserved. PII: S 0 3 0 3 - 2 6 4 7 0 0 0 0 0 7 7 - 0
In the present paper we propose a new physi- cal – mathematical model for the brain functioning
see Khrennikov, 1998a. This model is not based on the modern Newton – Einstein picture of
physical reality in particular, we do not use the real space R
3
as the mathematical basis of our model. We consider a new type of reality,
namely, reality of information. Cognitive systems are interpreted as transformers of information. For
transformers of information we develop the for- malism of classical mechanics on mental space
space of ideas. In particular, this theory de- scribes evolution of human ideas. The general
formalism of classical cognitive mechanics is de- veloped by analogue to the formalism of the
ordinary Newton mechanics which describes the motion of material systems. We propose cognitive
analogues of Newton’s laws of the classical me- chanics. Mathematically these laws can be de-
scribed by differential equations on mental spaces
1
. Starting with the initial idea x we can
phys
which is used in ordinary physical models. This is internal time of a cognitive system we can
call it mental or psychological time. The velocity 6
t of the evolution of an idea calculated as in Newton’s mechanics as the derivative: 6t =
dqt dt
has the meaning of the motivation to change the information state qt of a cognitive system.
Forces ft, q and potentials Vt, q are informa- tion mental forces and potentials which are ap-
plied to information states of cognitive systems. An information force changes the motivation and
this change of motivation implies the change of the information state q of a cognitive system.
The mathematical formalization of the classical cognitive mechanics cannot be done in the frame-
work of the real analysis. The real line R and the Euclidean space R
3
and even real manifolds are not directly related to cognitive information pro-
cesses. We use another number system, namely, the system of so called p-adic numbers integers
Z
p
see Borevich and Shafarevich, 1966; Schikhov, 1984; Khrennikov, 1994; Vladimirov et al., 1994
as the mathematical basis of our model. Here p \ 1 is a prime number which is the parameter of
the model. Mathematical details can be found in Section 7. This section contains also some biolog-
ical motivations namely, the ability to form asso- ciations to choose Z
p
as a mathematical basis of the model.
Geometrically we can imagine Z
2
as a tree starting with some symbol a root of the 2-adic
tree which can be interpreted as the signal to start the creation of the space of ideas of a cognitive
system. This root-symbol generates two branches
0 and 1 the first level of the tree; each vertex of the first level generates two branches to two
new vertices the second level of the tree. Thus there are now four branches 00, 01, 10, 11.
Such a process is continued by an infinite number of steps. As a result, there appears an infinite
2-adic tree with branches x = a
… which are
1
At first sight it is quite surprising that motions of material systems and mental systems ideas are described by the same
mathematical equations Newton or Hamilton equations. The only difference is that these objects evolve in different spaces
Newton real space and mental space, respectively. However, if we consider, instead of the motion of real material objects,
the motion of information about these objects, then such a coincidence of equations of motion for material and mental
systems would not seem so surprising.
We use p-adic trees for prime numbers p only by mathematical reasons see Section 7. The
same information model can be developed for any homogeneous tree with m branches on each level.
It is even possible to consider trees such that the number of braches m
j
depends on the level. The information processes in the brain de-
scribed by the classical cognitive mechanics are closely connected with neurophysiological pro-
cesses. Roughly speaking neurophysiology de- scribes ‘hardware’ of the brain and the classical
mechanics on mental spaces describes ‘software’ of the brain. Some mathematical models of this
software have been presented in Khrennikov, 1997; Albeverio et al., 1998; Khrennikov, 1998b;
Albeverio et al., 1999; Dubischar et al., 1999. The models of Khrennikov, 1997; Albeverio et al.,
1998; Khrennikov, 1998a; Albeverio et al., 1999. Dubischar et al., 1999, were discrete time models,
namely, it was assumed that the time parameter t for the evolution of ideas is discrete: t = 0, 1,
2, ,… thus chains of ideas x
, x
1
, … were studied in these models. In the present paper we study
‘continuous time’ evolution. On one hand, this gives the possibility to apply at least formally
the scheme of the standard formalism of the classical mechanics. On the other hand, in the
present model we can discuss carefully the mean- ing of ‘mental time’ and ‘mental velocity’.
The classical cognitive mechanics describes un- conscious cogniti6e processes. The phenomenon of
C
q is induced by an additional information potential quantum
potential on mental space or conscious potential Cq. The Cq could not be reduced to neuro-
physiolocal processes in the brain. It is induced by mental processes. The conscious potential Cq is
induced by a wave function Cq of a cognitive system by the same relation as in the ordinary
pilot wave theory for material systems.
In our model this wave function C is nothing than an information field conscious field. In the
mathematical formalism this field is described as a function C: X
men
X
men
, where X
men
is a mental space. The evolution of the C-function is de-
scribed by an analogue of the Schro¨dinger equa- tion on mental space.
In fact, our formalism of conscious forces and fields is a natural extension of the well known
theory of pilot wave developed by Bohm, 1951; De Broglie, 1964; Bell, 1987 and many others to
cognitive phenomena. Even in the theory of pilot wave for material systems especially in its variant
developed in the book of Bohm and Hiley 1993 the quantum wave function C is merely an infor-
mation field, but defined on real space R
3
of localization of material systems, this field acts to
material objects and the problem of information – matter interaction is not clear in this framework.
In our model a conscious field C-function is associated with purely mental processes and it
acts to mental objects, ideas.
By our model each classical unconscious in- formation state of a cognitive system the collec-
tion of ideas and mental processes in that these ideas are involved produces a new non-classical
field, conscious field C. This field induces a new information force f
C
which induces a permanent
Fig. 1. The 2-adic tree.
Of course, our formalism is just the first step to describe the phenomenon of consciousness on the
basis of a model of information reality. However, even this formalism implies some consequences
which might be interesting for neurophysiology, psychology, artificial intelligence complex infor-
mation systems, evolutionary biology and social sciences. Here we present briefly some of these
consequences.
Flows of cognitive information in the brain and other cognitive systems can be described
mathematically in the manner which is similar to the classical Newton mechanics for motions of
material systems. Therefore the motion of ideas notions, images in the brain has the determinis-
tic character of course, such a motion is per- turbed by numerous information noises, see
Dubischar et al., 1999, for the details. This mo- tion in mental space is not an evolution with
respect to physical time t
phys
, but with respect to mental time t. Information potentials can connect
different thinking processes in a single brain as well as in a family of brains. The consciousness
cannot be induced by a physical activity of mate- rial structures for example, groups of neurons. It
is induced by groups of evolving ideas. These dynamical groups of ideas produce a new infor-
mation field, conscious field, which induces a new information force, conscious force, which is the
direct analogue of quantum force in the pilot wave theory for quantum material systems. This
conscious force plays the great role in the infor- mation dynamics in the human brain and other
conscious cognitive systems. As in the classical cognitive mechanics, in quantum cognitive me-
chanics conscious potentials can connect thinking processes in different cognitive systems even in
the absence of physical potentials and forces. Therefore it is possible to speak about a collective
consciousness for a group of cognitive systems in particular, human individuals. We also note that
different conscious potentials conscious C-fields induce conscious forces f
C
of different informa- tion strength. The magnitude of the conscious-
ness can be measured at least theoretically. Thus different cognitive systems in particular, different
C
control or at least change their cogni- tive behaviours. From this point of view human
individuals and animals differ only by the behav- iors of their conscious C-fields.
As one of applications of our formalism to psychology, we try to explain Freud’s psychoanal-
ysis on the basis of our model as the process of the reconstruction of the conscious field of an
individual i having some mental decease via an information coupling of a psychoanalytic p on
the level of a collective C function of the system i, p.
2. Classical cognitive mechanics
