Complex-Dynamic Fractality and Its Applications in Life Sciences
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“unnatural” tensions in the “infinitely” large network of fractal interactions (eq. (24)),
among which only some part will explicitly appear in the observed properties of
organism dynamics. It is also evident that the problem can be solved only by essential
extension of unitary approaches to the unreduced, multivalued and fractal interaction
dynamics, taking into account all participating elements, as it is demonstrated by the
above analysis, which can uniquely transform the empirical, potentially destructive
unitary genetics into provably constructive complex-dynamical genetics.
Note finally the essential extension of mathematical concepts and approaches
involved with that urgently needed progress in applications, as the development of
fundamental science tools represents also its own interest, especially evident on the
background of persisting stagnation [9,11,18] and “loss of certainty” in fundamental
knowledge (cf. [19]). (i) First of all, one should mention the nonuniqueness of any real
problem solution, taking the form of its dynamic multivaluedness (section 2), and
related complex-dynamic existence of any system that replace the usual “uniqueness and
existence theorems” valid only for reduced, unitary models [9]. (ii) It follows that the
related unitary concept of “exact” (closed) solutions and its perturbative versions are
basically insufficient and fundamentally incorrect with respect to real world structures.
The true, dynamical meaning of the notions of “(non)integrability”, “(non)separability”,
“(non)computability”, “uncertainty”, “randomness”, and “probability” becomes clear:
we obtain now the nonintegrable and nonseparable, but solvable dynamics of a generic
many-body system (see eqs. (8)-(22)), while real world mathematics regains its
certainty and unification, but contains a well-defined, dynamic indeterminacy and
fractally structured diversity (i.e. it cannot be reduced to number properties and
geometry, contrary to unitary hopes). (iii) The property of dynamic entanglement and its
fractal extension (section 2) provides the rigorous mathematical definition of the
tangible quality of a structure, applicable at any level of dynamics, which contributes to
the truly exact mathematical representation of real objects, especially important for
biological applications. (iv) The irreducible dynamic discreteness, or quantization, of
real interaction dynamics expresses its holistic character and introduces essential
modification in standard calculus applications and their formally discrete versions,
including “evolution operators”, “Lyapunov exponents”, “path integrals”, etc. [9,11].
(v) The unceasing, probabilistic change of system realisations provides the dynamic
origin of time, absent in any version of unitary theory: in the new mathematics and in
the real world one always has a≠ a for any measurable, realistically expressed quantity
or structure a , while one of the basic, often implicit postulates of the canonical
mathematics is “self-identity”, a= a (related to “computability”). It has a direct bio-
inspired implication: every real structure a is “alive” and “noncomputable”, in the
sense that it always probabilistically moves and changes internally. In fact, any
realistically conceived a represents a part of a single, unified structure of the new
mathematics introduced above as dynamically multivalued (probabilistic) fractal (of the
world structure) and obtained as the truly exact, unreduced solution of a real interaction
problem (section 2). We can see in that way that such recently invented terms as
“biofractals” and “biomathematics” can have much deeper meaning and importance
than usually implied “(extensive) use of mathematics in biological object studies”.
Acknowledgement. The author is grateful to Professor Gabriele Losa for invitation to
the symposium and support.