0.1 Fundamental complexity diagrams
Categorical comparisons of different types of dynamical systems in diagrams provide useful
means for both their classification and understanding the relations between them. Such
powerful mathematical tools may also be considered as a further, practically useful
elaboration of Spencer’s philosophical principle ideas in biology and sociology in terms of the
emergence of higher complexity levels in living systems and societies. As explained by Barry
Mitchell (1965) and other category theoreticians, diagrams can be defined as functors,
and functor categories involve ‘meta-diagrams of diagrams’, or functors and natural
transformations, meta-categories of categories, and so on to higher dimensional algebras (HDA).
Therefore, category theory in higher dimensions/HDA appears as the natural setting for
considering the emergence of higher complexity levels–such as living organisms, the
human mind and societies in relation to the simpler, physicochemical/molecular/quantum
systems.
0.1.1 Diagrams of complexity levels
When viewed from a formal perspective of Poli’s theory of levels (Baianu and Poli, 2008), the two
levels of super– and ultra– complex systems are quite distinct in many of their defining properties,
and therefore, categorical diagrams that ‘mix’ such distinct levels do not commute. Considering
dynamic similarity, Rosen (1968) introduced the concept of ‘analogous’ (classical) dynamical
systems in terms of categorical, dynamic isomorphisms between their isomorphic state-spaces that
commute with their transition (state) function, or dynamic laws. However, the extension of this
concept to either complex or super-complex systems has not yet been investigated, and may be
similar in importance to the introduction of the Lorentz-Poincaré group of transformations for
reference frames in Relativity theory. Furthermore, one is always seeking the underlying
relational invariance or similarities in functionality among different organisms or between
different stages of development during Ontogeny (the development of an organism from a
fertilized egg), as well as during phylogeny, the evolution of organisms and species,
(encompassing also biomolecular evolution that may however be often ‘neutral’ with respect
to the the emergence of new phenotypes). In this context, the categorical concept of
‘dynamically adjoint systems’ was introduced in relation to the data obtained through
nuclear transplant experiments (Baianu and Scripcariu, 1974). Thus, extending the latter
concept to super– and ultra– complex systems , one has in general, that two complex or
supercomplex systems with ‘state spaces’ being defined respectively as A and A*, are
dynamically adjoint if they can be represented naturally by the following (functorial)
diagram:
with F ≈ F′ and G ≈ G′ being isomorphic (that is, ≈ representing natural equivalences between
adjoint functors of the same kind, either left or right), and as above in diagram (2.5), the two
diagonals are, respectively, the state-space transition functions Δ : A → A and Δ∗ : A∗ → A∗ of the
two adjoint dynamical systems. (It would also be interesting to investigate dynamic adjointness in
the context of quantum dynamical systems and quantum automata, as defined in Baianu,
1971a).
A left-adjoint functor, such as the functor F in the above commutative diagram between categories
representing state spaces of equivalent cell nuclei preserves inductive limits, whereas the
right-adjoint (or coadjoint) functor, such as G above, preserves projective limits (colimits). (For
precise definitions of adjoint functors the reader is referred to Brown, Galzebrook and Baianu,
2007, as well as to Popescu, 1973, Baianu and Scripcariu, 1974, and the initial paper by Kan,
1958).
0.1.2 Applications to mathematical models of: nuclear transplants, cloning, embryogenesis,
development and related mathematical biology problems
A left-adjoint functor, such as the functor F in the above commutative diagram between categories
representing state spaces of equivalent cell nuclei preserves limits, whereas the right-adjoint (or
coadjoint) functor, such as G above, preserves colimits. (For precise definitions of adjoint functors
the reader is referred to Brown, Galzebrook and Baianu, 2007, as well as to Popescu, 1973, Baianu
and Scripcariu, 1974, and the initial paper by Kan, 1958). Thus, dynamic attractors and genericity
of states are preserved for differentiating cells up to the blastula stage of organismic development.
Subsequent stages of ontogenetic development can be considered only ‘weekly adjoint’ or partially
analogous. Similar dynamic controls may operate for controlling division cycles in the cells of
different organisms; therefore, such instances are also good example of the dynamic
adjointness relation between cells of different organisms that may be very far apart
phylogenetically, even on different ‘branches of the tree of life.’ A more elaborate dynamic
concept of ‘homology’ between the genomes of different species during evolution was
also proposed (Baianu, 1971a), suggesting that an entire phylogenetic series can be
characterized by a topologically–rather than biologically–homologous sequence of genomes which
preserves certain genes encoding the essential biological functions. A striking example was
recently suggested involving the differentiation of the nervous system in the fruit fly and
mice (and perhaps also man) which leads to the formation of the back, middle and
front parts of the neural tube. A related, topological generalization of such a dynamic
similarity between systems was previously introduced as topological conjugacy(Baianu,
1986-1987a; Baianu and Lin, 2004), which replaces recursive, digital simulation with
symbolic, topological modelling for both super– and ultra– complex systems (Baianu and
Lin., 2004; Baianu, 2004c; Baianu et al., 2004, 2006b). This approach stems logically
from the introduction of topological/ computation and topological computers Baianu,
1971b), as well as their natural extensions to quantum nano-automata (Baianu, 2004a),
quantum automata and quantum computers (Baianu, 1971a, and 1971b, respectively);
the latter may allow us to make a ‘quantum leap’ in our understanding Life and the
higher complexity levels in general. Such is also the relevance of quantum logics and
LMn-logic algebra to understand the immanent operational logics of the human brain and
the associated mind meta–level. Quantum Logics concepts are introduced next that
are also relevant to the fundamental, or ‘ultimate’, concept of spacetime, well-beyond
our phenomenal reach, and thus in this specific sense, transcedental to our physical
experience (perhaps vindicating the need for a Kantian–like transcedental logic, but from a
quite different standpoint than that originally advanced by Kant in his critique of ‘pure’
reason; instead of being ‘mystical’- as Husserl might have said–the transcedental logic of
quantized spacetime is very different from the Boolean logic of digital computers, as it
is quantum, and thus non–commutative). A Transcedental Ontology, whereas with a
definite Kantian ‘flavor’, would not be at all ‘mystical’ in Husserl’s sense, but would
rely on ‘verifiable’ many–valued, non–commutative logics, and thus contrary to Kant’s
original presupposition, as well as untouchable by Husserl’s critique; the fundamental
nature of spacetime would be thus ‘provable’ and ‘verifiable’, but only to the extent
allowed by Quantum Logics, not by an arbitrary (‘mystical’) Kantian–transcedental
logic.
