This topic entry introduces one of the most general mathematical models of living organisms called
‘organismic supercategories’ (OS) which can be axiomatically defined to include both complete
self-reproduction of logically defined π-entities founded in Quine’s logic and dynamic system
diagrams subject to both algebraic and topological transformations.
0.1 Organismic Supercategories (OS)
OS mathematical models were introduced as structures in higher dimensional algebra
that are mathematical interpretations of the axioms in ETAS- a natural extension of
Lawvere’s elementary theory of abstract categories (ETAC) to non-Abelian structures and
heterofunctors.
When regarded as categorical models of supercomplex dynamics in living organisms OS provide a
unified conceptual framework for relational biology that utilizes flexible, algebraic and topological
structures which transform naturally under heteromorphisms or heterofunctors. One of the
advantages of the ETAS axiomatic approach, which was inspired by the work of Lawvere (1963,
1966), is that ETAS avoids all the antimonies/paradoxes previously reported for sets (Russell and
Whitehead, 1925, and Russell, 1937). ETAS also provides an axiomatic approach to recent higher
dimensional algebra applications to complex systems biology ([10], [11] and references cited
therein.)
0.2 Selected Examples of OS Applications to Relational and Complex Systems Biology
Whereas super-categories are usually defined as n-categories or in higher dimensional algebra,
organismic supercategories have flexible, algebraic and topological structures that transform
naturally under heteromorphisms or heterofunctors. Different approaches to relational biology and
biodynamics, developed by Nicolas Rashevsky, Robert Rosen and by the author, are compared
with the classical approach to qualitative dynamics of systems (QDS). Natural transformations of
heterofunctors in organismic supercategories lead to specific modular models of a variety of specific
life processes involving dynamics of genetic systems, ontogenetic development, fertilization,
regeneration, neoplasia and oncogenesis. Axiomatic definitions of categories and supercategories of
complex biological systems allow for dynamic computations of cell transformations, neoplasia and
cancer.
References
[1] Bacon, John, 1985, “The completeness of a predicate–functor logic,” Journal of
Symbolic Logic 50: 903–926.
[2] Paul Bernays, 1959, “Uber eine naturliche Erweiterung des Relationenkalkuls.” in
Heyting, A., ed., Constructivity in Mathematics. North Holland: 1–14.
[3] References [14] to [34] in the “bibliography of category theory and algebraic
topology”
[4] I. C. Baianu, J. F. Glazebrook, R. Brown and G. Georgescu.: Complex Nonlinear
Biodynamics in Categories, Higher dimensional Algebra and Łukasiewicz-Moisil
Topos: Transformation of Neural, Genetic and Neoplastic Networks, Axiomathes,16:
65–122(2006).
[5] Baianu, I.C. and M. Marinescu: 1974, A Functorial Construction of (M,R)–
Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19: 388-391.
[6] Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras:
The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.
[7] Baianu, I.C.: 1980, Natural Transformations of Organismic Structures. Bulletin of
Mathematical Biophysics 42: 431-446
[8] Baianu, I. C.: 1987a, Computer Models and Automata Theory in Biology and
Medicine., in M. Witten (ed.), Mathematical Models in Medicine, vol. 7., Pergamon
Press, New York, 1513-1577; CERN Preprint No. EXT-2004-072
[9] Baianu, I. C.: 2006, Robert Rosen’s Work and Complex Systems Biology, Axiomathes
16 (1–2): 25–34.
[10] Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum
Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks,
(M,R)–Systems and Their Higher Dimensional Algebra; PDF of Abstract and Preprint
of Report
[11] Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006,
Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and
Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic
networks, Axiomathes 16 Nos. 1–2, 65–122.
[12] Kuhn, Stephen T., 1983, “An Axiomatization of Predicate Functor Logic.”, Notre
Dame Journal of Formal Logic 24: 233–41.
[13] Willard Quine. 1976. “Algebraic Logic and Predicate Functors.” in Ways of Paradox
and Other Essays, enlarged ed. Harvard Univ. Press: 283–307.
[14] Willard Quine. 1982. Methods of Logic, 4th ed. Harvard Univ. Press. Chpt. 45.
