Algebraic Biology: Creating Invariant Binding Relations for Biochemical and Biological Categories

Abstract

The desire to understand the mathematics of living systems is increasing. The widely held presupposition that the mathematics developed for modeling of physical systems as continuous functions can be extended to the discrete chemical reactions of genetic systems is viewed with skepticism. The skepticism is grounded in the issue of scientific invariance and the role of the International System of Units in representing the realities of the apodictic sciences. Various formal logics contribute to the theories of biochemistry and molecular biology and genetics. Various paths of extension are invoked in these formal logics in order to express the information of biological apodicticism. Symbolizing the appropriate notations for invariant relations and for biological extensions of relations is fundamental to the exact generating functions of discrete algebraic biology. Aspects of philosophical perspectives of the relation scientific number systems are contrasted. The deep distinction between physical motion and biological motion is expressed in terms the roles of Aristotelian causes. The interior motion within perplex numbers is contrasted with the exterior motion of physical systems. The need for a new mathematics for biology is suggested.

Appendix

The perplex numerals are defined as relational objects in the sense of Leibniz and Peirce.

Logically, a perplex numeral is defined by two propositions as required by the symmetry of the equi-numerosity of the units and integers:

• An integer infers a set of units.

• A set of units infers an integer.

The usual notation of logicians for symbolic inference employs a horizontal bar to separate the proposition from the conclusion, the antecedent from the consequent. Again, we follow the symmetry principle of equi-numerosity. We can list the correspondence relations dually.

$$ {\frac{ 1*}{1}} ,{\frac{2}{1,1}},{\frac{3}{1,1,1}}, \ldots $$

Or, alternatively (dually), the list of inference of the correspondence of units to integers.

$$ {\frac{ 1}{1*}} ,{\frac{1,1}{2}},{\frac{1,1,1}{3}}, \ldots $$

(* Note that the symbol “1*” is used as a sign for the first integer to avoid possible ambiguity in calculations.)

The information content or “the difference that makes a difference” between successive terms of the series of perplex numerals is the stepwise increase in the size of the numerator and denominator so that the ratio of the integer to the units to the number of relations is constant. The uniformity of the transitive relation between successive integers is a necessary property for parity or equi-numerosity.

Graphically, each perplex numeral constitutes a teridentity, a labeled bipartite graph. The teridentity is a logical reference defined for inferences among terms. It serves as the axiomatic basis for compositions of relations among multisets of terms. A composition of a multiset of numerals preserves all the parts of each member of the multiset and extends the number of relations such that parity is preserved in the composed labeled bipartite graph.

A logical composition of two or more perplex numerals breaks symmetry. Formal logical operations on the graph edges create new relations such that a new object, a new labeled bipartite graph, comes into existence. It can be isomorphic to an existential entity, a reality (esse extra anima). The perplex numerals inform the perplex numbers, just as hydrogen and oxygen inform water and hydrogen peroxide.