BIOMATICS

It is evident that some form of computation takes place in biological systems and indeed within single molecules.  It thus follows that some form of mathematics occurs in these computations.  Whether it be basic set theoretical concepts such as subsets, intersection, and union or more complex manipulations such as Fourier transforms of visual data:

https://sites.northwestern.edu/elannesscohn/2019/07/30/developing-an-intuition-for-fourier-transforms/

Background Gottfried Leibniz considered the following thesis in the late 17th century:  (Some or all of) mathematics can be reduced to formal logic. It is often described as a two-part thesis.      1. All mathematical truths can be translated into logical truths.       2. All mathematical proofs can be recast as logical proofs.    In other words, that all mathematical truths and proofs can be restated in the vocabulary of logic. 

By the late 1800s Karl Weierstrass, Richard Dedekind and Georg Cantorhad all developed methods for defining the irrationals in terms of the rationals.  Giuseppe Peano had also gone on to develop a theory of the rationals based on his now famous axioms for the natural numbers.  Thus, by Gottlob Frege's (predicate calculus) day (1848-1925) it was generally recognized that a large portion of mathematics could be derived from a relatively small set of primitive notions. In 1910, Bertrand Russell and Alfred North Whitehead collaborated on Principia Mathematica, an attempt at a detailed deduction of mathematics from logic, which proved to be greatly influential yet controversial.    

In Bertrand Russell's words, it is the logicist's goal "to show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms". 

As a result, the question of whether mathematics can be reduced to logic, or whether it can be reduced only to set theory, remains open.  However, in light of modern theories of evolution, fractal geometry, physics, chemistry and computer science, some concepts are now self-evident.  Given that biological systems perform some sort of mathematics, and acceptance of evolution, it follows that these mathematical systems have evolved and therefore must have started from some initial state. Biomatics further raises at least the possibility that all of mathematics may be based on elemental algebraic structures as embodied in such molecules as the amino acids. 

Intramolecular Computation  Consider an algebraic system embodied in a molecule consisting of N atoms.  In the case where N = 3  we find the cube group(in terms of abstract algebra).  (Note that N = 1 and N = 2 can represent groups as well).  

Group theory (abstract algebra) is a well-developed branch of mathematics that provides many theorems and definitions.  The key concept is that it describes, formally, a small (fundamental?) mathematical system consisting of a set and an operation on the members of that set.  Could this then be nature’s way of evolving a system of mathematics and computation from a set of primitive notions?  It seems it must inevitably be so, for ultimately what separates the different species, from viruses to man, is the complexity of the molecules that carry the blueprint for the ontogeny of the species.  

As computer scientists, we seek and think in terms of information storage and processing.  We seek to compare and contrast biological manifestations of computer science paradigms including:    

• Algorithms
• Data Structures
• Theorems
• Computer Architecture
•  Switching elements (gates)
•       Circuitry
•  Finite State Machines
•  Mathematics

The seamless development of mathematics and computation from a few clearly stated axioms and rules of inference in pure logic as embodied in an atomic/molecular medium.  A gateway into the era of the molecule. 

The principles of mathematics, such as logic, set theory, and algebra, can be applied to analyze and reason about molecular systems. The concepts of variables, functions, equations, and transformations can be used to describe, manipulate and understand the properties and interactions of molecules.

The seamless development of mathematics and computation in an atomic/molecular medium highlights the universality and applicability of mathematical concepts across different domains. By harnessing the inherent properties of atomic and molecular systems, we can explore new avenues for computational modeling, data storage, information processing, and even the design of novel materials and technologies.

Overall, the integration of mathematics and computation within an atomic/molecular medium holds great potential for advancing scientific understanding, technological innovation, and problem-solving capabilities in a wide range of fields.                                                        

The term “Biomatics” consists of a cross contraction of Biological Mathematics.  The concept that is meant to be defined is the study of mathematics and computation as they occur in biological systems.  This is in contrast to the concept of mathematical biology, which is the use of mathematics to describe or model biological systems.  

A tentative definition might be:  The seamless development of mathematics and computation from a few clearly stated axioms and rules of inference in pure logic as embodied in an atomic/molecular medium.   Or, the study of "smart molecules". 

Immediately after fertilization the major role of the DNA machinery is to divide the cell and create an adult organism.  The DNA and the Histone protein with it's associated tails are programmed for this task. The path drawn by the end carbon atom of the free end of the histone tail may give significant geometric temporal clues as to the function of the DNA.  The tip serves as the Y in the equation Y=f(X), where X is some multidimensional vector.  At any given point in time the tip of the histone tail will be in some given location in a 3 dimensional space. This may yield significant clues as to the geometrical properties related to the genetic machinery...as demonstrated by the similarities demonstrated by the above image of the cross section of an embryo and a structure created by a simulation of a chain of carbon atoms similar to a histone tail.