Biomatics

It is evident that some form of computation takes place in biological systems and indeed within single molecules, making the study of computational biology essential. 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 like Fourier transforms of visual data, the interplay between mathematics and biology is clear.

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

By the late 1800s, Karl Weierstrass, Richard Dedekind, and Georg Cantor had all developed methods for defining the irrationals in terms of the rationals. Giuseppe Peano had also developed a theory of the rationals based on his now famous axioms for the natural numbers. Thus, by Gottlob Frege's time (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—including computational biology—some concepts are now self-evident. Given that biological systems perform some sort of mathematics, and the acceptance of evolution, it follows that these mathematical systems have evolved and therefore must have started from some initial state. Biomatics further raises the possibility that all of mathematics may be based on elemental algebraic structures, as embodied in molecules like 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 computational biology from a set of primitive notions? It seems it must inevitably be so, for ultimately what separates different species, from viruses to humans, is the complexity of the molecules that carry the blueprint for the ontogeny of the species.

As computer scientists, particularly those interested in computational biology, we seek and think in terms of information storage and processing. We aim 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

It is evident that some form of computation takes place in biological systems and indeed within single molecules, making the study of computational biology essential. 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 like Fourier transforms of visual data, the interplay between mathematics and biology is clear.

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

By the late 1800s, Karl Weierstrass, Richard Dedekind, and Georg Cantor had all developed methods for defining the irrationals in terms of the rationals. Giuseppe Peano had also developed a theory of the rationals based on his now famous axioms for the natural numbers. Thus, by Gottlob Frege's time (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—including computational biology—some concepts are now self-evident. Given that biological systems perform some sort of mathematics, and the acceptance of evolution, it follows that these mathematical systems have evolved and therefore must have started from some initial state. Biomatics further raises the possibility that all of mathematics may be based on elemental algebraic structures, as embodied in molecules like 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 computational biology from a set of primitive notions? It seems it must inevitably be so, for ultimately what separates different species, from viruses to humans, is the complexity of the molecules that carry the blueprint for the ontogeny of the species.

As computer scientists, particularly those interested in computational biology, we seek and think in terms of information storage and processing. We aim 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