PlanetPhysics/Topic on Axioms

\section{Topic on axioms in mathematics, logic algebra, mathematical physics and mathematical biophysics}

Introduction
In classical logic, an axiom or postulate is a `simple', fundamental proposition that is neither proven nor demonstrated (within a theory) "but considered to be self-evident"; furthermore, the choice of an axiom or system of axioms is justified by the large number of consistent consequences or mathematical propositions derived from such axioms. One needs, however, to distinguish between `physical axioms' (often called `postulates' that apply to various fields of physics), and mathematical axioms that have both a meaning and scope of applicability which is distinct from that of physical postulates (or physical axioms). On the other hand, physical axioms, or postulates, are ultimately also expressed in a mathematical form, albeit without becoming axioms of mathematics, or specific fields of mathematics. (In the remainder of this entry the attribute `axiomatic' will be employed only with the meaning of `physical-axiomatic', or `physically-postulated'.)

Furthermore, physical postulates, unlike mathematical ones, emerged as a result of numerous experimental studies and crucial physical experiments that can be logically and consistently explained on the basis of such fundamental, physical postulates; often, mathematical formulations of such fundamental physical postulates are referred to as (physical) `axioms', as in the case of `axiomatic' QFTs.

Axioms in Mathematics

 * Axioms of Set theory
 * Axioms of Number theory
 * Axioms of Geometry
 * Axioms of Topology
 * Axioms of Homology and cohomology theories
 * Axioms of topos theory
 * Axioms for categories; axioms of category theory
 * Axioms of algebraic K-theory
 * Axioms of higher dimensional algebra (HDA)
 * Axioms of Meta- and super- categories

Axioms of Logic and Logic Algebras

 * Axioms of Boolean logic algebra
 * Axioms of $$LM_n$$ logic algebras
 * Axioms of quantum logics
 * Axioms of XYZ

Axioms and Postulates in Mathematical Physics

 * Postulates of Relativity theories: Special and general relativity
 * Axioms of quantum geometry
 * Axioms of local quantum physics (algebraic quantum field theories (AQFT))
 * Axioms of XYZ

Axioms and Postulates of Mathematical biology/Mathematical Biophysics

 * Axiom of Organismic Selection
 * Axioms of Genetics
 * Postulate of Optimal Design
 * Postulate of Relational `forces'
 * Axioms of organismic complete self-reproduction
 * Axioms of organismic supercategories
 * Axiom of Fuzziness
 * epimorphism axioms and homology
 * adjointness axioms
 * Axioms of XYZ