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 [[../Predicate/|proposition]] that is neither proven nor demonstrated (within a theory) "but considered to be self-evident"; furthermore, the choice of an axiom or [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|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 [[../CosmologicalConstant/|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' [[../QuantumOperatorAlgebra5/|QFTs]].

Axioms in Mathematics

• Axioms of Set theory
• Axioms of Number theory
• Axioms of Geometry
• Axioms of Topology
• Axioms of Homology and [[../NoncommutativeGeometry4/|cohomology theories]]
• Axioms of [[../GrothendieckTopos/|topos]] theory
• Axioms for [[../Cod/|categories]]; axioms of [[../TrivialGroupoid/|category theory]]
• Axioms of [[../PAdicMeasure/|algebraic K-theory]]
• Axioms of [[../HigherDimensionalAlgebra2/|higher dimensional algebra]] ([[../2Groupoid2/|HDA]])
• Axioms of Meta- and super- categories

Axioms of Logic and Logic Algebras

• Axioms of Boolean logic algebra
• Axioms of $L{M}_n$ logic algebras
• Axioms of [[../LQG2/|quantum logics]]
• Axioms of XYZ

Axioms and Postulates in Mathematical Physics

• Postulates of Relativity theories: Special and [[../SR/|general relativity]]
• Axioms of [[../NAQAT2/|quantum geometry]]
• Axioms of [[../NewtonianMechanics/|local quantum physics]] ([[../CoIntersections/|algebraic]] [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theories]] ([[../MetaTheorems/|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 `[[../Thrust/|forces]]'
• Axioms of organismic complete self-reproduction
• Axioms of [[../SuperCategory6/|organismic supercategories]]
• Axiom of Fuzziness
• [[../Epimorphism2/|epimorphism]] axioms and homology
• [[../DualityAndTriality/|adjointness]] axioms
• Axioms of XYZ