topic on axioms | Physics Library

1 Topic on axioms in mathematics, logic algebra, mathematical physics and mathematical biophysics

1.1 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.

1.2 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

1.3 Axioms of Logic and Logic Algebras

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

1.4 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

1.5 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

1.6 Examples of Axioms

References

[1] S. Mac Lane. 2000. Ch.1: Axioms for Categories, in Categories for the Working Mathematician. Springer: Berlin, 2nd Edition.

[2] R. Brown R, P.J. Higgins, and R. Sivera.: “Non-Abelian Algebraic Topology”,(vol. 2. in preparation). (2008).

[3] W.F. Lawvere: 1963. Functorial Semantics of Algebraic Theories. Proc. Natl. Acad. Sci. USA, 50: 869–872

[4] W. F. Lawvere: 1966. The Category of Categories as a Foundation for Mathematics. , In Proc. Conf. Categorical Algebra–La Jolla, 1965, Eilenberg, S et al., eds. Springer –Verlag: Berlin, Heidelberg and New York, pp. 1–20.

[5] L. Löfgren: 1968. On Axiomatic Explanation of Complete Self–Reproduction. Bull. Math. Biophysics, 30: 317–348.

[6] Stephen Weinberg. 2000. Quantum Field Theory, vol. III. Cambridge University Press, Cambridge, UK.

[7] Detlev Buchholz and Rudolf Haag.1999. The Quest for Understanding in Relativistic Quantum Physics, pp. 38, arXiv : hep − th∕9910243v2, J.Math.Phys. 41 (2000) 3674–3697.

[8] Rudolf Haag. 1992. “Local Quantum Physics: Fields, Particles, Algebras”. Springer: Berlin.

[9] Hans Halvorson, Michael Mueger. 2006. Algebraic Quantum Field Theory. arXiv:math-ph/0602036, 202 pages; to appear in Handbook of the Philosophy of Physics, North Holland: Amsterdam.

[10] John E. Roberts. More lectures on algebraic quantum field theory. In Sergio Doplicher and Roberto Longo, editors, Noncommutative geometry, pages 263-342. Springer, Berlin, 2004.

[11] John E. Roberts. Lectures on algebraic quantum field theory. In Daniel Kastler, editor, The algebraic theory of superselection sectors (Palermo, 1989), pages 1-112. World Scientific Publishing, River Edge, NJ, 1990.

[12] I.C. Baianu: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics, 32: 539-561.

[13] I.C. Baianu : 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. Bulletin of Mathematical Biophysics, 33 (3), 339–354.

[14] I.C. Baianu: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.

[15] I.C. Baianu: 1980, Natural Transformations of Organismic Structures. Bulletin of Mathematical Biophysics 42: 431-446.

[16] I.C. Baianu, Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic networks, Axiomathes 16 Nos. 1–2, 65–122.

[17] R. Brown, J. F. Glazebrook and I. C. Baianu: A categorical and higher dimensional algebra framework for complex systems and spacetime structures, Axiomathes 17:409–493. (2007).

[18] R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom.Diff. 17 (1976), 343–362.

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