Axiomatic Reasoning And Axiomatic Methods

Axiomatic Reasoning and Axiomatic MethodsNP1402 (Mathematical Literacy)Dr. L. A. WaniThapar School of Liberal Arts and Sciences (TSLAS) Thapar Institute of Engineering and Technology Patiala, PunjabAxiomatic reasoning has a long history. It has been used in many dierent ways.Aristotle saw it as part of what he called rst philosophy. Euclid applied it to geometry.Archimedes used it in solving statics problems. Ptolemy studied astronomical models byit. Spinoza used it in his conceptual demonstrations, Newton in mechanics and optics,Peano in number theory, Hilbert in geometry and physics, Einstein in gravitation theory,Whitehead and Russell in logic. Many other scientists and philosophers used it in dierentelds. There was a common purpose in all the dierent uses. It was to provide a systematicinvestigation of the structures that are intended to be characterized by axioms. The struc-tures in question can be mathematical, physical (material), social, legal, pure structures,or what not.1 Axiom A formal statement or principle in mathematics, science, etc., from which otherstatements can be obtained. A self-evident truth that requires no proof. A universally accepted principle or rule.1Logic, Mathematics : It is a proposition that is assumed without proof for thesake of studying the consequences that follow from it.2 Examples Euclid's axioms form the foundation of