Mathematics What is it? What is it about?. Terminology: Definition Axiom – a proposition that is assumed without proof for the sake of studying the consequences.

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Presentation transcript:

Mathematics What is it? What is it about?

Terminology: Definition Axiom – a proposition that is assumed without proof for the sake of studying the consequences that follow from it Postulate – a proposition that requires no proof, being self- evident, or that is for a specific purpose assumed true, and that is used in the proof of other propositions Proof Conjecture – A guess or a hyphothesis Theorem – a theoretical proposition, statement, or formula embodying something to be proved from other propositions or formulas corollary – a proposition that is incidentally proved in proving another proposition

Nature: Symbolic, axiomatic and formal (deductive) Symbols manipulated according to defined rules, with no necessary connection to the external world.

Objects of study Numbers and shapes “Numbers” includes vectors “Shapes” encompasses N- dimentional systems

Applicability to knowledge of external world: Pure math: fortuitous Applied math: direct in many disciplines

Axioms in (and logic) May be inspired on experience, but are not empirically validated Caracteristics of a valid / elegant mathematical proof

Limitations? Mathematics cannot be completely derived from axioms. Mathematical systems cannot demonstrate their own consistency

Mathematics! Discovered or invented?