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Mathematics

Mathematics . What is it? What is it about?. Definition Axio m a proposition that is assumed without proof for the sake of studying the consequences that follow from it Postulate

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Mathematics

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  1. Mathematics What is it? What is it about?

  2. 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 Terminology:

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

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

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

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

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

  8. Mathematics! Discovered or invented?

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