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Understanding Mathematics: Definitions, Concepts, and Applications

Mathematics is a study of abstract concepts that involves axioms, postulates, proofs, and theorems. It begins with fundamental assumptions (axioms) that are taken as true without proof for theoretical exploration. The discipline encompasses symbolic manipulation, focusing on numbers, shapes, and higher-dimensional systems. Mathematics can be classified as pure or applied, affecting various fields. While it involves conjectures and theorems, it faces limitations in deriving absolute truths and proving its own consistency. The debate about whether mathematics is discovered or invented continues to inspire philosophical discussions.

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Understanding Mathematics: Definitions, Concepts, and Applications

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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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