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: fortuitousApplied 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?