mathematics what is it? what is it about?. terminology: definition axiom – a proposition that is...
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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?