Mathematics

1
Mathematics

• What is it?
• What is it about?

2
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

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?