Transcript Mathematics

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


Proof
Conjecture
–

Theorem
–
Postulate
–
a proposition that requires
no proof, being selfevident, or that is for a
specific purpose assumed
true, and that is used in
the proof of other
propositions

A guess or a hyphothesis
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 Ndimentional systems
Applicability to knowledge of
external world:
 Pure
math: fortuitous
 Applied 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?