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?