Transcript AOK - Mathematics

‘Pure Mathematics is,
in its way, the poetry
of logical ideas’
Einstein
‘Maths is like love,
a simple idea but
it can get very
complicated.’
Unknown
‘The highest form of
pure thought is in
Mathematics.’ Plato
‘Mathematics rightly
viewed, possesses not
only truth, but supreme
beauty; a beauty cold
and austere, like that of a
sculpture’ Bertrand
Russell
 Choose
3 words that for you describe the
essence of Mathematical knowledge.
 Do themes recur, are these a fair
reflection or a stereotype?
 In
a strict sense, mathematics differs from
science, if we accept that science is the
discipline that seeks understanding of
the physical world by means of the
scientific method. The reason
mathematics differs from this is because
mathematics does not, in a pure sense,
attempt to describe the physical world.
Mathematical theorems are not tested
against nature, but against logic.
 Basically
Mathematics is the derivation of
theorems from axioms.
• Mathematicians play games of ‘what if’ .
• They make up sets of rules for the game – these
are known as axioms
• And then explore the outcomes (theorems) of
playing the game.
• *video on Pythagoras and Euclid
 Different
fields of Maths such as
geometry, algebra, set theory etc. are all
axiomatic deductive systems.
 The axioms are used as the premises,
mathematicians apply valid deductive
reasoning to them, a process called
mathematical proof to obtain new
statements called theorems. These
theorems are used to build further
theorems which can come up with
additional premises…….
 What
is needed to make a true
conclusion?
• Valid reasoning AND your premises must be true
(remember valid reasoning is an argument that is
logically correct and your premises are what you
are basing your argument on premises =
axioms)
• Problem: How do we know if the axioms are true
– or are not the only possible truth?
 Read
the text about Euclid’s axioms.
 In geometry Euclid’s are more useful in
building a house, but Reimann’s in flying
an airplane.
 Once a Mathmatician adopts any specific
set of axioms, he can only play by them –
very, very strictly.
• Lessons from the IBO ‘Numbers and Numerals’
• Complete hand out and research one of the
numerical systems.
 In
pairs research one statement from the
TOK guide for Mathematics.
• Feedback to the rest of the class – 3 minutes
• Make a summary for the wiki site.