‘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.