AOK – Areas of Knowing Mathematics This should be watched and discussed after race and intelligence. Links to language Sapir/wharf, aborigines not having any numbers to create reality and therefore not having mental tools for IQ tests) , aesthetics, science, culture
Quotes ‘Mathematics is the abstract key which turns the lock of the physical universe’ John Pilkington ‘Math – the most logical of the sciences – shows us that the truth can be highly counterintuitive and that sense is hardly common’ K. C. Cole ‘To speak freely, I am convinced that it (mathematics) is a more powerful instrument of knowledge than any other’ Rene Descartes ‘Mathematics began when it was discovered that a brace of pheasants and a couple of days had something in common: the number two’ Bertrand Russell Who loves maths? Why do you love it? Who dislikes it? Why?
Definition A good definition of mathematics is ‘the science of rigorous proof’. History – Watch – The History of Maths (7) Or The story of One – (1hour 20) The idea of maths being the science of proof dates back to the ancient Greeks. The most famous of the Greek mathematicians being Euclid – He organised geometry into a rigorous body of knowledge and still prevail today. Geometry studied in high school is basically Euclean geometry.
The formal system Euclid developed had 3 key elements Axioms Deductive reasoning Theorems Students to explain what these are Is mathematics beautiful? Discuss
Maths as creative and imaginative How much string will you need to add to the original length of string if you were to tie it around the equator of the earth at 1 inch from its surface? Make an intelligent guess about how much string you would need to add Calculate how much string you would need to add (Hint pie=circumference/diameter
Perception of mathematics E = MC2 Does ones perception of mathematics depend on the knower? Does a child, a poet, a mathematician share the same view of maths?
Questions to consider Is emotion irrelevant to the acquisition of mathematical knowledge? Can we communicate in mathematics without words? Does mathematical knowledge aid or impede the acquisition of knowledge? How has culture and history influenced the development of mathematical thought? How has mathematical thought influenced culture and history? What is chaos theory? What is game theory? How have they affected a paradigm shift away from Darwinian theory (self interest)
The nature of Mathematics Empirical – mathematical truths are empirical generalisations Analytic – mathematical truths are true by definition Synthetic a priori – Platonism – mathematical truths give us a priori insight into the structure of reality. (this view can be traced back to Plato) Is maths discovered or invented?
Key Points Mathematics, which can be defined as ‘the science of rigorous proof’, begins with axioms and uses deductive reason to derive theroems Although proof is the logical matter of deriving theorems from axioms, mathematicians consider some proofs to be more beautiful than others According to the 3 different views about the nature of mathematical truths they are either 1. Empirical 2. True by definition 3. Rational insights into the universal truths While some people believe that mathematics is discovered, others believe it is invented, but neither view seems to be entirely satisfactory Kurt Godel proved that it is impossible to prove that a formal mathematical system is free from contradiction Mathematicians and philosophers are still perplexed by the extraordinary usefulness of mathematics
Reason – Can maths be reduced to logic? Religion – Do mathematical truths exist on the ‘mind of God’? Emotion – How important is intuition in maths? Mathematics Linking Questions Natural Sciences – Is the book of nature written in the language of mathematics? Language– How is mathematics like a language? The Arts – How important is beauty in Mathematics?