Aok – areas of knowing mathematics


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Aok – areas of knowing mathematics

  1. 1. AOK – Areas of Knowing Mathematics<br />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<br />
  2. 2. Quotes<br />‘Mathematics is the abstract key which turns the lock of the physical universe’ John Pilkington<br />‘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<br />‘To speak freely, I am convinced that it (mathematics) is a more powerful instrument of knowledge than any other’ Rene Descartes<br />‘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<br />Who loves maths? Why do you love it? Who dislikes it? Why?<br />
  3. 3. Definition<br />A good definition of mathematics is ‘the science of rigorous proof’.<br />History – Watch – The History of Maths (7)<br />Or<br />The story of One – (1hour 20)<br />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.<br />
  4. 4. The formal system Euclid developed had 3 key elements<br />Axioms<br />Deductive reasoning<br />Theorems<br />Students to explain what these are <br />Is mathematics beautiful? Discuss<br />
  5. 5. Maths as creative and imaginative<br />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?<br />Make an intelligent guess about how much string you would need to add<br />Calculate how much string you would need to add (Hint pie=circumference/diameter<br />
  6. 6. Perception of mathematics<br />E = MC2<br />Does ones perception of mathematics depend on the knower? Does a child, a poet, a mathematician share the same view of maths?<br />
  7. 7. Questions to consider<br />Is emotion irrelevant to the acquisition of mathematical knowledge?<br />Can we communicate in mathematics without words? Does mathematical knowledge aid or impede the acquisition of knowledge?<br />How has culture and history influenced the development of mathematical thought? How has mathematical thought influenced culture and history?<br />What is chaos theory? What is game theory? How have they affected a paradigm shift away from Darwinian theory (self interest)<br />
  8. 8. The nature of Mathematics<br />Empirical – mathematical truths are empirical generalisations<br />Analytic – mathematical truths are true by definition<br />Synthetic a priori – Platonism – mathematical truths give us a priori insight into the structure of reality. (this view can be traced back to Plato)<br />Is maths discovered or invented?<br />
  9. 9. Key Points<br />Mathematics, which can be defined as ‘the science of rigorous proof’, begins with axioms and uses deductive reason to derive theroems<br />Although proof is the logical matter of deriving theorems from axioms, mathematicians consider some proofs to be more beautiful than others<br />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<br />While some people believe that mathematics is discovered, others believe it is invented, but neither view seems to be entirely satisfactory<br />Kurt Godel proved that it is impossible to prove that a formal mathematical system is free from contradiction<br />Mathematicians and philosophers are still perplexed by the extraordinary usefulness of mathematics<br />
  10. 10. Reason – Can maths be reduced to logic?<br />Religion – Do mathematical truths exist on the ‘mind of God’?<br />Emotion – How important is intuition in maths?<br />Mathematics<br />Linking Questions<br />Natural Sciences – Is the book of nature written in the language of mathematics?<br />Language– How is mathematics like a language?<br />The Arts – How important is beauty in Mathematics?<br />