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Tok- Maths Presentation
The document discusses the relationship between mathematics and the natural sciences. It notes that mathematics has been remarkably successful at describing natural phenomena, which is unexpected given that mathematics is a product of human thought while nature exists independently. The document raises questions about how the "laws of nature" can be exact copies of the patterns humans discover in their abstract systems of thought. It calls this alignment of the two fields the "unreasonable effectiveness of mathematics."
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Tok- Maths Presentation
- 1. “Mathematics, rightly viewed,possesses not only truth, but supreme beauty – a beauty cold and austere, like that of a sculpture.”Bertrand Russell(1872 – 1970)The vain search for absolute truth
- 3. 1. What inyour view is Mathematics?2. Choose three words that best describe Maths. Post your words at http://www.wallwisher.com/wall/MrsVTOK1http://www.wallwisher.com/wall/MrsVTOK2
- 4. Four questions foryou to answer:Primes were invented (discovered?) by:a) The Ancient Greeksb) Insectsc) The Ancient Chinese
- 5. 2. The lengthof coastline in Britain is: a) 18 000 kmb) 36 000 kmc) infinited) all of the above
- 6. 3. The sumof the angles in a triangle is: a) 180 degressb) less than 180 degreesc) more than 180 degreesd) all of the above
- 7. 4. The mainpurpose of mathematics is to, a) be a tool for predicting real world events and real world problemsb) develop critical thinking skillsc) to create new mathematicsd) provide a challengee) provide a qualification for employmentf) be abstractg) give something for geeky dudes to do
- 8. New answers toold questions:
- 9. A Mathematician’s Answers1.b) The primes were invented by insects (we will discuss later whether the word invented is correctly used)2. d) Wikipedia gives 18000km for the coastline of Britain, but a Mathematician has a different answer – it depends on the accuracy of your measurement (How do culture and geography have an impact on the development of Mathematics Mathematics?)
- 10. A Mathematician’s Answers3.d) All of the above!4. c) The main purpose of Mathematics is to create new Mathematics
- 11. Formation of MathematicalKnowledgeWhat is the average of the numbers below?1 , 1 , 1 , 1 , 3 , 4 , 4 , 4 , 5 , 5 , 1200“Philosophy is a game with objectives and no rules. Mathematics is a game with rules and no objectives.” Anon
- 12. Euclid’s axiomatic system:1.Any two points can be joined by a straight line.
- 13. 2. Any straightline segment can be extended indefinitely in a straight line.
- 14. 3. Given anystraight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- 15. 4. All rightangles are congruent.
- 16. 5. If twolines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitable must intersect each other on that side if extended enough. (Parallel Postulate)
- 17. 5. Through apoint not on a given straight line, one and only one line can be drawn that never meets the given line. (Playfair’s Parallel Postulate XVIII)
- 18. 6. Things thatequal the same thing also equal one other.
- 19. 7. If equalsare added to equals, then the wholes are equal.
- 20. 8. If equalsare subtracted from equals, then the remainders are equal.
- 21. 9. Things thatcoincide with one other equal one other.
- 22. 10. The wholeis greater than a part.GirolamoSaccheri (1667-1733) – proof by contradiction5.Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.5a.There exists no parallel to a given line through a point not on that line.5b.There exist more than one parallel to a given line through a point not on that line.This alternative did not lead to any contradictionThis would lead to a contradiction with the rest of the axioms
- 23. Carl Friederich Gauss(1777 – 1855)The fifth postulate 5. There exist more than one parallel to a given line through a point not on that lineHyperbolic geometryJanos Bolyai (1802 – 1860)Nicolai Ivanovitch Lobachevsky (1793 – 1856)
- 24. Bernhard Riemann (1826– 1866)The fifth postulate 5. There exists no parallel to a given line through a point not on that line
- 25. The second postulate2. Any straight line segment can be extended indefinitely into a straight line 2. A straight line is unboundedElliptic geometry
- 26. Proof in MathematicsConsiderthe Euclidian geometry: what is the area of a triangle? Can you justify your answer?How did your teacher presented this claim?
- 27. Mathematics as aform of artA mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. (G.H. Hardy, 1877-1947)What is that mathematicians do? If the world would be divided into “poetic dreamers” and “rational thinker”, in which category would you place mathematicians?Mathematics is the purest of the arts, as well as the most misunderstoodAesthetic principle in Mathematics: simple is beautiful.It allows more freedom of expression than poetry, art of music – which depend on the the properties of the physical universe.
- 28. Exercise:Draw a circleand then indicate a point on this circle. How many regions do you see inside the circle? Answer: 1Consider one more point on this circle. Connect the two points with a line segment. How many regions do you see?Consider one more point on the same circle. Connect this last point with any of the two points. How many regions do you see?Repeat the procedure. You will have now 5 points. How many regions do you expect to see? Check your answer. Add another point. You have 6 points of the circle, Join the last point to each of the previous 5. How many regions do you expect to see?
Editor's Notes
- #3 Three man in an air balloon are lost in the Grand Canyon. In a desperate attempt, they scream simultaneously: “Where are we”? After 15 minutes, an answer comes back: “You are lost!” “This must have been a Mathematician”, says one of them. “How do you know?” asks another. “ The first one replies promptly: “Because the answer came very late, was true, but completely useless’”
- #4 My words are: Beautiful, creative,inspirational, logical, intuitive Students’ words will be probably different and this activity could generate in small groups interesting discussion. This activity will be repeated at the end of the unit. Has student’s view changed?
- #10 Answer to question 1 can be found in Marcus DuSautoy’sChristmas lectures – “Ch1 – The Story of Never Ending Numbers”, starts at 13:25, finishes 18:20.Answer to question 2 and a good introduction to fractals can be found in the same series, “Ch2 – The story of elusive shapes”, starts at 12:49 and ends at 18:06
- #12 The answer depends on what we mean by “average”. If we meant “add the numbers and divide by 11”, the answer is 96. If we meant “the most common number in the list”, the answer is 1. If we meant “the number in the middle”, the answer is 4. Which one is then true? Mathematicians name each of these averages differently. They are called mean, mode and median respectively. Some definitions will be then useful in order to make mathematical progress.Definitions are called axioms. Starting from axioms and performing a chain of logical steps Mathematicians build theorem. They represent the new knowledge.
- #14 Notice that all this had started as a intellectual game.Where the Euclidian geometry is wrong? The problem that someone find is that is does not correctly represent the three-dimensional universe that we live in.
- #17 At early age mathematical knowledge is formed more using intuition. With time students get more familiar with the concept of mathematical proof.
- #19 In the previous slide we can comment on the fact that at early stage the area of a triangle was introduced intuitively. Can I count on my intuition?Answers: 1, 2, 4, 8, 16, 31The students are surprised by the result. Aren’t all mathematical examples straightforward and predictable? This example makes us think. If a seemingly obvious conjecture cannot be make after 5 steps, how many of them should I take in order to make sure that it is true?
- #23 “My own interest in mathematics began with the story of Fermat’s Last Theorem, a problem invented in the 17th century by the French scholar and judge Pierre de Fermat. Fermat wrote in the margin of his book that he had a proof that could solve the problem, but, annoyingly, he explained that there was insufficient space to write down his proof. Following his death and the discovery of his marginal note, generations of mathematicians attempted to rediscover Fermat’s proof, which resulted in rivalries, rich prizes, tragedy, suicide, duelling at dawn, and three centuries of failure. Then, in 1963, a ten year old boy read about Fermat’s Last Theorem and promised himself that he would devote his life to finding Fermat’s proof. Andrew Wiles’s childhood dream dominated his life, and, eventually, in 1986, he realised a potential strategy for attacking the problem. He spent the next seven years working in secrecy, abandoning everything except mathematics and his family - his wife only learnt about his obsession during their honeymoon. In 1993, with his proof apparently complete, Wiles announced his success to the rest of the world, but then the discovery of an error during the refereeing process meant that his entire logical framework collapsed, leading to professional embarrassment and public humiliation. He was forced to return to his study, where he spent a year struggling to correct the mistake. Just when he was on the point of admitting defeat, a brilliant insight provided him with the fix he needed and his proof was complete. At last, he had achieved his childhood dream. For me, Wiles’s story includes the essence of a romantic tale: a lost treasure, a childhood dream, ruthless ambition, hope in the face of adversity, failure, and triumph. Furthermore, Wiles was not searching for riches, but for a solution to a purely intellectual problem. His desire was not fuelled by greed, but by curiosity. Pure mathematics has few applications in the real world, rather it consists of a series of conundrums which are challenges to the mathematician. Wiles’s success will not lead to patents, rather it is a tribute to the human spirit. In 1996 I and a colleague, John Lynch, made a BBC Horizon documentary on the subject, which begins with Wiles recalling the moment his odyssey was complete, at which point he is overcome with emotion and turns away from the camera. Mathematicians are not soulless.” from Simon Singh, “Mathematical Heroes”
- #25 Who are the critics of mathematics? They are mathematicians themselves, peers of the knowledge creators. The critics reached a higher level of abstraction – they are able to evaluate the entire area of knowledge, examining the knowledge claims from the point of view of their nature and bases. Generally the critics are philosophers highly reflective mathematicians.The Congress of Mathematics in 1900 – Hilbert’s 23 problems and the directions in mathematics for the next 100 years. “We must know, we will know”
- #26 Russell and Whitehead have started with constructing the set of real numbers using sets as mathematical tool. In 1901 they discovered a contradiction regarding the sets which are or are not members of themselves.
- #27 The barber’s problem had been used by Russell himself as a illustration of the paradox that he had found in the on sets which are or are not members of themselves.
- #28 When Nazi came to power in Germany, Godel emigrated to USA. He developed a strong friendship with Einstein and Godel has discovered some paradoxes in the solutions of the equations of general relativity. Godel gave his manuscript to Einstein as a present on his 70st birthday and this manuscript made Einstein have doubts about his theory.