Tok- Maths Presentation
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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 in your view is Mathematics? 2. Choose three words that best describe Maths. Post your words at http://www.wallwisher.com/wall/MrsVTOK1 http://www.wallwisher.com/wall/MrsVTOK2
- 4. Four questions for you to answer: Primes were invented (discovered?) by: a) The Ancient Greeks b) Insects c) The Ancient Chinese
- 5. 2. The length of coastline in Britain is: a) 18 000 km b) 36 000 km c) infinite d) all of the above
- 6. 3. The sum of the angles in a triangle is: a) 180 degress b) less than 180 degrees c) more than 180 degrees d) all of the above
- 7. 4. The main purpose of mathematics is to, a) be a tool for predicting real world events and real world problems b) develop critical thinking skills c) to create new mathematics d) provide a challenge e) provide a qualification for employment f) be abstract g) give something for geeky dudes to do
- 8. New answers to old questions:
- 9. A Mathematician’s Answers 1. 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 Answers 3. d) All of the above! 4. c) The main purpose of Mathematics is to create new Mathematics
- 11. Formation of Mathematical Knowledge What 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
- 13. 2. Any straight line segment can be extended indefinitely in a straight line.
- 14. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- 15. 4. All right angles are congruent.
- 16. 5. If two lines 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 a point 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 that equal the same thing also equal one other.
- 19. 7. If equals are added to equals, then the wholes are equal.
- 20. 8. If equals are subtracted from equals, then the remainders are equal.
- 21. 9. Things that coincide with one other equal one other.
- 25. The second postulate 2. Any straight line segment can be extended indefinitely into a straight line 2. A straight line is unboundedElliptic geometry
- 26. Proof in Mathematics Consider the Euclidian geometry: what is the area of a triangle? Can you justify your answer? How did your teacher presented this claim?
- 27. Mathematics as a form of art A 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 misunderstood Aesthetic 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 circle and then indicate a point on this circle. How many regions do you see inside the circle? Answer: 1 Consider 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?
- 29. Any mathematical conjecture faces the judgment of a FORMAL PROOF “I remember the time when I was kidnapped, and they sent a piece of my finger to my father. He said he wanted more proof.” Rodney Dangerfield, American actor
- 30. Theorem. The sum of any two even numbers is even. Proof: (try to produce a convincing argument) Definition. An even number is a number that is a multiple of 2. Let a and b be the two even numbers. a is a multiple of two, therefore a can be written as 2k, where k is a positive integer. b is a multiple of two, therefore b can be written as 2l, where l is a positive integer. Therefore a+b=2k+2l. Factorising the expression in the right hand side, a+b=2(k+l) Therefore a+b is a multiple of 2, so it is an even number. I am done!
- 32. is a deductive process
- 33. proof doesn’t give only certitude, but also understanding and very often new knowledge
- 34. conclusions that come through mathematical proof are certainwithin the system of axioms that it is built upon
- 35. Pierre de Fermat (1601 – 1665) Andrew Wiles (born 1953)
- 36. Back to the formation of mathematical knowledge: Forming knowledge claims the existence of a set of axioms axiomatic system (independent) consistent complete An axiomatic system is called consistent if it lacks contradiction, i.e. within the system cannot be proven both a statement and its negation. The system is called to be complete if any statement can be proven or disproven within the system. Independence is not a requirement for the axiomatic system. One axiom is called independent if cannot be proven from the other axioms. In an independent system all axioms are independent.
- 37. The vain search for absolute truth At the end of 19th century the critics of mathematics reached a shocking conclusion Mathematical knowledge has limitations. Before the development of non-Euclidian geometries, mathematicians thought that they possessed the absolute truth. The battle to restore the status of mathematics as the area of knowledge providing the eternal truth has begun “We must know, we will know” (Hilbert, 1900)
- 38. Could Mathematics reach the state of completeness? Can all statements be proved or disproved from the axioms within the system? Can the consistency of axioms be proved? Attempt to construct the entire field of mathematics from the principles of logic alone. “Not ignorance, but ignorance of ignorance is the death of knowledge.” Alfred North Whitehead “Many people would sooner die than think. In fact, they do so” Bertrand Russell
- 39. Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves. Under this scenario, we can ask the following question: Does the barber shave himself?
- 40. Within any consistent axiomatic system there are true propositions that cannot be proven to be true. Kurt Godel (1906 – 1978)
- 41. Mathematics and Natural Sciences http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html http://www.youtube.com/watch?v=dK_narib4do
Notes de l'éditeur
- 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’”
- 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?
- 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
- 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.
- 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.
- At early age mathematical knowledge is formed more using intuition. With time students get more familiar with the concept of mathematical proof.
- 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?
- “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”
- 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”
- 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.
- 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.
- 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.