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.