Lesson 1The nature and importanceof Mathematics
Definitions How would you define Mathematics? Choose 3 words that describe Mathematics for you.
What is Mathematics? Dictionary: “The science of spatial and numerical relationships” ToK Guide: “Mathematics is the study of patterns and relationships between numbers and shapes.” Ivan Nomav: “What people spend eternity being forced to do if they end up in hell.” Science Student: “A gate crasher to a Science party.” English Student: “An eleven letter word” Mathematics Student: “Everything”
How important is Mathematics? Read the Time article about the life of John Nash. Does this article challenge you to believe that there is more to the subject of Mathematics than your school Mathematics curriculum would suggest? Make a list of the reasons why Mathematical knowledge is important. Euclid of Alexandria: “The laws of nature are but the mathematical thoughts of God.” What is the meaning and significance of this?
1+ 5 Nature’s Beautiful Number : The Golden Section 2 Applications in nature, music, design, architecture, art.
An example of Mathematical knowledge Starting in the red square, try to find the correct path through the following maze, moving horizontally, vertically or diagonally In pairs, see if you can write the next square in this sequence If you manage to work it, reflect on how!
Key question to discuss about thenature of mathematics: Is Mathematics invented or discovered? Why is this an important question? Discuss this in pairs.
“God created the integers, all the rest isthe work of man” Comment.
Mathematics in the beginning… If two dinosaurs joined two other dinosaurs in a clearing, there would be four dinosaurs, even though no humans were around to observe it, and the beasts were too stupid to know it. If the four then had a race, medals could have been awarded for first, second, third and fourth. The idea of cardinality, being able to place things in order, existed and was discovered. But numbers were invented to describe and manipulate cardinality. What is the earliest evidence of human counting?
There is a 35,000 year-old baboon’s thigh bone discovered in the Lebombo Mountains of Africa marked with 29 notches representing a calendar stick. Is this evidence of counting?
One opinion: “I believe that mathematical reality lies outside us, that our function is to discover and observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’ are simply our notes of our observations.” (G. H. Hardy, Apology)
A counter argument? “Did Shakespeare “discover” his sonnets? Surely all finite sequences of English words “exist”, and Shakespeare simply chose a few that he liked.” “A block of stone contains every possible statue that can be carved from it. When a sculptor selects one of these statues, it’s said to be an act of creativity. If mathematics already exists, then so do all possible mathematics, including an infinity of incorrect, worthless, boring, irrelevant, useless, ridiculous, and incomprehensible mathematics. Shouldn’t the finding of worthwhile mathematics be given the same consideration as finding a work of art in a stone be called creative?”
Knowledge issues: So, was ‘4 + 2 = 6’ created or discovered? Is it true or false? What if I define + to mean multiply? What if my symbol ‘4’ means something different to someone else’s ‘4’?
When are the following true? 10 + 10 = 20 10 + 10 = 100 10 + 10 = 8 10 + 10 = 10
Answers…? 10 + 10 = 20 When using normal (decimal) arithmetic 10 + 10 = 100 When using binary arithmetic 10 + 10 = 8 When dealing with time – 10 o’clock add 10 hours is 8 o’clock 10 + 10 = 10 When adding 10 litres of water to a bucket that can only hold 10 litres of water
Therefore: The idea of cardinality (being able to measure elements of a set i.e. numbers of dinosaurs) existed and was discovered. But numbers themselves were invented to describe and manipulate cardinality. Who is to say 4 + 2 won’t equal 8 if someone redefines the rules and is able to convince the whole world of this new definition.
Knowledge questions: We need rules to define things, but who defines these rules? To what extent are these mathematical rules ‘knowledge’? List your own examples of mathematical rules and reflect on the extent to which these are examples of ‘knowledge’
BBC Radio Programmes If you have time, listen to the first 5 to 10 minutes of one or more of the following The number zero The number Pi Infinity ( and beyond ) The imaginary number i
Homework Review your notes from today and read p.53-61 from Alchin, and make any extra notes you need to in your journal In your journal list some examples of mathematical knowledge you have acquired and comment on how this knowledge has been acquired. Has your knowledge been acquired through an experiential, practical or propositional method, or a combination? Why do some people learn mathematical knowledge very easily and outperform their peers by years, whereas others find it almost impossible to learn, however hard they try. Describe your personal experience in this subject.
Justifying a mathematical knowledge claim In groups discuss one piece of mathematical knowledge you have acquired and try to justify this knowledge claim. Is your knowledge based on experiential, practical or propositional evidence? As a group, choose one of these knowledge claims to justify to the rest of your class Reflect on how this differs from proof claims in other areas of knowledge
What constitutes proof? Discuss the attempts to justify mathematical knowledge in the following clip from ‘Ma and Pa Kettle’: http://www.youtube.com/watch?v=Bfq5kju627c What constitutes proof in relation to mathematical knowledge? How does this compare to other areas of knowledge?
What constitutes proof? Using the minimum number of colours, shade your map of Africa so that no countries that share a common border have the same colour. How many colours were required? Can you prove that this is the minimum number of colours required for any map? Could there be a map that requires 5 colours? The States of mainland USA? Testing as many maps that we could, and finding that the use of 4 colours was sufficient in each, would not constitute proof for mathematicians – why? How is this different to the Scientific Method used to establish a scientific theory? For example; from the atom then to protons, electrons and neutrons, then to quarks and smaller sub atomic particles.
How does this story highlight the different use of empirical evidence for the three individuals. Who has used induction and who has used deduction? Explain the similarities between this story and our inability to prove the four colour map problem.
Knowledge claims: theorems and axioms The theorems used today are the accumulation of the discoveries made by many cultures. Sometimes these discoveries have been lost and then found again. Over the centuries, existing theorems lead to further discoveries and the ever expanding field of mathematical knowledge. Pythagoras and his brotherhood is thought to have started formal proof around 500 BC. Euclid ( 330 BC ): Wrote a collection of 13 books called The Elements, detailing his work, the work of the Pythagorean Brotherhood and a compilation of the mathematical knowledge to date.
Knowledge claims: theorems and axioms Based on axioms (postulates), self evident truths, and deductive reasoning: rationalism. Using these axioms, a mathematician follows deductive reasoning to formulate new theorems. While empirical evidence may be gathered to infer a mathematical truth, on its own it is not sufficient for the proof of this observation. Rigid deductive proof is required of new mathematical knowledge before it can be claimed to be true. New proofs must be peer assessed by an authority from the mathematical community before they can be claimed to be true.
Undeniable Proof? “Archimedes will be remembered when Aeschylus is forgotten, as languages die and mathematical ideas do not. Immortality may be a silly word but probably a mathematician has the best chance of whatever it may mean.” G H Hardy “In most sciences one generation tears down what another has built and what one has established another undoes.” “In mathematics alone each generations adds a new story to the old structure.” Hermann Hankel How might these quotes be used to assess how a mathematician feels about knowledge claims in their subject?
An example of mathematical proof:180 degrees in a triangle Definition; 180 degrees in a straight line. (Thank the Babylonians for this. They Line 1 liked multiples of 60) a k y Euclid’s 5th postulate (axiom) implies that if line b Line 2 x 1 and line 2 are parallel then a+b=180 and x+y=180
Base angles in the triangle must be a and y as they are on the straight line and must agree with the sum of 180 degrees as stated by the 5th Postulate (Axiom) Now, angle sum in the Line 1 triangle is a+k+y a k y But, the straight angle at the top of the triangle shows that a+k+y=180 b a y x Line 2 Therefore; angle sum in the triangle is 180 degrees QED "quod erat demonstrandum", which means "which was to be demonstrated (or proved)"
Using the previous proof for the triangle b c y a x z A diagonal of a quadrilateral divides the quadrilateral into 2 triangles. Angle sum of all triangles is 180 degrees. ( Proved earlier using Euclid’s Axiom) Therefore, the internal angles of a quadrilateral sum to 2 x 180 = 360 degrees And one for you. How many degrees in a polygon with any number of sides?
This would be fine except for… What about a triangle that is drawn on a curved surface? Pick three points on a disk and connect the points with straight lines. Notice that the sum of the angles of this triangle is always less than 180 degrees. This observation forced mathematicians to redefine the use of Euclid’s Axioms and then introduce a new field of mathematics called Hyperbolic Geometry. (Non-Euclidean Geometry)
Every Secondary school student has probably met “Pythagoras’ theorem”, haven’t they? What does it state? Rational solutions for x +y =z 2 2 2 Eg 32 + 4 2 = 52 , 52 +12 2 = 132 Rather than squaring each number, the French Mathematician Fermat tried raising each number to a different power. The example below is nearly correct when the numbers 6, 8 and 9 are cubed. 6 − +1 = 333 89 Fermat claims his new equation differs from Pythagoras original equation, which has infinitely many rational solutions, to an equation which has no solutions for integers n greater than 2, ie x y z n += n n
Fermat scribbled in his copy of Arithmetica. “ I have a truly marvellous demonstration (Proof) of this proposition which this margin is too narrow to contain.” Before he could demonstrate his proof, if it really did exist as he often boasted about his mathematical talents, he was to die several months later. And so the challenge began to prove this statement using the existing axioms and theorems. Prize offered. Thousands of attempts throughout the world over the next 300 years to find a proof. The “Winner” Andrew Wiles, 1995, after over 20 years of work Wiles uses 20th century mathematical knowledge in his proof. Did Fermat actually prove this theorem in the 17th century?
“God has a transfinite book with all the theorems and their best proofs. You don’t really have to believe in God as long as you believe in the book”. (Paul Erdos quoted in Bruce Schecter, My Brain Is Open: The Mathematical Journeys of Paul Erdos What is the difference between believing in God and being able to prove it and believing in Mathematics theorems and being able to prove them? Discuss.
Mathematics; A knowledge systemwithout doubt? How do we choose the axioms underlying mathematics? Is this an act of faith? Read Godel: Incompleteness Theorem Alchin pp 64-66 While Godel showed that it may be impossible to prove that all mathematical theorems can be proven using an axiomatic system, this did not necessarily mean that these theorems were false. If the axiomatic system, which appeared to be devoid of doubt, cannot be proven to be infallible, imagine what might happen if the man you are about to watch is thought of as the mathematician who proves that one of the founding axioms is not true. House of Cards
Plenary This is the end of this lesson. You should have some idea about the following questions. Are all mathematical statements either true or false? Can a mathematical statement be true before it has been proven? Is certainty possible in the axiomatic system that underscores mathematical knowledge? What are the roles of empirical evidence and deductive reasoning in establishing a mathematical claim? How does proof differ for mathematicians with say scientific proof?
Homework: Respond to the following Review your notes from this lesson and read the rest of chapter 4 from Alchin’s book making supplementary notes where relevant Consider your own response to this question: What counts as knowledge in ethics as compared to mathematics? To what extent would you agree that knowledge claims are as well-supported in ethics as in mathematics? Write down why some people might agree with this belief and why others might disagree. Make sure you refer to specific knowledge claims in each of these two areas of knowledge.