Transcript TOK
Areas of Knowledge: Mathematics Mathematics The Mathematical Paradigm • axioms • deductive reasoning • theorems Axioms: Euclidean Geometry Euclid’s five axioms:1. It shall be possible to draw a straight line joining any two points 2. A finite straight line may be extended without limit in either direction 3. It shall be possible to draw a circle with a given centre and through a given point 4. All right angles are equal to one another 5. There is just one straight line through a given point which is parallel to a given line Deductive Reasoning All men are mortal (1) Socrates is a man (2) Therefore Socrates is mortal (3) Theorems 1. Lines perpendicular to the same line are parallel 2. Two straight lines do not enclose an area 3. The sum of the angles of a triangle is 180 degrees 4. The angles on a straight line sum to 180 degrees A problem Given a + c =180 Prove b = c ao bo co A proof and therefore therefore a + c = 180 given a + b = 180 a+c=a+b b=c angles on a straight line (theorem 4) by substitution QED Proofs and conjectures The sum of the first n odd numbers = n2 (where n is any number) Goldbach’s conjecture Every even number is the sum of two primes: 2 =1+1 4 =2+2 6 =3+3 8 =5+3 10 = 5 = 5 Problem 1 There are 1,024 people in a knock-out tennis tournament. What is the total number of games that must be played before a champion can be declared? Problem 2 What is the sum of the integers from 1 to 100? 1 2 3 4 5... 46 100 99 98 97 96 …55 47 54 48 53 49 52 50 51 Problem 3 The Basis of Mathematics The Basis of Mathematics: Options • Empiricism- mathematical truths are empirical generalisations • Formalism- mathematical truths are true by definition • Platonism- mathematical truths give us a priori insight into the structure of reality Is Mathematics discovered or invented? Riemannian Geometry A Two points may determine more than one line (instead of axiom 1) B All lines are finite in length but endless- ie circles (instead of axiom 2) C There are no parallel lines (instead of axiom 5) Riemannian Geometry: Theorems 1 All perpendiculars to a straight line meet at one point 2 Two straight lines enclose an area 3 The sum of the angles of any triangle is greater than 180 degrees Lateral Thinking Problem A hunter leaves his house one morning and walks one mile due south. He then walks one mile due west and shoots a bear, before walking a mile due north back to his house. What colour is the bear? Applied Mathematics ‘The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.’ - Eugene Wigner ‘The Unreasonableness Effectiveness of Mathematics’ Linking Questions