Mathematics- Scope

• What is the area of knowledge about?

• What practical problems can be solved through applying this knowledge?

• What makes this area of knowledge important?

• What are the current open questions in this area?

• Are there ethical considerations that limit the scope of inquiry?

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 points2. A finite straight line may be extended without limit

in either direction3. It shall be possible to draw a circle with a given

centre and through a given point4. All right angles are equal to one another5. 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

a + c = 180 given

and a + b = 180 angles on a straight line (theorem 4)

therefore a + c = a + b by substitution

therefore b = c 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 47 48 49 50

100 99 98 97 96 …55 54 53 52 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’

WOKs

Memory Sense Perception Language Reason Emotion Intuition Imagination Faith

Knowledge Questions

Linking Questions