1. 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?

2. The Mathematical Paradigm
axioms
deductive reasoning
theorems

3. Axioms: Euclidean Geometry
Euclid’s five axioms:-
It shall be possible to draw a straight line joining any two points
A finite straight line may be extended without limit in either direction
It shall be possible to draw a circle with a given centre and through a given point
All right angles are equal to one another
There is just one straight line through a given point which is parallel to a given line

4. Deductive Reasoning All men are mortal (1) Socrates is a man (2)
Therefore Socrates is mortal (3)

5. 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

6. A problem
Given a + c =180
Prove b = c ao bo co

7. 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

8. Proofs and conjectures
The sum of the first n odd numbers = n2 (where n is any number)

9. 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

10. 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?

11. Problem 2 What is the sum of the integers from 1 to 100? …

12. Problem 3

13. The Basis of Mathematics

14. 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

15. Is Mathematics discovered or invented?

16. 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)

17. 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

18. 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?

20. 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

21. ‘The Unreasonableness Effectiveness of Mathematics’

22. WOKs Memory Sense Perception Language Reason Emotion Intuition
Imagination
Faith

23. Knowledge Questions

24. Linking Questions