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