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Mathematics- Scope What is the area of knowledge about? What practical problems can be solved...

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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?
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Page 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.

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?

Page 2: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

The Mathematical Paradigm

• axioms

• deductive reasoning

• theorems

Page 3: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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

Page 4: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

Deductive Reasoning

All men are mortal (1)

Socrates is a man (2)

Therefore Socrates is mortal (3)

Page 5: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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

Page 6: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

A problem

Given a + c =180

Prove b = c

ao bo co

Page 7: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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

Page 8: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

Proofs and conjectures

The sum of the first n odd numbers = n2 (where n is any number)

Page 9: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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

Page 10: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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?

Page 11: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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

Page 12: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

Problem 3

Page 13: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

The Basis of Mathematics

Page 14: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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

Page 15: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

Is Mathematics discovered or invented?

Page 16: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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)

Page 17: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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

Page 18: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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?

Page 19: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.
Page 20: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

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

Page 21: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

‘The Unreasonableness Effectiveness of Mathematics’

Page 22: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

WOKs

Memory Sense Perception Language Reason Emotion Intuition Imagination Faith

Page 23: 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 Questions

Page 24: Mathematics- Scope What is the area of knowledge about? What practical problems can be solved through applying this knowledge? What makes this area of.

Linking Questions


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