Mathematics- Scope What is the area of knowledge about? What practical problems can be solved...
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Transcript of Mathematics- Scope What is the area of knowledge about? What practical problems can be solved...
![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.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649ea05503460f94ba38ba/html5/thumbnails/1.jpg)
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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The Mathematical Paradigm
• axioms
• deductive reasoning
• theorems
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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
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Deductive Reasoning
All men are mortal (1)
Socrates is a man (2)
Therefore Socrates is mortal (3)
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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
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A problem
Given a + c =180
Prove b = c
ao bo co
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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
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Proofs and conjectures
The sum of the first n odd numbers = n2 (where n is any number)
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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
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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?
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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
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Problem 3
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The Basis of Mathematics
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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
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Is Mathematics discovered or invented?
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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)
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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
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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?
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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
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‘The Unreasonableness Effectiveness of Mathematics’
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WOKs
Memory Sense Perception Language Reason Emotion Intuition Imagination Faith
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Knowledge Questions
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Linking Questions