TOK: Mathematics

Unit 1 Day 1

Introduction

Opening Question

Is math discovered or is it invented? Think about it. Think real hard. Then discuss.

"Mathematics is the language in which God has written the universe."

-Galileo Galilei

Activity!

Go with your groups and find an example of math in the real world and brainstorm as many real world examples as you can that involve math.

Math

Math might be characterized as the search for abstract patterns

Patterns are everywhere

For example: 2+2=4

It doesn't matter what it is, if you have two of something then add two more, you have four of the

same item

Circles: If you divide the circumference by the diameter, you always get pi.

Mathematical Paradigm

Defined: "The science of rigorous proof"

Dates back to the Greeks

Euclid first person to consider

Formal system of reasoning.

Three key elements

Axioms

Deductive reasoning

Theorems

Axioms

Starting points or basic assumptions. Premises.

Can’t prove axioms (Infinite regress)

Four traditional requirements for a set of axioms.Consistency

Once proven inconsistent, you can prove almost anythingIndependence

You should not be able to deduce any more axioms from an axiomSimplicity

Clear and simple as possibleFruitfulness

Should be able to prove as many theorems as possible using the fewest number of axioms

Euclid's 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 another5. There is just one straight line through a given point which is parallel to a given line.

Euclid later used these five axioms to form theorems

Deductive Reasoning

Ex. Syllogism(1) All humans beings are mortal(2) Socrates is a human being (3) Therefore Socrates is mortal

If we say that (1) and (2) are true, then (3) must be true.

Theorems

Theorems are like Conclusions

Can be used to construct complex proofs

Derived by Euclid based on his five axioms and deductive reasoning

Lines perpendicular to the same line are parallel

Two straight lines do not enclose an area

The sum of the angels of a triangle is 180 degrees

The angles on a straight line sum to 180 degrees

Example Theorem

Pythagorean Theorema² + b² = c²The Pythagorean (or Pythagoras') Theorem is the statement that the sum of (the areas of) the two small squares

equals (the area of) the big one.

Proofs and Conjectures

Conjectures

A hypothesis that seems to work

Not necessarily true

Proofs

A theorem is shown to follow logically from the relevant axioms

Necessarily true

Inductive Reasoning

Particular to general

Not completely certain

Don't jump to conclusions (Ex. on pg 193)

Goldbach's Conjecture

Famous mathematical conjecture

Every even number is the sum of two primes

Proven up to an 18 digit number

Still can't be considered a theorem

Beauty, Elegance, and Intuition

An elegant proof might even be described as beautifulPaul Erdos spoke of “the BOOK” of mathExercise 1 (Pg. 195)

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?Exercise 2 (Pg. 195)

What is the sum of the integers from 1-100?