Theory of Knowledge:

1. Theory of Knowledge: Mathematics

2. What is maths? • In order to discuss what maths is, it is helpful to look back at how maths as a discipline developed. • The ancient world had made impressive discoveries in mathematical methods. The Egyptians had discovered the 3:4:5 triangle had a perfect 90° angle, the Babylonians the quadratic equation and the concept of place value. • They used these methods to build and administrate their empires, but their maths was more like a recipe book in that they had a method that they followed blindly, confident that it would lead them to the correct answer. Why and how it worked was immaterial, only that it did.

3. The Father of Mathematics • While numerical manipulation and geometric knowledge are impressive they are no more maths than cleaning a test tube is chemistry, or bringing a paintbrush into contact with a canvas is Art. • Pythagoras (in the 6th century BCE) was the first person to look beyond tricks that worked, and to examine why they worked. The Chinese and Babylonians has discovered the relationship c2 = a2 + b2, now known Pythagoras’ Theorem over 1000 years before he was born. The reason why it is called Pythagoras’ Theorem is that he was the 1st person to prove that it must be true for all right-angled triangles. • The concept of proof, and consequently of mathematical truth was the birth of mathematics as a discipline in its own right.

4. Mathematical Proof vs. Scientific Method What makes Maths a vastly different area of study to science is how a mathematician and a scientist go about constructing theorems and theories. Let us look at how an idea becomes a theorem/theory in each discipline. c a b

5. Scientific Method: A scientist would examine a number of right-angled triangles and observe that their sides seem to follow the rule c2 = a2 + b2. S/he would then advance the formula, being the best model available, as a hypothesis. Next s/he would make a number of predictions about possible triangles that fit the rule, construct them, then verify that the resulting triangle is right-angled. After many fellow scientists had also reproduced the same results and verified similar prediction the hypothesis would become accepted by the community as a scientific theory. The theory would remain valid and useful as long as all future results obtained continued to fit the theory.

6. Mathematical Method: There are literally dozens of mathematical proofs of Pythagoras’ Theorem, here is a simple geometric one:

7. Chessboard Problem A chessboard has two squares removed from opposite corners. The challenge is to cover the board completely with dominos, each one being equal in size to two of the board’s squares.

8. Absolute Proof & Axioms Classical proof as invented by Pythagoras starts with Axioms. An Axiom is a statement which can be assumed to be true, or is self evidently true. By then arguing logically it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless then the conclusion will be undeniably valid and can be termed a Theorem.

9. Pythag forever... Science is constantly evolving, but maths is forever. Pythagoras’ theorem was proved 2500 years ago, and unlike the science of the day is as valid now as it was then. Due to the way in which theorems are constructed, any that are proved to be flawless are absolutely and eternally true, and hence maths is the only area of knowledge in which you can find genuine certainty, and truth.