TOK
Area of Knowledge
Diana Denisova & Victoria Starkova
Invented: Golden Ratio
History of Mathematics
"There is not a shred of evidence to back up the claims, and every reason to assume they are both false. Simply a myth"
Invented: Zero
"The symbol "0" is an extremely great invention . It helped in writing smallest to biggest numbers with ease. The place value notation is great. The entire progress of science and technology was due to invention of zero."
2000 BC Egyptians
Moscow papyrus:
Rhind papyrus (instruction manual):
- ×, ÷, +, -
- fractions, area formulas
- composite and prime numbers
- arithmetic and geometric means and series
- first linear equations
- second order algebraic equations
3000 BC Babylonians
- multiplication tables on clay tablets
- no decimal point
- no positional notation (O!)
- true place value system
- (digits on the left -> larger value)
- base 60 numerical system
- (60 seconds, 60 minutes, 360°)
"Man has fingers equal to 9+1. They wanted to make the "( 9+1) th " number repetetive . They deliberately wrote "10" instead of "9+1"."
"Amusingly, in 1994, a relation was 'discovered' even between the 'number of the Beast' and the Golden Ratio"- Just a coincidence resulting from residual fascination in numerology
“[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.”
–Galilei, Galileo (1564 – 1642), Opere Il Saggiatore p. 171.
Discovered: Golden ratio
"Another spurious claim is that if you measure the distance from the tip of your head to the floor and divide that by the distance from your belly button to the floor, you get GR. But this nonsense."
"Once upon a time there was no zero. Of course people knew if they had nothing, but there was no mathematical notation for it. Zero was independently invented only three times."
1.61803398875
600 BC Greeks
Geometry
Use of deductive reasoning
- (before only inductive -> repeat observations to establish rules)
- Logic!!
Thales' Theorem
Pythagoras
- "All is number" school
- Pythagorean theorem
Plato
- Platonian Academy
- analytical method
Eudoxus & Archimedes
- "method of exhaustion" -precurser to integration
Euclid
- "Elements"
- axiomatic method: definition, axiom, theorem, proof
- logical framework of mathematical knowledge
Ptolemy
- some use of O
- "How can nothing be something?"
Discovered: Zero
Renaissance
Algebra and accounting intertwined
Luca Pacioli
- "Summa Arithmetica" for merchants' bookkeeping
- 1st published book on algebra
- "plus and minus" signs
Solutions to cubis and quartic eguations
Trigonometry emerges for navigation
- chart of sines and cosines published
"There is no doubt that anybody who grew up in a western or mideastern civilization is a pupil of the ancient Greeks, when it comes to mathematics, science, philosophy, art, and literature.
Medieval
- God had ordered "all things in measure, and number, and weight".
- Quadrium: arithmetic, geometry, astronomy, and music
- Boethius, Fibonacci, Oxford scholars
- physical problems: velocity, force, resistance, uniform acceleration
Ancient joke: "He found out about zero; we will have to kill him."
"Nothing comes from nothing"
(ex nihilo nihil fit)
MODERN DAY
"KING LEAR: ..what can you say to draw
A third more opulent than your sisters? Speak.
CORDELIA: Nothing, my lord.
KING LEAR: Nothing?!
CORDELIA: Nothing.
KING LEAR: Nothing will come of nothing, speak again.
“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.
–Johannes Kepler
During Scientific Revolution
logarithms first used
Galileo, Brahe, Kepler-astronomic discoveries
Rene Descartes- analytical geometry -"Discourse on the Method
Leibnitz & Newton - calculus
Pascal & de Fermat - probability
Euler:
- graph theory- "Seven Bridges of Königsberg" Problem
- standardized math terms and notations "π" " i"
Gauss- geometry, series, fundamental theorems
Is mathematics discovered or invented?
What is mathematics?
Pure mathematics is, in its way, the poetry of logical ideas.
~Albert Einstein
Methodology
n. "the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations"
-Merriam Webster
First known use:
1573
Black holes result from God dividing the universe by zero. ~Author Unknown
Deductive reasoning
Logic
rationalistic thinking
axiomatic method: definition, axiom, theorem, proof
language for science
precise & accurate
"always the right answer"
little room for emotion