1. Methodology - Kenneth
1.1. Proof by Substitution
1.1.1. Making assumption by group
1.1.1.1. If a cat is an animal and all animals are organisms, then a cat is an organism
1.2. Proof by Contradiction
1.2.1. By assuming the opposite view of the subject is true and proving that wrong.
1.3. Axioms
1.3.1. Ideas which are widely accepted in the field of mathematics and are used to deduce other theories
1.3.1.1. Number Theorem
1.3.1.2. Set Theorem
1.4. Theorems
1.4.1. An idea proved by other truths
1.4.2. Examples
1.4.2.1. Pythagoras Theorem
1.4.2.2. Fermat's Last Theorem
1.5. Proof by Mathematical Induction
1.5.1. Making assumption that this is true for all cases if it is true for a few case.
1.6. The Pigeon Hole Theorem
1.6.1. If there are more items than holes to fit it into, at least one of the holes have 2 or more items
1.7. Concept of Good Mathematics
1.7.1. To make something which seems hard to do much easier to accomplish
1.7.1.1. Equations
1.8. How do you do research?
1.8.1. Blue skies research
1.8.1.1. Researching for the sake of researching, no clear goal in mind
1.8.2. Relating different areas
1.8.2.1. Using other theorems to patch up theorems
1.8.3. Attacking a famous problem
1.8.3.1. Coming up with different approaches to a problem
2. Scope & Application - Shahmeer
2.1. What's Mathematics and how can we define it?
2.1.1. The 3 views
2.1.1.1. Platonist
2.1.1.1.1. Mathematics is seen as code to understand the world around us
2.1.1.2. Constructivist
2.1.1.2.1. The nature of Mathematics is artificial and abstract
2.1.1.3. Formalist
2.1.1.3.1. An abstract game played according to invented rules
2.1.2. Pure Mathematics
2.1.2.1. Mathematics focused on mathematical thinking
2.1.2.2. Examples
2.1.2.2.1. Algebra
2.1.2.2.2. Analysis and PDE's
2.1.2.2.3. Geometry
2.1.2.2.4. Number Theory
2.1.2.2.5. Probability and Statistics
2.1.2.3. The basic foundation of maths that allows us to help solve problems. We use pure mathematics and apply it to scenarios.
2.1.3. Applied Mathematics
2.1.3.1. Mathematics with an intent of practical external benefit.
2.1.3.2. Use of pure mathematics in real life.
2.1.3.3. Examples
2.1.3.3.1. Prime number theory
2.1.3.3.2. Combinatorics
2.1.3.3.3. Computational Biology
2.1.3.3.4. Theoretical Computer Science
2.1.3.3.5. Theoretical Physics
2.2. Is Mathematics an invention or a discovery?
2.2.1. Would mathematics exist if people didn't?
2.2.1.1. Plato argues that mathematical concepts are concrete within the universe, regardless of our existence
2.2.1.2. Euclid believed that nature itself was the physical manifestation of mathematical laws
2.2.1.3. Others believe that although mathematics does exist in the universe, it is the representation of these mathematical concepts (ie numbers) that cannot exist without humans.
2.3. What is the scope of Mathematics and what is in focus of the research?
2.3.1. Role of logic
2.3.1.1. Axioms
2.3.1.1.1. Role of axioms
2.3.1.2. Dedyctuve Reasoning
2.3.1.3. Theorems
2.3.1.3.1. A statement that is/can be proved to be true on the basis of axioms or other already established theorems
2.4. Application: Algorithms
2.4.1. The math in which computers use to decide and calculate certain things.
2.4.1.1. https://www.youtube.com/watch?time_continue=93&v=ENWVRcMGDoU - This video illustrates good examples of the real life application of maths.
3. Historical Development - Kate
3.1. Disagreements in Mathematics
3.1.1. Conjecture is an opinion or conclusion formed on the basis of incomplete information
3.1.1.1. Conjecture supported by a proof can become a theorem
3.1.1.1.1. Eg. What should be used to cover an enormous field with equal pieces without leaving any gaps? In 300 Pappus of Alexandria said the best is to use Hexagons because they cover the same surface and have the smallest border. However, he did not demonstrate it. In 1999 Thomas Hales demonstrated it and and proved that Pappus was correct.
3.1.1.2. Conjecture can be replaced by a better one
3.1.1.2.1. Eg. What should be used to fill the space space (3D) with equal pieces without leaving any gaps? In 1887 Lord Kelvin said it is best to use truncated octahedron. In 1993 Weaire and Phelan suggested a better structure to fill the space - Weaire-Phelan structure. This structure will remain the best until something better is suggested.
3.1.2. Theorem is a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths
3.1.3. Geometry
3.1.3.1. Euclid (300 BCE) wrote "Elements" (13 volumes) in which he structured and supplemented the work of many mathematicians that came before him. "Elements" consists of:
3.1.3.1.1. Deffinitions
3.1.3.1.2. Common notions
3.1.3.1.3. Postulates
3.2. Paradigm shifts - revolutions in which the entire point of view changes
3.2.1. Paradigm shifts occur when the old paradigm runs out of intellectual steam.
3.2.2. Bourbakism
3.2.2.1. Bourbakists set out to place the whole of mathematics on a unified, general, and very abstract, basis. Their idea was to lay down general principles from which all the special cases that had intrigued previous generations of mathematicians can be derived.
3.2.2.1.1. To understand Bourbakism and its messages you have to know the general principles. Mathematicians knew those principles but not the rest of the world. This led to misunderstanding.
3.2.2.2. Bourbakism had ranged triumphantly over the topology of multidimensional spaces, solving almost all of the important problems in 5 or more dimensions.
3.2.2.2.1. Toplology
3.2.2.2.2. Mathematics of knots
3.3. Significant events in the history of Mathematics
3.3.1. Brahmagupta and the Invention of Zero (628 AD)
3.3.1.1. Solved the problematic arithmetical operations when using Roman numerals; much easier using Arabic numerals
3.3.1.2. Zero was thought of as a placeholder in positional systems (Babylonians, Mayans, Greeks and Chinese)
3.3.1.3. ‘When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes a zero.’
3.3.1.3.1. He did not figure out the division with zero: 0/0=0 à indeterminate - undefined result: L’Hopitale’s Rule (17th century)
3.3.2. Axiomatisation (Euclid proves which truths are actually true
3.3.2.1. if a=c and b=c, then a=b
3.3.3. Fibonacci, F.Sequence and Phi
3.3.3.1. Brought Arabic numerals to Europe
3.3.3.2. In his book included a number of mathematical problems: the most famous one is about breeding of rabbits, and the pattern of the sequence of this reproduction
3.3.3.2.1. When one divides a number with the one before it, one comes closer and closer to the irrational number PHI (Euclid's Ratio)
3.3.3.3. PHI can be also found in the Equiangular Spiral, also called Bernoulli's Logarithmic Spiral
3.3.4. Al-Khwarizmi & The Invention of Algebra
3.3.4.1. Al-Khwarizmi: Hisab Al-jabr w’al-muqabala (The Compendious Book on Calculation by Completion and Balancing)
3.3.4.2. Focus: solving equations related to shapes
3.3.4.3. Language: words to explain the problem and pictures to solve them
3.3.4.4. Methods (used to simplify his equations): al-jabr ‘completion’, and al-muqabala ‘balancing’ à algebraic operations used nowadays: letters of the alphabet represent the unknown numbers and we use the arithmetic operators
3.3.4.4.1. These geometric proofs were one of the first methods figuring out which numbers the letters actually represent
3.3.4.4.2. Central part of computer programming à actual use of words
3.3.5. Rene Descartes and Analytic Geometry
3.3.5.1. Combination of algebra and geometry
3.3.5.2. If a letter represents number, then two numbers can represent a point in space à Cartesian coordinates (x,y) à equations can define curved lines etc.
3.3.5.2.1. essential tool for science and engineering
3.3.6. Fermat’s Last Theorem
3.3.6.1. Fermat left a note in the margins of Diophantus’ Arithmetica: ‘I have discovered a truly remarkable proof which this margin is too small to contain.’ (The author’s guess is Fermat’s proof would not have been correct.)
3.3.6.1.1. Solved in 1994 by Andrew Wiles; 150-pages-long proof
3.3.7. Pascal's Triangle (*not really Pascal's)
3.3.7.1. Exploration of number theory and triangular numbers
3.3.7.1.1. Enables us to expand Binomials
3.3.8. Counting in Binary and Computers
3.3.8.1. Using bits to count; bits are represented in Arabic numerals 0 and 1
3.3.8.2. The origins are ancient, but first investigated by Gottfried Leibnitz, who saw the two numerals as representation of Creation, 1 standing for God and 0 for Void
3.3.9. Babbage Difference Engine
3.3.9.1. Developed this first automatic machine producing more accurate tables (avoiding human error) using 2 orders of difference
3.3.9.2. He also designed a bigger machine of 6 orders of difference, but it was not constructed till 200 years later by London Science museum (the technology in his times just wasn't there
3.4. Is there an absolute certainty in Mathematics?
3.4.1. eg. Peano Arithmetic
3.4.1.1. 7 axioms (including induction)
3.4.2. Properties, principles
3.4.2.1. Powerful maths systems are sufficiently expressive
3.4.2.2. Completeness - ability to prove every true statement
3.4.2.3. If we can prove something, we should not be able to prove the opposite of that thing
3.4.2.4. Consistency - proving at most one of a statement and its opposite
3.4.2.4.1. Inconsistency is disastrous for the system
3.4.3. Gödel's First Incompleteness Theorem
3.4.3.1. "Any sufficiently expressive math system must be either incomplete or inconsistent"
3.4.4. Gödel's Second Incompleteness Theorem
3.4.4.1. "A consistent math system cannot prove its own consistency"
3.4.5. Russel's Paradox (Barber's Paradox)
3.4.5.1. He discovered that something can be both true and not true at the same time - this would mean that Maths is faulty.
3.4.6. Turning Machine
3.4.6.1. Turing investigated the universality of logical arithmetics statements being true or false, and proved it was impossible. His proof led to the invention of Turing Machine - a theoretical work behind electronic computers.
4. Language & Concepts - Krishna
4.1. Is Mathematics a universal language?
4.1.1. How can the knowledge of numerical systems and their history help is to understand Math in relation of TOK?
4.1.1.1. A Brief history of Numerical Systems
4.1.1.1.1. 1, 2, 3, 4, 5, 6, 7, 8, 9, 0
4.1.1.1.2. Early humans likely counted animals in a flock or members in a tribe
4.1.2. How does 'labelling' affect the language used in Maths?
4.1.2.1. Contrasts between British English and American English
4.1.2.1.1. Billions and Trillions
4.2. Metaphors and Symbols
4.2.1. Our whole number system is based on being able to change what we count as "one"
4.2.1.1. Change Units
4.2.1.1.1. Compose
4.2.1.1.2. Partition
4.2.1.1.3. Process can be on going
4.2.1.2. "One isn't always one"
4.2.1.2.1. Is a hundred, one hundred, ten tens or a hundred ones?
4.2.1.2.2. 1 changes its value
4.2.2. Schema
4.2.2.1. A mental structure; abstract structures used in Mathematics
4.2.2.1.1. Infinities and Continuum Hypothesis
4.2.2.1.2. Numbers and Set Theory
4.2.3. Variables and Constants
4.2.4. Metaphors
4.2.4.1. "Understanding is based on the ability to change perspectives"
4.2.4.2. Finding a pattern, a connection, a structure or some regularity
4.3. Linear Notation and Higher Dimensional Algebra
4.3.1. New kind of "higher dimensional algebra"
4.3.1.1. Symbols are related not just to those to the left and those to the right, but also up and down, or out of the page, as well
4.3.1.2. Becomes closer to and more able to model some geometric situations
4.3.1.3. Leads to the formulation and proofs of new theorems
5. Personal Knowledge - Tiana
5.1. The importance of mathematics to personal experience
5.1.1. Search for understanding of the world: the need for survival and the wish to know what there is
5.1.1.1. need a science of structure, method of knowing what is true; mathematics underlies this and is the basis for the other subjects
5.1.2. Mathematics brings humility
5.1.2.1. in deciding the truth about one apparently simple and clear statement, being aware of the limitations of mathematical truth, that not all that is true can be proved
5.1.2.1.1. mathematicians do not write the final solution or the unified theory which will solve everything
5.1.2.2. Mathematicians look for the surprises which show us a new view of the world, and new riches to explore
5.1.2.2.1. the world holds many features to be explored using this language - eg. vein patters in leaves
5.2. "Imagination is more important than knowledge" - Albert Einstein
5.2.1. Relates to Maths because the language lets you explore unearthly and invisible elements
5.2.1.1. could be giant / minute so imagination as a WOK is necessary
5.2.2. Mathematicians are good at understanding and imagining moving things around.
5.2.2.1. Eg: moving one side of an equation to another/ changing a pattern in space
5.2.3. The objects and ideas that mathematicians use are a variety of remembered processes. The representation of these ideas in writing is by contrast often bare and sparse
5.2.3.1. the difficulty of learning the use and application of maths
5.2.3.2. However, writing out maths allows for each person to make the interpretation and internalisation most appropriate to themselves