# Mathematics

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Mathematics

## 1. Methodology (Amelie)

### 1.1. Formal Axiomatic Approach

1.1.1. Law of excluded middle: every statement must be true or false

1.1.1.1. Always right or wrong, no in betweens, not like NS / HS

1.1.2. Logical deduction: based on tautologies

1.1.2.1. tautologies: statements always true

1.1.3. Proof by substitution

1.1.3.1. all cats are animals, all animals are cute, therefore all cats are cute

1.1.4.1. contradictions are always false no matter what

1.1.5. AXIOMS: initial statements / previous truths that provide a rule / definition

1.1.5.1. LAYER 1 Axioms of mathematical theory

1.1.5.2. LAYER 2: Axioms of set theory (generally accepted, some highly intuitive)

1.1.6. THEOREM: a true statement based on other truths

1.1.6.1. Supported by truth

1.1.6.2. eg 3D pythagorean theorem based on 2D pythagorean theorem

1.1.6.3. make assumptions that other truths are definitely true

### 1.2. Types of Proofs

1.2.1. Direct Proof: If... Then...

1.2.2. Proof by negation

1.2.3. Principle of Math Induction

1.2.3.1. eg if f(x) is real then f(x+1) is also real / can provide a real answer

1.2.4. The Pigeon Hole Principle

### 1.3. Technology and Formal Proof

1.3.1. All assumptions are explicit

1.3.2. Nothing is left out, every possible situation is included

1.3.3. Theorems are often complex

1.3.4. Sufficient evidence needs to be developed in the proof to create a theorem

1.3.5. Computers and humans check the proof

1.3.6. Computers are automated to check the eligibility and application of the proof

1.3.7. Formal proof starts off as traditional proof

1.3.8. PROOF IS ALWAYS QUESTIONED + EVALUATED

### 1.4. Quotes / Passages

1.4.1. “Everything should be made as simple as possible, but not simpler.” - Einstein

1.4.1.1. Best mathematics makes mathematics easier, however it is hard to do this

1.4.2. "I would have written a shorter letter, but I did not have the time." - Blaise Pascal

1.4.2.1. shows that often time is spent on discovering new theorems and testing rather than improving existing theorems

1.4.3. Methodology / theorems in mathematics often start off as very complicated, but upon further evaluation can be made more simple.

1.4.4. Aim is to make math easy to understand

## 2. Scope and Application (Angela)

### 2.1. Use proofs and theorems like blocks to build mathematics

2.1.1. Theories are tested and built on proofs

2.1.2. Euclid formalised mathematics

2.1.2.1. Revolutionised the way it was written, presented, and thought about

2.1.2.2. Axioms

2.1.2.2.1. Used to prove what you think is true

2.1.2.2.2. A starting point for reasoning, a premise that is accepted without proof

2.1.3. Purpose of studying proofs

2.1.3.1. Proofs are everywhere

2.1.3.1.1. Underlie architecture, art, computer programming, internet security

### 2.2. Use of logic

2.2.1. Deductive Reasoning

2.2.1.1. The process of reasoning from one or more statements (premises) to reach a logically certain conclusion

2.2.2. Main AOK: Reason!

2.2.3. Theorems

2.2.3.1. Aa statement that is/can be proved to be true on the basis of axioms or other already established theorems

### 2.3. Views of mathematics

2.3.1. Platonists

2.3.1.1. Mathematics seen as a code to understand the world around us

2.3.1.2. Mathematical objects exist, and are abstract & independent of human minds and linguistic practices

2.3.2. Constructivists

2.3.2.1. The nature of math is artificial and abstract

2.3.2.1.1. Mathematical Humanism

2.3.2.2. It is necessary to find/"construct" a mathematical object to prove that it exists

2.3.2.2.1. Proof by contradictions aren't valid

2.3.3. Formalists

2.3.3.1. An abstract game played according to invented rules

### 2.4. Since ancient times humans have debated whether math is invented or discovered

2.4.1. We created mathematical concepts to help us understand the universe around us

2.4.1.1. Truth values of mathematical statements are based on rules that humans created

2.4.1.2. Abstract relationships based on patterns discerned by brains

2.4.1.2.1. No existence outside mankind's conscious thought

2.4.1.2.2. To create useful but artificial order from chaos

2.4.2. Math is the native language of the universe itself, existing whether we find its truths or not

2.4.2.1. Pythagoreans, Plato, and Euclid believed this

2.4.3. Are numbers, shapes and equations truly real or just a theoretical concept?

### 2.5. Pure Mathematics

2.5.1. No particular intent of use

2.5.2. Mathematics focused on mathematical thinking

2.5.3. Examples

2.5.3.1. Algebra

2.5.3.2. Numer Theory

2.5.3.3. Geometry

### 2.6. Applied Mathematics

2.6.1. Mathematics with an intent of practical external benefit

2.6.2. Examples/Purpose

2.6.2.1. Use mathematical knowledge to solve practical problems

2.6.2.2. Prime Number Theory - ‘public key cryptography’

2.6.2.3. Theoretical Physics

2.6.2.4. Computational Science

2.6.2.4.1. Algorithms

2.6.2.5. Investigate patterns/trends

2.6.2.5.1. In the natural world and human world

2.6.2.5.2. In data

2.6.2.5.3. Allow us to make predictions/educated guesses about the future

2.6.2.6. Engineering

2.6.2.7. Economics/Financial management

2.6.2.7.1. Specifically for decision making

2.6.2.7.2. Eg. Demand and supply curves

## 3. Language and Concepts (Josephine)

### 3.1. Evolution of numeral systems

3.1.1. dependent on culture

3.1.1.1. 1) extension of tally marks; used new symbols to represent larger magnitudes

3.1.1.1.1. i.e: Egyptian, Greek, Hebrew, Roman

3.1.1.1.2. Issue: the larger the number, the more symbols you would have to write

3.1.1.2. 2) Positional notation

3.1.1.2.1. reusing same symbols but assigning different values based on position in sequence

3.1.1.2.2. i.e: Babylonians, Mayans, Ancient Chinese, Indian

3.1.2. current notation

3.1.2.1. current digits evolved from North African Maghreb region

3.1.2.1.1. + '0' developed by Mayans

3.1.2.2. Hindu arabic numeral systems globalised in 15th c.

3.1.2.2.1. global agreement reduces subjectivity

3.1.2.2.2. How do you know which notation is best fit?

3.1.2.3. other notations

3.1.2.3.1. Binary = base 2 (digital programming)

3.1.2.3.2. Duodecimal = base 12 (baker's dozen, Chinese Zodiac)

### 3.2. Labels

3.2.1. can cause misinterpretations, unless value is completely agreed upon

3.2.1.1. ie: billion

3.2.1.1.1. old British = a million million

3.2.1.1.2. American English = a thousand million

3.2.2. units are important

3.2.2.1. 1 cm ≠ 1 m

3.2.2.2. composed

3.2.2.2.1. 1 egg x 12 = a dozen eggs

3.2.2.3. partitioned

3.2.2.3.1. 1 loaf of bread / 12 = 12 slices

### 3.3. Purpose

3.3.1. Used to represent patterns

3.3.1.1. always creating new symbols

3.3.2. Equations: x+x = 2x

3.3.2.1. analogy between two things

3.3.2.1.1. each term = a new perspective

### 3.4. Concepts

3.4.1. ie: number, length, area, volume, curvature, symmetry

3.4.2. combined into mathematical structures

3.4.2.1. to study relationships between concepts

3.4.3. Set Theory

3.4.3.1. organises objects into collections (sets) and observes properties

3.4.3.1.1. natural

3.4.3.1.2. integers

3.4.3.1.3. real

3.4.3.1.4. complex

3.4.3.2. just like we group words in verbs, nouns...

### 3.5. WOKs for understanding

3.5.1. Reason

3.5.2. Memory

3.5.2.2. formulas (unless given)

3.5.2.3. can solve problems faster

3.5.2.3.1. requirement to understand before memorisation

3.5.3. Imagination

3.5.3.1. abstract concepts

3.5.3.1.1. i.e: infinity

## 4. Personal Knowledge (Siyu & Gaurika)

### 4.1. WOKs in maths

4.1.1. language

4.1.1.1. mathematical symbols form a new "language" to express meanings

4.1.1.1.1. every mathematicians in the world can understand it despite language barriers

4.1.1.1.2. Has limitations

4.1.1.1.3. The idea of using symbols as a representation of ideas can be applied across the various AOKs

4.1.1.2. mathematical language is well structured, not expressed casually

4.1.2. intuition

4.1.2.1. Making educated guesses or predictions

4.1.2.1.1. e.g. when proving a theory

4.1.2.1.2. may not always be reliable, so need reason to support it

4.1.2.1.3. can be influenced by memory

4.1.3. memory

4.1.3.1. remembering what is given and what is not in the question

4.1.3.2. remembering theorems

4.1.3.3. remembering what you did last time to solve a similar problem

4.1.3.3.1. practice makes perfect

4.1.3.3.2. Using similar methods to solve similar types of problems

4.1.3.4. remembering formulas

4.1.3.4.1. which formulas to apply to specific types of questions

4.1.3.5. Becoming a better mathematician involves remembering previous mistakes, errors and difficult questions that one may encounter

4.1.4. sense perception

4.1.4.1. How long is this segment?

4.1.4.2. How large is this angle?

4.1.4.3. What is the approximate ratio?

4.1.4.4. could interfere with imagination

4.1.4.4.1. e.g. 3D geometry

4.1.4.5. could confuse reason

4.1.4.5.1. confuse what is known with what is merely perceived

4.1.4.6. Helps to visualize different scenarios

4.1.4.6.1. Creating diagrams or visual representations

4.1.5. imagination

4.1.5.1. applied when picturing abstract ideas

4.1.5.1.1. e.g. 3D geometry

4.1.5.1.2. convert algebra into geometry

4.1.5.1.3. e.g. Linear algebra

4.1.5.1.4. use realistic items to represent abstract ideas

4.1.5.2. Used to come up with creative ideas to discover new proofs, methods etc.

4.1.5.2.1. Math teachers always push students to think "outside the box"

4.1.6. reason

4.1.6.1. the major WOK in maths

4.1.6.1.1. induction

4.1.6.1.2. deduction

4.1.6.1.3. implication

4.1.6.2. applied in

4.1.6.2.1. mathematical proving

4.1.6.2.2. mathematical analysis

4.1.6.3. Logical analysis

4.1.6.3.1. study of pattern and structure

4.1.6.3.2. Very methodical and step-by-step

4.1.7. emotion

4.1.7.1. emotion shapes people's interest in maths

4.1.7.1.1. if people do well in maths, they are more likely to deem it important

4.1.7.1.2. A lot of students find math tedious and boring which results in lost interest

4.1.7.1.3. the sense of achievement after I have solved a problem makes me enjoy maths

## 5. Historical Development (Gaurika and Sid)

### 5.1. Conjectures and Theroems

5.1.1. Conjecture

5.1.1.1. mathematical statement that has not yet been rigorously proved

5.1.1.2. Conjectures have some extent of truth to them however they more closely relate to a "theory" in the NS world

5.1.1.3. Conjecture supported by proof becomes theorem

5.1.1.4. example of a Conjecture

5.1.1.4.1. In the year 300, Pappus of Alexandria claimed hexagons are most efficient

5.1.1.4.2. Lord Kelvin had a conjecture stating the best shape to use to fill gaps is a truncated octahedron

5.1.2. Theorem

5.1.2.1. statement that can be demonstrated to be true by accepted mathematical operations and arguments

5.1.2.2. Theorems relate to a "law" in the NS world as it has evidence to back it up and therefore has some basis to a claim

5.1.2.3. example of a Theorem

5.1.2.3.1. in 1999 Thomas Hales used evidence that created the honeycomb theorem

5.1.2.3.2. this showed that Pappus of Alexandria was in fact right

5.1.2.3.3. Weaire and Phelan found a new structure better than the truncated octahedron and disproved Kelvin's conjecture

5.1.2.4. Theorems however are also the best only until something better shows up

5.2.1. a set of ideas used to explain something is true

5.2.2.1. a fundamental change in underlying beliefs, assumptions, theory or approach

5.2.2.2. Paradigm shifts occur when older paradigms run out of "intellectual steam"

5.2.3. Thomas Kuhn

5.2.3.1. stated that science is the subject of Paradigm Shifts

5.2.4. People usually think Math does not go through Paradigm Shifts as it is a subject of "perfect truths"

5.2.4.1. however, this isn't true. Paradigm shifts are about external viewpoints, not internal logic

5.2.4.2. see the same in a different and new way (a new framework)

5.2.5. Bourbakism (50s and 60s)

5.2.5.1. an example of a paradigm shift

5.2.5.2. what computer scientists now call data compression

5.2.5.2.1. a method for deducing special facts from general principals

5.2.5.3. there was controversy between mathematicians and physicists because of this paradigm shift

### 5.3. Key moments in the History of mathematics

5.3.1. Brahmagupta & The Invention of Zero

5.3.1.1. found that using Arabic numbers was more efficient than Roman numbers

5.3.1.2. 628 AD - in India, zero was recognised as a number

5.3.2. Axiomatisation

5.3.2.1. Euclid determining which truths are true

5.3.3. Fibonacci, F.Sequence and Phi

5.3.3.1. Arabic numbers brought to Europe

5.3.4. Al-Khwarizmi & The Invention of Algebra

5.3.4.1. relation between shapes and equations

5.3.5. Rene Descartes and Analytic Geometry

5.3.5.1. combining algebra and geometry

5.3.6. Fermat’s Last Theorem

5.3.6.1. solved in 1994

5.3.7. Pascal's Triangle

5.3.7.1. number theory, binomial expansion and triangular numbers

5.3.8. Counting in Binary and Computers

### 5.4. Absolute certainty in Mathematics?

5.4.1. are there loopholes in logic how do you know that your evidence is fully justifiable?

5.4.2. 1931 - Kurt Goedel stated that math will forever be incomplete