Mathematics

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Mathematics by Mind Map: Mathematics

1. Scope & Application (Candice)

1.1. 3 Bias View

1.1.1. Platonists

1.1.1.1. Mathematics is used as a code to understand the world around us

1.1.2. Constructivists

1.1.2.1. Mathematics is artificial and abstract (mathematical Humanism)

1.1.3. Formalists

1.1.3.1. An abstract game played according to invented rules

1.2. Two main concepts:

1.2.1. 2. Patterns

1.3. Pure mathematics

1.3.1. Focused on mathematical thinking

1.3.2. No particular intent of use

1.3.3. E.g. Algebra * Algebraic Geometry, Analysis & PDEs, Mathematical Logic & Foundations

1.4. Applied mathematics

1.4.1. With an intent of practical external benefit

1.4.2. E.g. Prime Number Theory, Combinatorics Biology, Physical Applied Mathematics, Theoretical Physics

1.4.3. Algorithm

1.4.3.1. A set of instructions with which to solve problems

1.5. Advocates for the Independent Reality of Mathematics

1.5.1. Pythagoreans of 5th Century Greece believed numbers were both living entities and universal principles

1.5.1.1. One- "the monad” the generator of all other numbers and source of all creation

1.5.1.2. Numbers were active agents in nature

1.5.2. Plato argues mathematical concepts were concrete which as real as the universe itself

1.5.3. Euclid, the father of geometry believed nature itself, was the physical manifestation of mathematical laws

1.5.4. Other argues, numbers may or may not exist physically. Their truth value are are on rules that humans created

1.5.5. Mathematics related to natural science

1.5.5.1. Mathematical knot theory first developed around 1771 to describe the geometry of position

1.5.5.1.1. In the late 20th century was used to explain how DNA unravels itself during the replication process

1.6. Mathematical Theorems

1.6.1. Proofs

1.6.1.1. Providing a solid foundation for mathematicians, logicians, statisticians, economists, architects and so on to build their theory on

1.6.1.2. Well-established tule to PROVE beyond a doubt that some theorem is true

1.6.2. Euclid

1.6.2.1. Father of Geometry

1.6.2.2. Set out to formalise mathematics by establishing the rules of the game

1.6.2.2.1. Axioms (the rule of the game)

1.6.2.3. Rule is use them to prove what you think is true

1.6.2.3.1. If you can't, your theorem or idea be false

1.6.2.4. QED- What was to be proved

2. Methodology (Yuma)

2.1. Mathematics is commonly known as the subject with the highest modality, because every answer is either right or wrong.

2.1.1. Logical Deduction

2.1.1.1. proof by Substitution: working backwards to find the solution eg all cats are animals; all animals can communicate therefore all cats can communicate.

2.1.1.1.1. Eg

2.1.1.2. proof by Contradiction: all contradictions are always false without exception.

2.1.1.3. Axioms: Initial statements providing us with truths. eg: Logarithm rules Differentiation Rules (Power, Chain.....)

2.1.1.4. Theorem: A true statement based on other accepted truths

2.1.1.4.1. Proofs: A Proof of a theorem is a finite sequence of claims being derived logically.

2.1.1.4.2. David Hilbert (1862-1943) one of the founders of proof theory and mathematical logic; responsible for further axiomatisation of Geometry; his approach signalled the shift to modern axiomatic method in algebra

2.1.1.4.3. Pigeon Hole Theorem

2.1.1.5. Research in Mathematics

2.1.1.5.1. Blue Skies Research: research for the research sake, without any clear goal

2.1.1.5.2. Relating different areas of research: Filling in the gaps

2.1.1.5.3. Attacking a famous problem: one needs to research as what has been done and come up with new ideas, possibly techniques.

2.1.1.5.4. Apply a standard method to a standard type problem: the aim is to simplify the problem, and possibly in the future apply the outcomes to a new research.

3. Language & Concepts (Jong)

3.1. Universal language

3.1.1. only language shared by all human beings regardless of culture, religion, or gender

3.1.1.1. E.g. ㅠ is 3.14159.... no matter what country a person is from - uniform

3.1.2. Importance of labelling or units

3.1.2.1. representing the things that we measure

3.1.2.2. make it easy to express very large or very small numbers

3.2. Numbers and Set Theory

3.2.1. Natural numbers

3.2.1.1. positive integers (1,2,3...)

3.2.2. Integers

3.2.2.1. natural numbers + their inverses + 0

3.2.2.1.1. -2,-1,0,1,2...

3.2.3. Real numbers

3.2.3.1. Rational numbers

3.2.3.1.1. numbers that can be represented in a fraction; ratio between two integers

3.2.3.2. Irrational numbers

3.2.3.2.1. numbers that cannot be represented in a fraction form

3.2.4. Complex numbers

3.2.4.1. Imagniaries

3.2.4.1.1. i=√-1

3.3. Infinities and Continuum Hypothesis

3.3.1. Aleph naught

3.3.1.1. lowest infinite number

3.3.1.1.1. cardinality of a set of natural numbers

3.3.2. Aleph one

3.3.2.1. cardinality of the set of all countable ordinary numbers

4. Historical Development (Audrey)

4.1. Brahmagupta & The Invention of Zero

4.1.1. solved the problematic arithmetical operations when using Roman numerals; much easier than Arabic numerals

4.1.2. Zero was thought of as a placeholder of positional systems (Babylonians, Mayans, Greeks, Chinese)

4.1.3. 628AD - India: zero is recognised as a number

4.1.3.1. a number that remains unchanged

4.1.3.2. a number multiplied by zero becomes zero

4.1.3.3. How does a zero affect negative and positive numbers, back then discussing fortunes and debts

4.1.3.3.1. "A debt subtracted from zero is a fortune." He did not figure out hte division with zero (undetermined)

4.1.3.3.2. L'Hopitale's Rule (17th century)

4.2. Axiomatization

4.2.1. Euclid proving which truths are actually true

4.2.2. If a=c and b= then a=b

4.3. Fibonacci, F. Sequence and Phi

4.3.1. brought Arabic numerals to Europe

4.3.2. number of mathematical problems

4.3.2.1. most famous: breeding rabbits and the pattern of the sequence of reproduction

4.3.3. this sequence was known by the academics in India, but Fibonacci brought it further

4.3.3.1. one can observe that when one divides a number with the one before it, one comes closer and closer to the irrational number PHI (Euclid's Ratio)

4.3.4. PHI can also be found in the Equiangular Spiral, called Bernoulli's Logarithmic Spiral

4.4. Al-Khwarizmi & The Invention of Algebra

4.4.1. Focus: solving equations related to shapes

4.4.2. Language: words to explain the problem and pictures to solve them

4.4.3. Methods (used to simplify his equations):

4.4.3.1. al-jabr ‘completion

4.4.3.2. al-muqabala ‘balancing’algebraic operations used nowadays

4.4.3.2.1. letters of the alphabet represent the unknown numbers and we use the arithmetic operators

4.5. Rene Descartes and Analytic Geometry

4.5.1. combination of algebra and geometry

4.5.2. if a letter represents number, then two numbers can represent a point in space Cartesian coordinates (x,y) equations can define curved lines etc.

4.5.3. essential tool for science and engineering

4.6. Fermat’s Last Theorem

4.6.1. Fermat left a note in the margins of Diophantus’ Arithmetica

4.6.1.1. ‘I have discovered a truly remarkable proof which this margin is too small to contain.’

4.6.1.2. Solved in 1994 by Andrew Wiles; 150-pages-long proof

4.7. Pascal's Triangle

4.7.1. exploration of number theory and triangular numbers

4.7.2. enables us to expand Binomials

4.7.3. Pascal was a religious person and at times attributed his work to "divine intervention"

4.7.4. was not really Pascal's Triangle, as many before Pascal were intrigued by these "patterns and secrets"

4.7.5. !

4.8. Counting in Binary and Computers

4.8.1. using bits to count; bits are represented in Arabic numerals 0 and 1

4.8.2. 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

4.9. Babbage Difference Engine

4.9.1. developed this first automatic machine producing more accurate tables (avoiding human error) using 2 orders of difference

4.9.2. 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)

4.10. 1931: Kurt Goedel stated that Mathematics will be forever incomplete

4.10.1. wrote Goedel's Incompleteness Theorems

4.10.1.1. Turing investigated the universality of logical arithmetics statements being true or false, and proved it was impossible

4.10.1.1.1. . His proof led to the invention of Turing Machine - a theoretical work behind electronic computers.

4.10.2. "For any consistent formal theory that proves basic arithmetical truths, it is possible to construct an arithmetical statement that is true, but not provable in the theory. That is, any consistent theory of a certain expressive strength is incomplete."

5. Personal Knowledge (Iris)

5.1. Search for understanding the world

5.1.1. Method of knowing what is true

5.1.1.1. Limitations of mathematical truth: Not all that is true can be proven

5.1.2. Gives a science of structure

5.1.3. Show new view of the world and new riches to explore

5.1.3.1. Experiences lead us to expect these to appear

5.1.3.1.1. Humans have the ability to reflect on what they do

5.2. Math underlies and is necessary for all other subjects

5.2.1. Mathematics is used every day. We are always using: numbers, graphs, addition, multiplication etc.

5.2.1.1. buildings, business, information systems

5.2.1.2. trying to find underlying patterns

5.3. Importance of imagination and interpretation

5.3.1. Understanding of the effects of repetitive processes and methods

5.3.2. Mathematicians are good at imagining moving things around

5.3.2.1. ie. from one side of an equation to another; changing a pattern in space

5.3.2.2. use of movement of arms and hands to convey what is happening --> trying to share knowledge

5.3.3. bare and sparse notation of mathematical notes allows for individual interpretation and internalisation