Mathematics

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Mathematics by Mind Map: Mathematics

1. Scope & Application (Mahima)

1.1. 3 views

1.1.1. Platonists: Math=code to understand world around

1.1.2. Constructivists: Math is artificial and abstract

1.1.3. Formalists: Math is an abstract concept with imaginary rules

1.2. 2 types

1.2.1. Pure maths

1.2.1.1. Math focused on Math thinking

1.2.1.2. No real use in life

1.2.2. Applied Math

1.2.2.1. has practical benefit

1.3. Math: Invented or Discovered?

1.3.1. invented

1.3.1.1. Ancient Greece

1.3.1.1.1. numbers = living entities + universal principles

1.3.1.2. Plato

1.3.1.2.1. Math concrete

1.3.1.3. Euclid

1.3.1.3.1. Math in nature

1.3.1.4. Others

1.3.1.4.1. Numbers may be imaginary

1.3.1.4.2. math concepts are not imaginary

1.3.1.5. Math is invented logic

1.3.1.5.1. no existence outside conscious thought

1.3.1.5.2. used to create artificial order from chaos

1.3.1.6. Many different outcomes using same rules

1.3.2. discovered

1.3.2.1. math proven to be framework necessary to understand universe

1.3.2.2. often math with seemingly no real context can find be used in real life more than a century later

1.3.2.2.1. knot theory developed 1771 used in late 20th century to understand DNA structures

1.4. Proof

1.4.1. important: gives many professions a solid foundation to build and test their theories on

1.4.2. Math is a game need axioms or rules to play

1.4.2.1. if you're theorem doesn't follow the rules then it is false

1.4.3. well-established rules to PROVE beyond a doubt that some theorem is true

1.5. Algorithms

2. Language & Concepts (Nairo)

2.1. History of numerical systems

2.1.1. different civilisations had different numerical systems

2.1.1.1. Didn't use positional notation

2.1.1.1.1. Used extensions of tally marks • new symbols were added to represent larger magnitudes of value • each symbol was repeated as many times as necessary and all of them were added together

2.1.1.2. Used positional notation

2.1.1.2.1. could use the same symbol but could still assign different values based on their position in the sequence

2.2. Numbers and Symbols

2.2.1. Whole numbers

2.2.1.1. "whole number system"

2.2.1.1.1. depends on being able to change units

2.2.2. Rational and Irrational numbers

2.2.2.1. Rational

2.2.2.1.1. a number that can be represented as a ratio — a fraction

2.2.2.2. Irrational

2.2.2.2.1. a number that cannot be represented as a ratio — a fraction

2.2.2.2.2. developed by the philosopher Hippasus

2.2.3. Infinity

2.2.3.1. there are infinite number of infinities of different sizes

2.2.3.1.1. examples

3. Methodology (Ashley)

3.1. Formal Axiomatic Approach

3.1.1. Law of the excluded middle

3.1.1.1. every statement must be either true or false

3.1.2. Logical deduction in mathematics

3.1.2.1. based on tautologies (statements true under all possible circumstances)

3.1.3. Proof by:

3.1.3.1. Substitution: all cats are animals; all animals can communicate; --> therefore all cats can communicate

3.1.3.2. Contradiction: contradictions are always false --> under any circumstances

3.1.4. Axioms: initial statements, previous truths

3.1.4.1. Axioms of mathematical theory

3.1.4.1.1. axioms together with statements that can be deduced from the axioms using the rules of inference provided by a system of logic

3.1.4.2. Axioms of the set theory

3.1.4.2.1. a statement that serves as a starting point from which other statements are logically derived

3.1.5. Theorem

3.1.5.1. a true statement based on other accepted truths - supported by a proof

3.1.5.1.1. Direct truth

3.1.5.1.2. Proof by negation

3.1.5.1.3. The principle of mathematic induction

3.1.5.1.4. The pigeon hole principle

3.2. Research

3.2.1. Blue skies research

3.2.1.1. research for the research sake, without any clear goal

3.2.2. Relating different areas of research

3.2.2.1. filling the gaps

3.2.3. Attacking a famous problem

3.2.3.1. one needs to research as what has been done and come up with new ideas, possibly techniques

3.2.4. Applying standard method to a standard type of problem

3.2.4.1. the aim is to simplify the problem, and possibly in the future apply the outcomes to a new research

4. Historical Development (Will.K)

4.1. The two main ways of Mathematical law production.

4.1.1. Theorems

4.1.1.1. The theorem is when a conjecture is proven right and is therefore seen as being a universal truth forever.

4.1.1.1.1. This can be done in the form of demonstrations or in the form of mathematical proofs.

4.1.2. Conjectures

4.1.2.1. Conjectures are essentially animal eggs which need to crack the shell the theorems

4.1.2.1.1. The shell being the "proof"

4.2. Paradigm shifts

4.2.1. As well as many Theorems and conjectures along the history of math, there were also a series of paradigm shifts, similar to those in the AOK of NS

4.2.1.1. Examples

4.2.1.1.1. Deductive Mathematics from the Ancient Greeks 800 B.C. and 500 B.C, which gave way to maths using axioms and proofs to come up with mathematical facts. Can be argued that this is where conjectures and theorems first came up.

4.2.1.1.2. Brahmagupta & The Invention of Zero 628 AD which helped solve the tedious problem of using roman numerals in math.

4.2.1.1.3. The bringing of arabic numerals to Europe by Leonardo Fibonacci which changed the way we calculate starting in the 12th Century

4.2.1.1.4. Pascals triangle in the year 1623 which allowed the exploration of number theory and triangular numbers.

4.2.1.1.5. Gottfried Leibniz inventing the binary number system in 1689 which is used in the core basis of machine code used in computers and many other technological devices.

5. Personal Knowledge (Nat & Jerry)

5.1. Contribution of individuals in own experience with Mathematics

5.1.1. G11 mathematics teacher advocated, encouraged "endless sea of questions" method of practice

5.1.1.1. Though it may sound boring, in a subject where one must intuit which concept to apply to a question quickly, practicing high volumes of questions can help train this intuition

5.1.1.2. Trains brain to think in a mathematical, logical manner, helps with problem-solving inside and outside of maths

5.1.2. Assessments on previous topics and pop quizzes

5.1.2.1. These test your long-term knowledge and understanding

5.1.2.2. As Mathematics is a hugely practice based subject, pop quizzes can test the reliability of our knowledge in a particular topic

5.2. Responsibilities of Future Mathematicians

5.2.1. Given that the field of mathematics has constantly expanded over the years, future mathematicians have the responsibility to continue this

5.2.1.1. Expansion of the field can be done by...

5.2.1.1.1. Proving / disproving existing theorems

5.2.1.1.2. Proposing new theorems

5.3. Reliance on Assumptions

5.4. Implications of Mathematics on personal perspectives

5.4.1. Existence of coherent, unified, complex mathematical principles can be the basis of theist's argument from design

5.4.1.1. Links WOKs - reason + faith

5.4.1.2. Relies on the assumption that coherence, unity, complexity are allusions to intentional and purposeful design, rather than something atheistic, such as natural selection

5.4.1.3. Can be argued, but not necessarily proven

5.4.1.3.1. Mathematical principles exist apart from human agency, humans merely discover them and make inferences based on them

5.4.2. Grocery shopping

5.4.2.1. Grocery shopping requires a broad range of math knowledge from multiplication to estimation and percentages

5.4.2.2. Each time you calculate the price per unit, weigh produce, figure percentage discounts, and estimate the final price, you're using math in your shopping experience

5.4.3. Baking

5.4.3.1. Conversions between imperial and metric measuring units

5.4.3.2. E.g

5.4.3.2.1. °F to °C

5.4.3.2.2. Ounces to grams

5.4.3.2.3. Fluid ounce to mL

5.4.4. Travelling

5.4.4.1. Estimating the amount of fuel you’ll need to planning out a trip based on miles per hour and distance traveled

5.4.4.2. In areas without GPS signal, one can only depend on physical maps and compass directions

5.4.4.2.1. Even though this is more related to geography, compass direction calculations still require some degree of mathematical calculation

5.5. Links to other AOKs, WOKs

5.5.1. Inductive reasoning, which is the verifying of fact through a series of particular premises, is especially relevant to Mathematics, specifically mathematical induction

5.5.1.1. Mathematical induction does rely on the assumption that p(k) is true before using it to prove p(k+1) to be true

5.5.2. Use of mathematical language in the Natural Sciences to visualise causal relationships

5.5.2.1. Correlation is not necessarily indicative of causation, which is what mathematical formulas try to visualise

5.5.3. Mathematical equations are definitely hugely important in the natural sciences.

5.5.3.1. Physics

5.5.3.2. Chemistry

5.5.4. Use of game theory in the Human Sciences to mathematically model decision making situations

5.5.4.1. Application of game theory in the Cuban Missile Crisis

5.5.4.2. Relies on the assumption that all players are rational and acting in their best interests