1. Scope & Application (Katie)
1.1. 3 Basic Views
1.1.1. Platonists
1.1.1.1. Mathematics seen as a code to understand the world around us
1.1.2. Constructivists
1.1.2.1. the nature of Mathematics is artificial and abstract (Mathematical Humanism)
1.1.2.1.1. In focus: exploring mathematical structures in order to investigate patterns, but also to use this knowledge to solve practical problems
1.1.3. Formalists
1.1.3.1. an abstract game played according to invented rules
1.2. Numbers may or may not exist physically
1.2.1. Mathematical statements don't
1.2.1.1. Values are based on rules designed by humans
1.2.1.1.1. This suggests thats maths is:
1.3. Highly debated topic: Maths
1.3.1. Invented or discovered?
1.3.2. Artificial construct or universal truth?
1.3.2.1. Maths debate then becomes spiritual
1.3.2.2. Answer can depends on the specific concept being looked at
1.3.3. Human product or natural possibly devine creation
1.4. What is proof? and why is important in maths?
1.4.1. Provide stable foundations for many different areas of work to test their theories on
1.4.1.1. Mathematicians
1.4.1.2. Logicians
1.4.1.3. Staticians
1.4.1.4. Economists
1.4.1.5. Architects
1.4.1.6. Engineers
1.4.1.7. Etc....
1.4.2. Euclid: Lived 2300 years ago
1.4.2.1. Revolutionised the way in which proof is written, presented and thought about
1.4.2.2. Set out to formalize mathematics by establishing a set of rules or Axioms
1.4.2.2.1. Use axioms to help prove what you think is true
1.4.3. Proofs are:
1.4.3.1. Well-established rules to PROVE beyond a doubt that some theorem is true
1.5. what is the scope of Mathematics and what is in focus of the research:
1.5.1. What role does LOGIC play in Mathematics?
1.5.1.1. Axioms
1.5.1.2. Deductive reasoning
1.5.1.3. Theorems
1.5.2. What is the role of AXIOMS?
1.5.2.1. A starting point for reasoning
1.5.2.2. A premise that is accepted without proof
1.5.3. What are THEOREMS?
1.5.3.1. A statement that is/can be proved to be true on the basis of axioms or other already established theorems
1.5.4. What are CONJECTURES?
1.5.4.1. A hypothesis that appears to work but which has not yet been proved
2. Language & Concepts (Hamza)
2.1. Origins of Numbers
2.1.1. Counting has been a fundamental part of human learning and knowledge production/acquisition. Today, there is a global standard of base 10 (see Origins of Numbers bubble) with a different image to represent each of the base ten numerals. However, this system has evolved from a range of other systems from different countries.
2.1.1.1. During early human existence, the need for counting originated from tribes needing to count their resources (cattle, wood etc). Most used the simple tally system. However, as society advanced, the need to count larger objects became prominent, but the tally system was impractical.
2.1.1.1.1. Greek, Hebrew and Egyptian systems, began the idea of using different images for every larger value of a fixed increment, and filling the values in between with the base numbers (depending per region). These were just an extension of the tally system, and after a while, became insufficient.
2.1.1.1.2. After the tally extension systems, a new concept was conceived, known as 'positional notation'. In this system, the same symbol for a value could be reused throughout the number, but it's magnitude depended on its position in the term.
3. Methodology (Juval)
3.1. The Law of the excluded Middle: every statement must be either true or false
3.1.1. An axium is a statement that you base your research on. You may be trying to prove that it is true or untrue. (I think)
3.1.2. A theorem is something that has been proven to be true
3.1.2.1. There are 4 types of proofs
3.1.2.1.1. Direct Proof (If... then ...)
3.1.2.1.2. Proof by Negation (assume P is true, Q is false; if you arrive at a contradiction, then P is true and Q is true)
3.1.2.1.3. The Principle of Mathematical Induction (If P(n) is true then P(n+1) is true)
3.1.2.1.4. The Pigeon Hole Principle: if n items are put into m containers, with n > m, then at least one container must contain more than one item
3.1.2.2. For something to be considered "proven true" there must be no doubt that it is true
3.1.2.3. Formal vs. traditional proofs
3.1.2.3.1. A traditional proof can be a simplified generalisation
3.1.2.3.2. A formal proof must be expanded until there are no more possible outcomes to be considered true
3.2. "Everything should be made as simple as possible, but not simpler." - Albert Einstein, apparently
3.2.1. A good math paper should be easy to understand
3.2.2. Ironically, notation is used to simplify math
3.2.3. Keeping this in mind, we can prove that something is true by using these laws:
3.2.3.1. Proof by SUBSTITUTION: all cats are animals; all animals can communicate; therefore all cats can communicate
3.2.3.1.1. This can lead to errors if one of the base cases is incorrect
3.2.3.2. Proof by CONTRADICTION: contradictions are always false under any circumstances
3.2.4. Simplifying things (equations and explanations) is often difficult and time consuming
3.3. Generalisations are how we understand math
3.3.1. Eg. Pythagoras' theorem
3.3.1.1. Always true
3.3.1.2. Can be applied to any numbers
3.3.2. Eg. Formulas
3.4. Reasons for research
3.4.1. 1. Blue Skies Research
3.4.1.1. Research for the sake of it, even if there's no explicit goal
3.4.2. 2. Relating different areas of research
3.4.2.1. filling the gaps
3.4.2.1.1. Between two formulas
3.4.2.1.2. Between two areas of math
3.4.3. 3. Attacking a famous problem
3.4.3.1. Researching what has already been done and coming up with new techniques
3.4.4. 4. Apply a standard method to a standard type problem
3.4.4.1. Simplify the problem, with the hopes that in the future it can be applied to the outcomes for a new research embarkment
4. Historical Development (Yuval)
4.1. Key Points in the History of Mathematics
4.1.1. Brahmagupta and The Invention of Zero
4.1.1.1. problematic arithmetical operations
4.1.1.1.1. much easier using Arabic numerals than using Roman numerals
4.1.1.2. zero was thought of as a placeholder in positional systems
4.1.1.2.1. (Babylonians, Mayans, Greeks and Chinese)
4.1.1.3. 628 AD – India
4.1.1.3.1. zero recognised as a number
4.1.1.4. ‘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.’
4.1.1.5. ‘A debt subtracted from zero is a fortune
4.1.1.5.1. 0/0=0 à indeterminate
4.1.2. Axiomatization
4.1.2.1. Euclid proving which truths are actually true
4.1.3. Fibonacci, F.Sequence and Phi
4.1.3.1. brought Arabic numerals to Europe
4.1.3.1.1. made a huge difference to the way we calculate
4.1.3.2. in his book included a number of mathematical problems
4.1.3.2.1. breeding of rabbits, and the pattern of the sequence of this reproduction
4.1.3.3. sequence was already known by the academics in India
4.1.3.3.1. Fibonacci brought it further and one can observe that when one divides a number with the one before it, one comes closer and closer to the irrational number PHI
4.1.3.4. can be also found in the Equiangular Spiral
4.1.4. Al-Khwarizmi & The Invention of Algebra
4.1.4.1. solving equations related to shapes
4.1.4.2. Language
4.1.4.2.1. words to explain the problem and pictures to solve them
4.1.4.3. Methods
4.1.4.3.1. ‘completion’
4.1.4.3.2. balancing
4.1.4.3.3. letters of the alphabet represent the unknown numbers and we use the arithmetic operators
4.1.4.4. geometric proofs
4.1.4.4.1. one of the first methods figuring out which numbers the letters actually represent
4.1.5. Rene Descartes and Analytic Geometry
4.1.5.1. combination of algebra and geometry
4.1.5.2. a letter represents number
4.1.5.2.1. two numbers can represent a point in space à Cartesian coordinates
4.1.5.3. essential tool for science and engineering
4.1.6. Fermat’s Last Theorem
4.1.6.1. 'a truly remarkable proof which this margin is too small to contain.’
4.1.6.1.1. Solved in 1994 by Andrew Wiles;
4.1.7. Pascal's Triangle
4.1.7.1. exploration of number theory and triangular numbers
4.1.7.2. enables us to expand Binomials
4.1.7.3. Pascal was a religious person and at times attributed his work to "divine intervention"
4.1.8. Counting in Binary and Computers
5. Personal Knowledge (Anna)
5.1. Study of patterns and sequences
5.1.1. helps us understand the world and make sense of it
5.1.2. mathematics in the end underlies and is necessary for all these other subjects
5.1.2.1. e.g Sciences and scientific equations
5.2. limitations of mathematical truth, that not all that is true can be proved
5.2.1. Kurt Godel was a logician and mathematician
5.2.2. he created godel numbering
5.2.2.1. matching symbols to natural numbers
5.2.2.1.1. example: Gödel number for the symbol "=" is 5