What's inside an empty bag? Nothing – but that's something on which all mathematics is founded...
What, then, is a number? Mathematical logicians realised that to define the number 2, you need to construct a standard set which intuitively has two members. To define 3, use a standard set with three numbers, and so on. But which standard sets to use? They have to be unique, and their structure should correspond to the process of counting. This was where the empty set came in and solved the whole thing by itself.
Key words - analytic, axioms, empiricism, formalism, idealisation, synthetic, theorem, pure mathematics, applied mathematics
Key concepts
Mathematical paradigm - described as the 'science of rigorous proof'
Mathematics is the study of pattern - abstract pattern that places concepts in a systematized relationship to one another, expressed in a symbolic system that we can manipulate using reason alone.
Nature of mathematical truths - Different views are 1. empirical 2. true by definition 3. rational insights into universal truths
Axioms - Starting points or basic assumptions - four traditional requirements 1. Consistent 2. independent 3. Simple 4. Fruitful
Godel's incompleteness theorem - Kurt Godel proved that it is impossible to prove that a formal mathematical system is free from contradiction.
Goldbach's conjecture - Every even number is the sum of two primes
Proofs and conjectures - A formal system begins with axioms and uses deductive reasoning to prove theorems
Mathematics and certainty -
Discovered or invented - while Platonists believe that mathematical entities are discovered and exist 'out there', formalists argue that they are invented and exist only 'in the mind'
Riemannian geometry - the mathematician Riemann came up with the idea of replacing some of Euclid's axioms with their contraries
Pure and applied mathematics - Mathematics moves both towards the abstractions of the mind, and also towards the connection wiht the world.
Pure mathematics - researches in pure mathematics, are not concerned with the direct practical applications of their labour.
Applied mathematics - Mathematics that is used to solve problems in the real world.
Fermat's last theorem -
Mathematics in social context - Alan Bishop, in his studies, identified six forms of recurring mathematical ideas spanning vastly different cultures 1. Counting 2. Locating 3. Measuring 4. Designing 5. Playing 6. Explaining
?? Is mathematics invented or discovered?
?? If mathematics is an abstract intellectual game, then why is it so good at describing the world?
?? Why should elegance or beauty be relevant to mathematical value?
?? Is mathematics independent of culture?
?? Is there a distinction between truth and certainty in mathematics?
?? What is the relationship between truth and mathematics ?
?? Is mathematical creativity the same as other types of creativity? Is there any difference between beauty in maths and beauty in music?
?? To what extent do you think people's beliefs about the value of mathematics are determined by their ability in the subject?
?? To what extent do you think the geometric paradigm can be applied to other areas of knowledge?
?? To what extent do you think governments should fund 'useless' research in pure mathematics?
'Mathematics is the abstract key which turns the lock of the universe' John Polkinghorne
'Everything that can be counted does not count. Everything that counts cannot be counted' Albert Einstein
'A mathematician is a machine for turning coffee into theorems' Paul Erdos
'When you have satisfied yourself that the theorem is true, you start proving it.' Arthur Koestler
'If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics' Kurt Godel
'On each decision, the mathematical analysis only got me to the point where my intuition had to take over' Robert Jensen
Mathematics is the most beautiful and most powerful creation of the human spirit. Stefan Banach
The essence of mathematics lies in its freedom. Georg Cantor
The mathematics is not there till we put it there. Arthur Eddington
Logic and mathematics are nothing but specialised linguistic structures. Jean Piaget
Film is one of the three universal languages, the other two: mathematics and music. Frank Capra