Mathematics can be highly abstract & far from reality. In economic models growth can be infinite. Indeed, capitalism likely requires unending economic growth to continue.
Natural numbers: 0,1,2…
Rational numbers: integers + fractions …-1,-1/2,0,1/2,1…
Real numbers: rational + numbers which can be represented as infinite decimal expansions, e.g. √2, π
Computable numbers: real numbers whose expansion can be generated by algorithms (i.e. by a rule), e.g. 1/3 + 0.3333… & 93/74 = 1.2567567567…
Imaginary numbers: i2 = -1 square roots of negative numbers
Complex numbers: a + ib where a is the real part & b the imaginary part
ℵ0 = number of natural numbers, integers & rational numbers
Cantor showed that there are more real numbers than rationals (using the diagonal slash)
C = number of real numbers (continuum) – not countable
Does C = ℵ1? where ℵ1 is the next infinite number greater than ℵ0 this is the continuum hypothesis
Russell paradox: R is the set of all sets which are not members of themselves. Is R a member of itself or is it not? If it is not a member of itself then it should belong to R, since R consists precisely of those sets which are not members of themselves. Thus R belongs to R after all – a contradiction. On the other hand, if R is a member of itself, then since ‘itself’ is actually R, it belongs to that set whose members are characterised by not being members of themselves, i.e. it is not a member of itself after all – again a contradiction
Real numbers have the property that between any two of them, no matter how close, there lies a third. Is this also the case for physical distances or time? or is there some kind of granularity?
Gödel showed that any precise mathematical system of axioms & rules of procedure whatever, provided it is broad enough to contain descriptions of simple arithmetical propositions & provided that it is free from contradiction, must contain some statements which are neither provable nor disprovable by means allowed within the system.
Reflection principle: insight? reflecting upon meaning
Intuitionism (Brouwer): They demand a definite (mental) construction be presented before it is accepted that a mathematical object actually exists. In this way they deny the law of the excluded middle & the procedure of reductio ad absurdum
Argand plane: the geometrical representation of complex numbers
Mandelbrot set – un/bounded points of c Complex dynamic systems
Why get so excited that mathematical concepts can seem ‘timeless’? Why should we expect everything to change with time? Is there a psychological appeal of immortality with maths?