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Friday, 16 March 2012

Mathematics in Logic and Language

Natural numbers - words used to count things. To count is to create an abstract category or group.
On average, natural numbers go up to 7.
A number such as 73,962 is abstract and mysterious, impossible to visualise or recognise.

Any complex number can be analysed

Analytic philosophy - idealist or synthetic - breaking things down to find their logical basis.

Plurality - more than one
Plurals of plurals. Eg: 9 = 3 x 3

'Special words' - have been used in the past by tribes to count stuff.
Eg- 'one thing', 'more than one thing' and 'many things'
All that would be needed is those 3 phrases.

Syntax and Grammer - Symbols = progress

Zero = Nothing = Something which cotnradictrs Aristotles law of contradiction, the foundation of all logic.
Bertrand Russell says '0 (nothing) is impossible'.
0 is not nothing, for example it can mean 'there is nobody in the room'.
0 + 1 = 1 but 0 x 1 = 0 ---- Could this be considered a contradiction?

The first manipulable number was 2, in Greece.
Greeks believed the number 3 had magical powers. Even basing a religion on worshipping the number 3.

Geometry - Platonic theory of numbers
The world consists of geometrical shapes.
Pie is a retio derived from a geometrical idea.

Music - Every note is defined by a ratio length.
Eg- From middle C

RUssell thought that number and arithmetic were neither Platonic ideal forms, nor empirical generalistion.

Principia Mathematica
Founder - Campaign for nuclear disarment

His intellectual career began as a dedicated Hegelian idealist, he retains some of this in social theories, especially his popular scientific work.

Mathematics appeared to be a contradiction of idealism, because numbers appear to have an objective in some sense, and their nature is not apparatnly determined or affected by the act of obseriving them.

Peano's Axioms
0 is a natural number (it is not nothingness)
x = x - Every numver is its own equivalent
Every natural number jas a successor number.
But no natural number has 0 as a successor.

Russells Paradox
The key terms Russell uses are class and belonging to a class and similarity

Number in general is 'the class of classes similar to a given class'

So number three is a word nominally corresponding a logical class composed of all possible classes.


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