Natural numbers and used to count things. To count things is to create an abstract category. On average, natural numbers go up to 7. This can be shown if you stand in a room with 7 other people in it, you know there are 7 people almost without thinking. If you put a few more in, then you would have to count the amount of people in the room. A number such as 34251 is impossible to recognise or visualise as it is abstract/mysterious.
There are 3 fundamental attitudes to languages:
1 - numbers are natural and can be empirically observed.
2- intuitions of harmonic perfect platonic other world e.g. Descartes and geometry.
3- they are abstract logical objects, constructed purely from syntax (Frege).
Plurality is when there is more than one of something. An example of plurals of plurals is 3x3 = 9
A large number e.g. 2316 is a predicate symbol of more basic symbols according to syntax. Chomsky said that language must be innate - wired in brain.
Words/phrases have been used to count things e.g. 'one thing', 'more than one thing' and 'many things'. All that is needed is those 3 phrases. This can be related to a monkey and bananas, if there is one banana in front of a monkey, it knows there is just one banana, but the monkey doesn't want just one. If there are 2 bananas, the monkey is satisfied because there is more than one. If there are no bananas in front of the monkey, there is an absence of 'a thing'.
The problem with zero/nothing
Zero = nothing = something. This contradicts Aristotle's law of contradiction, which is the foundation of all logic. Bertrand Russell said that nothing is impossible.
0 is not nothing e.g. there can't be 'no one on the road'. 0 is a natural number, 1+0=0 (nothing).
1+0 = 1 (everything) this is a huge leap forward.
1+1 = 2, doubles in size e.g. if a baby is alive for one hour and then the next hour, his/her life has doubled in size.
N-+1 is an infinitely small increment.
Pythagoreanism/Platonism attitude to numbers
Numbers are the communication of an 'alien' world. Prime numbers are evidence for God's existence? All other numbers are rational combinations of prime numbers.
Orphic religion (Nietzsche's Apollonian religion - BoT). The prime number 3 has special religious significance (Holy Trinity and Jesus rising on the 3rd day). Not just limited to religion - rule of thirds in photography and 3 movements in symphonies.
Other primes have significance - 1 in Islam but hated by Greeks.
Pythagoras regarded only plurals as natural numbers sot hey began counting with two.
Numbers as logical objects
Frege solved the problem of zero 1000 years after the problem was first 'discovered'. His book 'The Foundation of Arithmetic' linked logic and arithmetic in an overall system of philosophy of language. It was adapted by Bertrand Russell and Whitehead, they called it 'Principia Mathmatica'. They attempted to demonstrate the logical basis for numbers, arithmetic and maths. It rejected JS Mill's numerological empiricism.....
Frege's method
'All things which are identical are equal to themselves'. This is regardless of what the things actually are e.g. a pillow and a book. Larger numbers can be built up as logical constructs
Zero as a class of all possible objects which are not equal to themselves, zero is now a logical object - zero or not zero. A 'nul' is unexplainable to a computer as it is not an empirical thing. Even a 0 in binary is something, it is just part of a code.
No comments:
Post a Comment