Logic and mathematics: Frege believes there are natural numbers; used to count things, these natural numbers are just concepts, many things, more than one thing etc. Counting creates an abstract category or group - for example with plurals - you can have a pride of lions, a murder of crows, an unkindness of ravens (natural numbers, abstract concepts) This is used to refer to a number of things where you cannot physically count them all out, like trying to count the number of people in a football stadium, you just know there is a hell of a lot of people there.
There are three attitudes to language, most importantly numbers:
1) they are natural and can be empirically observed
2) they are intuitions of a harmonic, platonic other world - you can never find the noumena, the essence of a number
3) they are abstract logical objects constructed purely from syntax*
*Syntax - results of modifying the meaning of one object to another. For example, verbs and adjectives - house can be syntactically altered to blue house.
Numerical naturalism / Evolutionary psychology: Apes and Neanderthal tribes appeared to be able to judge simple empirical plurality, typically for example, the absence of a banana or something else of importance.
Noam Chomsky: People are born with an innate understanding on syntax and language, he argues otherwise how do we know this? Contra John Locke who thought we are born with a blank slate - we have nothing innate.
Pythagorean-ism / Platonism: Numbers have heavily influenced Christianity, for them numbers are like an insight into God (prime numbers) the belief behind this is that 7 essentially cannot be thought up, primes can't be divided. Numbers are believed to have like special powers, people hold them in high regard like choosing numbers for the lottery, you, for some obscure reason, believe picking certain numbers that you believe to be special from the others increases your chances of winning when really it doesn't work like that - pretty much superstitious nonsense.
There is a religious significance regarding the number 3. Three is apparently the magic number - three acts in a play, three movements in a symphony and in a waltz, the big three in religion being the father, the son and the holy ghost. This also creeps into journalism, with terms like rule of thirds.
Primes are held in high regard by Islamic believers, Islam exhibits cults around plural primes four, five and seven. Babylonians exhibit a similar obsession with numbers - the 12 Zodiacs, each representing a month of the year, commonly known as star signs.
Pythagoras regarded plurals as the only real natural numbers, starting numerically at 2 because a number that is 1, not one or nothing are completely different categories. Odyssey telling the tale of Odysseus & the Cyclops; the cyclops asks if there is anyone there to which the reply is there is no-one there. This utterly confuses the cyclops because nothing being there is a fully ineffable concept - the process of there being nothing there is flawed, there cannot be nothing as the nothing is something. Used again in this example which perhaps illustrate the point more - there is nothing on the road, well yes there is because the road is there and also nothing, being something, is present. This does not mean the same thing as the road being empty, or clear of obstructions.
Problem of zero and nothing: Zero came from India, later via Islam. Whole Arabic numerical systems were introduced in the middle ages after the fall of Rome. Zero is an intrinsically difficult concept, as expressed earlier; zero = nothing but nothing = something.
Contra to Aristotle's law of contradiction solved by Leibniz's monads - an object can contain it's own negation. Modern philosophers of mathematics assert that zero is a natural number. This is because if you have: zero + one = one, making something out of nothing.
Common sense view of numbers: There are logical objects according to Frege, his book the Grundlagen is a philosophy of the logic of numbers. For Frege, maths is just a language and all the same analytically.
- Languages have three things:
1) vocabulary of objects (words and numbers)
2) syntax - modify the meaning comb-grammar
3) grammar
Frege's work was adapted - Bertrand & Whitehead - Principia Mathematica - going to assume this is the principle of mathematics
Frege's Method: Axiom - all things identical are equal to themselves, this is asserted apriori; deductive, true by definition. Follows all things which are pairs are identical to all other pairs (regardless of what they are pairs of) they are still pairs nonetheless. The class of all things which are pairs - logically can call it two, it does not matter. Large numbers can be built as logical constructs as along the lines of 'the class of all things which are pairs of pairs' - we can attach any symbol we like, for example four. Furthermore, one is the class of all things that are not in a pair, eg, lost sock = not a pair, just a single sock.
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