Based on Frege, Russell and Whitehead.
Natural numbers are simply words used to count things - to count is to create an abstract category or group.
Syntax - logical system using rules of inference to alter the meaning of symbols --> house = noun, blue house = syntactic adjective. Is syntax learnt or innate? (Chomsky)
There are 3 fundamental attitudes towards language - especially numbers.
- They are natural and can be empirically observed.
- They are intuitions of a harmonic perfect Platonic other world.
- They are abstract logical objects that are constructed purely from syntax.
Numerical Naturalism/Evolutionary Psychology.
0 = absence of a thing
1 = one banana/enough bananas
2 = a lot of bananas (more than one)
This came from apes and stoneage tribes who seemed to be able to judge and decipher simple empirical plurality --> 'one thing', 'more than one thing' and 'many things' are all they need.
Even for people from advanced cultures, small number words are functionally different to large number words.
If you go into a room and there is 1 or even 3 people there you don't count them you just know it is occupied, you categorise them as plurality. So, the number 7,434 is a predicate symbol of more basic symbols organised according to known syntax. (a predicate can be analysed).
Prime numbers are pre-existing supernatural forms necessary pre-conditions for consciousness.
All other numbers are just rational combinations of prime numbers - this contradicts Kant - "existence in not a predicate".
The prime number 3 has significance - its a 'magic number', 'third time lucky', 'rule of thirds', 'three chord triad'. The beginning, middle and the end. The Father, the Son and the Holy Spirit.
Special problem of nothing and zero:
It is naturally impossible to have nothing (0).
1 and not 1 are categories.
0 is nothing but nothing is something (contradicts Aristotle's law of contradiction - the fundamental axiom of all logic).
Aristotle's logic - the sun is the sun, the moon is the moon therefore they are not the same thing.
What does +1 mean?...
0+1=1 (infinitely large increment)
1+1=2 (double in size)
N+1 (infinitely small increment)
Numbers as logical objects:
The problem of zero and nothing remained unsolved for 1000 years until Frege (1848-1925).
Frege's method:
Axiom - all things which are identical are equal to themselves (asserted a priori, deductive, definitional truth).
All things which are pairs are identical to all other pairs.
The class of all pairs, contains all pairs and this can be given a purely nominal symbol (eg two) a word or a numeral, it doesn't matter.
Zero is a class of all possible objects which are not equal to themselves. There are no such objects (NULL). Computers never had to key in a zero, they used Frege's 'null class'.
Epistemology.
The study of knowledge and justified belief.
It is concerned with the following questions:
- What are the necessary and sufficient conditions of knowledge?
- What are its sources?
- What is its structure?
- What are its limits?
Frege was concerned to set out the relationship between epistemology and other related disciplines. He too over but adapted Kant's distinction between a priori and a prosteriori knowledge.
Psychology is interested in the cause of our thinking whereas mathematics is interested in the proof of our thoughts.
Frege was similar to Descartes. However, not in the ego - Descartes' ego was a non-ideal subject of thinking but Frege's ego is a non-ideal object of thought.
System of logic - John Mill described as a textbook of the doctrine that derives all knowledge from experience. He set a system of formal logic.
Mill wanted to disassociate his work with the work of Hobbes.
Mill was unlike Frege in that Frege believed that arithmetic and logic were both a priori.
Frege was the second founder of logic - after Aristotle. He looked at logic and systemised it which led to the conclusion that logic was a priori and analytic. For Frege the most important part of logic was validity and invalidity of a particular form of inference, shown through syllogism.
For example...
All cats have fur
some cats are black
some cats have black fur
this is an example of a valid inference
BUT
All cats have fur
some cats are black
all cats have black fur
is an example of an invalid inference.