Modern Logic

Frege devoted much of his career to trying to show how mathematics can be reduced down to logic. Natural numbers are words used to count things. To count things is to create an abstract category or group. To create words and abstract symbols for more than one category requires a system of a number of words. It also requires a logical syntax in order to combine these number-words to imply or further establish number words.

-They are natural and can be empirically observed. This was a view held by Mill, and the average person.

-They are intuitions of harmonic perfect other world. This view was held by Descartes, and Pythagoras.

-They are abstract objects, constructed mostly by syntax. View held by Frege and Russell.

Numerical Naturalism follows the idea that there is 0, 1 or more than 1. These simple pluralities are fairly primitive and therefore are used by apes and tribes. Apparently, the highest natural number is 7 (this obviously varies from person to person) but a human being can walk into a room and instantly recognise that there are 7 people in there without counting.

Prime numbers according this view are “pre-existing, eternal, supernatural forms”. They also show necessary preconditions for consciousness. All numbers are combinations of prime numbers. Prime numbers often have religious or magical significance. These includes the holy trinity, “three is the magic number”, the fact that symphonies are performed in three movements.
According to Nietzsche, Russell and Frazer, this is all Orphism. Christians worship the number 3, Muslims worship the number 1.
Babylonians- 12 (12 months, 12 signs of zodiac)
This is all Orphism (according to Nietzsche, Russell and Frazer) e.g. Christianity's trinity of Father, Son and Holy spirit. The Greeks (including Pythagoras) regarded only plurals as natural numbers and started the counting system with 2. There could never be a zero, because there could never be “nothing.”

The concept of 0 came from India, and much later from Sufi-Islam. The fact that zero meant nothing but then nothing meant something was a pretty big problem. Since then, modern philosophers of mathematics have asserted zero as a natural number (logically derived as 1-1=0) this problem remained unsolved until Frege came along. He regarded arithmetic as a language. He rejected Mill’s numerical empiricism as you cannot find 0 in nature.

Frege's Method
Axiom - all things which are identical are equal to themselves.

All things which are pairs are identical to all other pairs, regardless of what they are pairs of. The class of pairs can therefore be given a nominal value (2). Larger numbers can be built as logical constructs. “The class of all things which are pairs of pairs” (4) 1= the class of all things not associated with other things. 0= Class of all possible objects that are not equal to themselves. No such objects.