Natural numbers are simply words used to count things, and counting is nothing more than an activity in which objects are categorised dependent on plurality. All that is required for creating these categories is a language of number words and a logical syntax for their combination.
There are three fundamental approaches to numbers.
1) They are natural and can be empirically observed (Mill)
2) They are intuitions of a perfect and harmonic platonic world (Pythagoras and Descartes)
3) They are abstract logical objects, constructed purely from syntax (Frege, early Russell)
1- Numerical Naturalism
Basic plurality approach, as observed in stone age tribes. And monkeys.
They could observe:
0 - absence of a banana.
1- a single banana.
2+- more than one banana/many bananas/an abundance of bananas.
This is simple plurality, and in many situations is all that is needed - 'one thing', 'more than one thing' and 'many things'. For example, if I see one car in a car park, I do not need to count it, I can immediately categorise it as one. If there is more than one (anywhere between 2 & 6), I know how many are there without counting. Beyond 6 or 7, I would need to count specifically to know exactly how many were there, but I would simply categorise it as many cars. I would also be able to say the car park was empty (zero, the null class). These classes are the 'natural numbers'. A number such as 22793 is simply basic symbols arranged to create a more complex one, according to known syntax. But if there were that many cars in a car park, I would not bother counting. I'm confident if I said there were many cars, you would get the picture. That's also a rather large car park.
2- Pythagoreanism/Platonism]
Prime numbers are pre-existing, supernatural forms and are necessary for consciousness. All other numbers are simply rational combinations of prime numbers. This contradicts Kant's claim that "existence is not a predicate", for Platonism, existence is a predicate of numbers and other forms. Prime numbers exist along with the perfect form, outside of the human dimension, and are eternally true.
3, the first plural prime, has significant religious connotations (unfortunately de la soul were not the first to observe three as the magic number, though arguably they did it more enjoyably than most). There is the holy trinity, Jesus rose again on the third day, the trimurti in hinduism etc. It is also hugely significant in secular society, with art containing the rule of thirds, the three chord triad in music and the three acts in most dramas. There are other primes with historical and religious significance, but none to the extent of three.
Pythagoras, and all the Greeks, only regarded plurals as natural numbers, and so counting commenced from two. 'One' and 'not one' were different logical categories.
Nothing and Zero - The concept of Zero arrived much later, and turned out to be something of a nuisance. Zero is nothing, but nothing is something, and therefore zero is something? It's a toughie.
This problem of the law of contradiction was solved by Leibniz, stating that an object can contain its own negation, so that cleared that up. Modern mathematicians have asserted that zero is in fact a natural number, which makes everything a little easier. But only a little, as 0+1=1, but 0x1=0, which caused the problem of how to define the instruction '+1'.
3- Numbers as logical objects
Fortunately, Frege turned up to clear up the zero problem. 1000 years later. Still, better late than never.
Frege linked logic and arithmetic in an overall philosophy of language. Frege's approach refuted both Mill and Platonism before him.
Axiom - All things that are identical to themselves are equal. It then follows that all pairs of things are equal to all other pairs, regardless what they are pairs of.
Pairs therefore get their own symbol (2), and from here it is established that all things that are not related to other things have their own symbol (1). Zero then is the class of all things which are not equal to themselves, and no such objects exist. Therefore zero is logically defined.
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