Do number's barn-dance?

We talk of the set of all the natural numbers, and numbers going on for ever, even though these notions don't look arithmetical. Numbers are always generated by addition, multiplication, and other funcions.

Numbers don't barn-dance. All contemporary theories of number, like finitism and ultra-finitism, suggest otherwise. But numbers don't do collections, or orderings, or listings ... numbers don't arrange, period. We may fool ourselves into thinking that 'numbers' do these things when we aren't looking or calculating, in a Platonic realm perhaps.

For example, we say that 2 'follows' 1. But 2 never follows 1. Whoever heard of a number coming 'after' another number, except in the context of being a teaching expedient? What sort of function allows a number to usher in a 'next' number?

Numbers don't do lists, amorphous collections, sequences... look what happens if we think they do. Take the 'sequence' of 'natural numbers'. We count each number once, irrespective of its value. So we have one 'one', one 'two', one '65536', etc. It isn't our count of these numbers that arranges them, though. The numbers must magically arrange themselves through their own efforts, amorphously in one set, or sequenced, etc. But there is no method by which a number can cross the divide from being function-generated to being a member of an arrangement. In an arrangement all objects are, as far as the arrangement goes, identical.

SPECULATION and CONCLUSION

My point? My point is that we have generated myths about 'numbers'; myths whose sources are found among school rote teaching methods. These methods worked only if we believed in the idea that numbers are independent, individual entities independent of the functions that create them. So, numbers possibly could, we believed then (and now), order themselves.

We confuse arithmetical possibility with grammatical possibility. It IS grammatically admissable to speak of isolable numbers only because grammatically it is how we were taught to speak about 'numbers'. It started in infancy when we saw isolated 'numbers' on big coloured cubes, and numbers with personalities in 'Sesame Street'. The indoctrination continued in school through the employment of a grammatically correct, roted and sequenced, addition, subtraction, and times-table. Later, we saw numbers neatly boxed-up in logarithmic tables and - as if through some natural, innate ordering property - lined up on calculator screens.

Numbers are not countable. They are neither isolable nor individual objects. The term 'numbers', 'a number', 'all' numbers, etc., are arithmetically inadmissable, even if they are grammatically par for the course.
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