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, the
ones we work with, 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 some 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 and sit by its side?

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.