> 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.