Re: Successor Axiom: on what grounds TF?

Helene.Boucher_at_wanadoo.fr
Date: 02/06/05 

Date: 5 Feb 2005 22:31:33 -0800

Barb Knox wrote:
>
> The natural numbers are an abstraction of counting concrete things
(sheep,
> pebbles, whatever).

But all our countings are, in fact, rather small. So while it may be
that abstraction of different countings indeed result in the same
number, these numbers are small. I count ten things here, I count ten
things there, voila I abstract to get the number 10.

I can also accept that we learn that we can always add 1 to our numbers
- you see this in small children, who very early learn to ask "what
number comes after?" when they are learning to count. So are you
saying that, since they are always given an answer (fortunately I can
still count past my children!), they abstract to the idea that there is
always one more number after any given number? I think this is more an
induction, rather than an abstraction.

>Imagining an arbitrary flock of sheep or pile of
> pebbles, there seems nothing preventing adding one more sheep or
pebble to
> it.

In what way can you imagine it? If it is by some kind of pictorial
image in your mind, then I would suggest that you are really imagining
some large (but fixed) number of sheep plus one. If it is some other
kind of imagination, then I would need to know more what kind you are
talking about, to be able to say if you can or if I think it is
relevant that you can.