On Wed, 17 Sep 2008 22:25:51 +0100, John Jones wrote:

Let's try and trace this.

Two guys are going to trade (the beginning and traditional function of
numbers, I believe). Each has a table. One guy puts up three apples on
his, the other two oranges on his. They agree. The trade is made.

There is no need for numbers here (nor name for that matter). To them,
there were not 3 apples and 2 oranges but rather only some apples and
some oranges by concept or name, the amount being self-evident with sight.

Later, they are away from their tables but want to trade. They both have
agents or workers who actually look over and handle the apples and
oranges. They now need a way to describe, in abstraction, an amount. The
actual act of trading or physical exchange is being abstracted through
their workers.

3 becomes assigned to the tangible apples as a way to evaluate the
abstracted trade. So, 1 apple is 'followed' by 2 apples and so on.. as a
way to evaluate the apple/orange trade.

Since understanding this progression and the basic operation of adding or
subtracting apples can be applied to oranges, horses, jars of wine and,
dang!, every one thing, it becomes important to learn this process
outside of the apples, the number process itself removed from any object.

No matter how well this number process is understood, however, it remains
an abstraction. If traders met and one said, "I have '3', I will trade it
for '2'" the other saying, "No I will trade 2 for 2" and they agree on
this, nothing at all happens. Number without object is meaningless
outside of number only.

1 apple plus 1 apple has a tangible meaning. 1+1 only does as an
operational principle but this is all. 1+1 takes as a given that some or
all objects can operate under it so there is no need to constantly
reference any object.

Somewhere and somehow this has led to the number becoming it's own object
of reference.