Re: Laws & Theories

Posted by Peano on Mar 07, 2002 at 21:15
(67.211.115.248)

Re: Laws & Theories (Chevalier)

In the beginning there were no numbers. People only had three quantites: none, few, and many.
The addition table back then was quite simple:

none+none=none
none+few=few
none+many=many
many+none=many
many+few=many
many+many=many
few+none=few
few+many=many

few+few=?

This is the question. WHEN does few+few become many? This is where numbers arise. Numbers themselves are simply ideas. We use these ideas to describe when few+few becomes many so that everyone has a universal understanding of what is meant by none, few, and many.

The most common numbers in use today are the
"natural numbers." These are the numbers
"1, 2, 3, 4, etc." But, simply as ideas, what
are they? Well, they are built from the Peano Axioms. The first 3 Peano axioms simply state this:

I. 1 is a natural number
II. Every natural number has a successor
( a number which comes after it)
III. 1 is not the successor of any number

There is also a notion of "mappings" - this is
when we take elements of one set and associate them with elements of another set. Think of it as though you have a box of stuffed animals, and a basket of name tags, and you decide to put a name tag on each animal. This is how mappings work.

There are two types of mappings that primarily
interest mathematicians & scientists:
1. one-to-one, and
2. "onto"

A "one-to-one" map is one in which every element
from the starting set goes to ONE element of
the ending set. The limitation, though, is that
the ending set CAN have more members than the
starting set. Since every element of the starting
set must go to some member of the ending set,
there are members of the ending set which are not
connected to any members of the starting set
(too many stuffed animals - not enough name tags.)

The second type of mapping, an "onto" map, is one
in which every member of the ending set comes from
a member of the starting set. If there are more
members in the starting set than in the ending
set, two members of the starting set get assigned
to the same member of the ending set
(too few stuffed animals - too many name tags -
some animals are wearing more than one name tag.)

What is not allowed, though, is for a single member of the starting set to go to two members of the ending set (A name tag cannot be put on
more than one stuffed animal simultaneously.)
While such a correspondence is a mapping, it
is NOT a function.

A mapping that is one-to-one and onto is said
to be a BIJECTION.

The next question becomes, "What is counting?"

Counting is a bijection from a set (fingers, blocks, houses, money, etc.) to the natural numbers.