We can count by bins and not know it. The Peano Axioms apply word for word to bins. The bins can contain 2 items and we count bins of 2 starting with a bin labeled 0.

Fractions are a way of re-interpreting or looking inside bins that we labeled with whole numbers.

We can think of a horizontal line on the bottom of counting by ones and twos. These are in effect labels. Above it we count by bins of two.

(0,1),(2,3)(4,5),…

0, 1, 2,…

So when we count by one along the top line, we don’t get to bin 1 until we get to 2 along the top line.

We can think of such lines of bins for bins of 3, bins of 4, etc.

These can be thought of as applying at the same time to the bottom row.

Within a bin line, we have Peano Axioms at the same time for the bins and for counting by one ignoring the bins.

If we want to add 1/2 + 1/2 = 1, we go to the bin 2 line and we simply do 1+1 = 2. This corresponds to the 1 bin on the bottom lie.

For 1/2 + 1/3, we go to the bin 6 line and relate 1/2 to 3/6 and 1/3 to 2/6 and then count by ones on the bin 6 line to do 2+3=5.

This corresponds to bin 0 on the bottom line.

If we add 1/6, i.e. add one on the bin 6 line, we get to 6/6 which is bin 1 on the bottom line.

The key insight is that the Peano Axiom rules apply to bins of 2 or bins of 3, etc. Thus when we count whole numbers, the whole numbers can themselves be bins, i.e. sets.

This is the basis of fractions. Existing methods are partly using this idea. They don’t explicitly say though that the Peano Axioms apply to each bin type and that we can count along a bin line by the individual items using the Peano Axioms for that line.

When we recognize the Peano Axiom rules apply to counting bins and not just some type of atomic numbers, we get a new insight.

We can think of atomic numbers as being some bin line where this stops. But we can also think of them as the base line and then we split the atom. The two points of view are both useful.

In the split the atom point of view, you are never counting truly fundamental numbers. You are always counting bins.

http://en.wikipedia.org/wiki/Auguries_of_Innocence

To see a world in a grain of sand

And a heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour.

Each bin that we are counting as whole numbers can be an entire universe. Each bin as a unit of time can be an entire eternity.

To see a world in a bin of one

And a heaven in a number line tick

Hold infinity in the bin of one increment,

And eternity in a single node.

When we count whole numbers, we don’t know what is inside the bins. The Peano Axioms work for bins not just atomic numbers. Thus we never know if we are counting bins or atomic numbers, or what is in the bins.

When we do fractions, we look inside the bins and find we have multiple number nodes in each bin.

“Beauty is truth, truth beauty,” – that is all

Ye know on earth, and all ye need to know.

“The bin has many, many in the bin,” – that is all

Ye know in counting, and all ye need to know.

“The bin has many, many in the bin,” – that is all

Ye never know in counting how many are in your bin

until you need to look in to do fractions.