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Topic: Constructing Arithmetic from Geometry
Replies: 1   Last Post: Jul 2, 2014 9:13 PM

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Charlie-Boo

Posts: 1,585
Registered: 2/27/06
Constructing Arithmetic from Geometry
Posted: Jul 2, 2014 6:05 PM
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Peano axiomatized Arithmetic by defining the naturals, Addition and Multiplication. We can see the naturals as simply a "ruler" (as in a 12 inch piece of wood) that is infinite in one direction, with arbitrary symbols along it where the location of each symbol is a function.

Addition and Multiplication are ways to fill up one quadrant of the lattice squares on a plane. For Addition we have:

0 1 2 3 . . .
1 2 3 4 . . .
2 3 4 5 . . .
3 4 5 6 . . .
. . . . . . .
. . . . . . .

Imagine the ruler across the top and along the left side, and each square inside is the value of the sum of the two coordinates.

Starting with a blank quadrant, how can we take the infinite ruler and paste the correct numbers down on it? There are at least 2 ways to do it in 2 steps. For example, we can lay a copy of the ruler at the top of the quadrant as one step.

Multiplication is much trickier! We need to construct:

0 0 0 0 . . .
0 1 2 3 . . .
0 2 4 6 . . .
0 3 6 9 . . .
. . . . . . .
. . . . . . .

How can we use the ruler and the first quadrant above to fill in the right numbers here?

C-B




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