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Constructing Arithmetic from Geometry
Posted:
Jul 2, 2014 6:05 PM


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?
CB



