In article <email@example.com>, Charlie-Boo <firstname.lastname@example.org> wrote:
> 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
There's a standard theory of this. For quite a neat brief treatment you can go to the fountainhead: Hilbert "Foundations of Geometry" 1971 translation, section 15 (pp.51-3). There are more details in that book and in other books about coordinatizing an affine plane (e.g. the Euclidean plane) or a projective plane.