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

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Ken.Pledger@vuw.ac.nz

Posts: 1,377
Registered: 12/3/04
Re: Constructing Arithmetic from Geometry
Posted: Jul 2, 2014 9:13 PM
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In article <0735c3c0-7a73-4584-9dd2-47fb79d55ce3@googlegroups.com>,
Charlie-Boo <shymathguy@gmail.com> 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.

Ken Pledger.



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