The Grid

The main point I want to make in the Pythagoras article is in the section titled "Imposing the Grid. The demonstration showing that the hypotenuse of the unit square is not commensurable with the edges shows that it is natural to create systems that are not commensurable with each other. If you took two pieces of graph paper and turned one at a 45 degree angle to the other, you would find that you cannot match up the vertices.



(I made a bigger view of the image that shows more vertices and has the PHP code for the graph.)

The fact that there are entire systems which are not commensurable with eachother is extremely important. In the case of perspective. I could create a mathematical model that talks about everything relative to me. You could have a mathematical model with everything relative to you. Since we see things from different angles, the rational numbers in my system will appear as irrational numbers to you.

The fact that we are looking at the world through different world views does not mean that things are helpless. It means we have to be wary of the grids we impose on the world.