Is Base Unit Geometry Lacking?

Question

This article somewhat on the rationality of pi defines some geometric terms differently, such that there are no perfect circles, and no irrational numbers, there are only approximations. At least, that's how I understood it.

Then, the author claims:

Base-unit geometry loses no explanatory power, eliminates an infinite number of unnecessary objects, and gives a logical foundation on which to build a stronger theory.

So he claims that his system of thinking about geometry makes everything simpler, as well as making it more practical since we don't encounter perfect shapes.

Is there anything lacking about this system of thinking? If we don't have real or irrational numbers or perfect shapes, as I think the article claims, then where (if anywhere) in practical or applied mathematics does the system become inadequate?

_{As a side node, I don't entirely know how to tag this question. Feel free to adjust tags if something fits better.}

Answer 1

The author is a classic crackpot. The only intellectual content of this paper is that it is viable to reconstruct geometry as geometry on a discrete lattice, and that this is somehow more desirable than the usual formulations. However, having never studied the foundational math underlying the usual formulations of geometry, he has no standing for saying whether the lattice viewpoint is more (or less) desirable).

Worse yet, if you decide geometry is axiomized on a lattice, you then ought to give some examples of theorems that you can now prove, which are different than the usual geometric theorems. However, the author is apparently not strong enough in math to provide any such theorems (and they do indeed exist, and can be quite beautiful, though somewhat messy). Attacking the matter of "lattice pi" may well lead to the study of class numbers, for example, because the existence of rational approximations to pi which are "too good" is related to that field. But first you have to refine your definition of pi -- on a lattice, the definitions in terms of circle area and circle perimeter differ!