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.