Rob
This paper is stunning, a breakthrough in understanding the reasoning processes necessary for science. Even though I helped edit your paper, I still struggle with the simplicity of it. Here is why I think your paper is destined to be seen as a seminal work. It introduces a whole series of major ideas, including:
1) The introduction of a law of the included middle into mathematics, science and logic as a process of forming and symbolizing new distinctions with which to reason.
2) The reintroduction of number as a relationship or ratio between quantities.
3) Extending the concept of relationships to unlike things or quantities.
4) The recognition that "axioms are required to encode what law describe".
5) Identifying the structural relationships that emerge from 3) and 4) as the proper objects of scientific reasoning.
6) A shift from arithmetic to geometry requiring only a redefinition of the number one.
The Law of the Included Middle (1) appears to be entirely new, although the idea that new concepts emerge through the drawing of new distinctions is familiar from "pop ontology".
Number-as-relationship (2), as MacDuff notes, was clearly stated by Isaac Newton, but subsequent scientists and mathematicians have not taken it up and developed it, and it has apparently been largely forgotten.
While Newton may have been consciously using the extension of this concept of number-as-relationship to relationships between unlike things (3) in developing his laws, I have found no sign that he ever articulated this.
While references "axioms of nature" are frequent in the time of Galileo, Newton and Locke - for example Francis Bacon in Novum Organum (Google Books edition, p.67) refers to Newton's Laws as such - I've seen no evidence that these have ever been used as the foundation of a formal system of reasoning. Thus 4) appears to be entirely new.
5) appears to be entirely new.
6) appears to be entirely new, although the idea of a relationship between line segments is as old as Euclid (Book V). Once this shift is made the Pythagorean theorem becomes a consequence of the law of the included middle.
It is in our nature as human beings to recoil when confronted with something that contradicts our deeply-held beliefs. For those of us who have been frustrated by the opacity, complexity and clumsiness of mathematics used in physics, finding a way to push past these barriers may come as a relief. Hopefully, MacDuff's consciously cognitive-dissonance inducing "circle and square" exercise will convince readers of the limits of current mathematics. Your simple examples offer the reader an opportunity to take a first step the process of using identities to construct the formulas of science.
Thank you for this paper,
Chris Horton, Ph. D. (Physics)