Theory of Ratio

The role of mathematics in empirical science is puzzling, mysterious, and in my opinion has defied rational explanation. Why mathematics should be so enormously productive and effective in the handling of empirical phenomena is clearly an important foundational problem in science, but one for the most part that has been ignored. This is somewhat surprising given the rich theory and the abundance of results that have been developed in the philosophy and foupdations of mathematics. While there have been and still are great debates about the acceptability of the use of certain inf'mitistic and nonconstructive procedures in mathematical proofs, very little of these concerns have spilled over into science. The empirical sciences seem to have accepted without question the validity of the Platonic/Cantorian development of mathematics, and freely employ infinitistic and nonconstructive methods in their modeling of empirical phenomena. What is particularly surprising about this is that most empirically oriented scientists shy away from strong metaphysical commitments, and at the foundational level of their science usually promulgate the use of rather conservative methodologies. There are obviously many ways of approaching the problem of explaining the role of mathematics in empirical science. The one employed in this paper, which falls under an interdisciplinary branch of science generally known as measurement theory, tries to explicate the underlying assumptions for the proper assignment of numbers to empirical variables. While this is only one aspect of justifying mathematical procedures, it is an important and necessary first step. However, depending upon one's philosophy of science and mathematics, even this restricted problem has a variety of approaches. The approach I take is heavily colored by a vision I have of what a good theory of measurement would look like. The part that is relevant to this paper goes as follows: I see the empirical world as a finite collection of