R. DUNCAN LUCE AND LOUIS NARENS
Everyone is aware that measurement is a cornerstone of science, one that in some cases is highly controversial. Much complex technology underlies the refined measurement of certain physical quantities, some of which can be estimated to surprisingly large numbers of significant figures; one of the more elaborate businesses spawned by the social sciences, a business that affects all of our lives, attempts to measure intellectual ability and/or achievement; and elaborate computer programs are widely used to provide numerical representations (and simplifications), e.g., by factor analysis and multidimensional scaling, of complexes of data. Behind all of this activity is a belief, often sustained by a mixture of intuition and successful--if ill understood--procedures, that certain bodies of data can be represented in some fashion by numbers and their relations to each other. The goal of the semiphilosophical, semimathematical field of our title is to lay bare the types of empirical structures that admit such numerical representations.
The reason for the term "axiomatic" in the title is that this is how the structures involved are described. The task is to isolate axioms that, on the one hand, are empirically and/or philosophically acceptable for at least one important scientific interpretation of the primitives and that, on the other hand, permit us to prove mathematically that the structure is closely similar (usually, isomorphic or homomorphic) to some numerical structure. Ultimately, one aims for a finite collection of different classes of structures that span all the scientifically interesting cases.
At present, our knowledge appears quite adequate for the better developed parts of classical physics. It is interesting to note that many influential writers
© 1981 American Mathematical Society____________________
The research for this paper was partially supported by a grant (ITS-79-24019-1) from the National Science Foundation.