Reference: Simon, H. A.; Kalagnanam, J.; & Iwasaki, Y. The Mathematical Bases for Qualitative Reasoning. KSL, August, 1990.
Abstract: Much reasoning about quantities takes place without use of mathematical formalisms, but solely in terms of ordinary language. A good deal of such qualitative reasoning makes implicit use of the properties of ordinal variables and monotonic transformations. In this paper, we attempt to provide the formal foundations of qualitative analysis and to show how qualitative reasoning arises naturally and simply out of the structure of systems of algebraic equations and ordinary differential equations. Our goal is to explicate, using familiar mathematical formalisms, the practices of researchers in many fields who use qualitative reasoning, and therby to gain an understanding of the formal assumptions and mechanisms that underlie such analysis. We sketch out some of the properties of functions, and especially continuous differentiable functions, that are invariant under monotonic transformations of the variables, and show how these properties can be used to analyze phenomena where the variables are only defined ordinally.