ABSTRACT
Chapter 3 examined addition, subtraction, and the negative operation as ways of constructing new values from old. We now consider these operations in a different light—as relations among values already given within an encompassing background. Cartesian models, which are Cartesian coordinate systems interpreted as arithmetical value structures, constitute that background. This has the great advantage of portraying value structures not only geometrically, as Hasse diagrams do, but also algebraically. We thus have two separate routes to problem-solving and understanding. Cartesian coordinate systems can have any finite number of dimensions. The information contained in the “values” (n-tuples of real numbers) of such models is rich enough to eliminate all indeterminacies in the addition and subtraction tables of Chapter 3. Cartesian models can be used to illustrate the extraordinary ways in which both comparability and incomparability emerge in the addition of values. Even stranger and less familiar are values that are incomparable with zero. These come in various gradations and proliferate unnervingly in higher-dimensional models. (Chapter 10 will calm nerves by explaining how to handle them in decision making.) A final section introduces some basic concepts of model theory.
