Meaningless statements

Increasingly, discrete mathematics is influenced by connections with other fields. We give one example of a broad topic in the field of discrete mathematics that comes from the social sciences, and in particular from the theory of measurement that has been developed to put measurement, especially in the social sciences, on a firm mathematical foundation. The key concept is that of meaningful statement, a statement whose truth or falsity remains unchanged after admissible transformations of scales of measurement. We apply this concept to combinatorial optimization, graph coloring, scheduling, linear programming, 0-1 optimization, and multiperson games. 1. The Future of Discrete Mathematics Increasingly, discrete mathematics is influenced by connections with other fields. In the mathematical sciences, it is influenced by connection with probability, geometry, algebra, analysis, topology, number theory, . . . Outside the mathematical sciences, it is influenced by connection with biology, chemistry, physics, manufacturing, engineering, . . . In this paper, I will give one example of a broad topic in the field of discrete mathematics that comes from the social sciences, but has been developed not only by economists and psychologists, but also by philosophers of science, physicists, logicians, and, of course, mathematicians. It is motivated by the theory of measurement that has been developed to put measurement, especially in the social sciences, on a firm mathematical foundation. In this paper, I will give a brief overview of measurement theory and define a central concept of the theory, that of a meaningful statement. I will then apply the theory of meaningfulness to discrete mathematics in various ways. In particular, I will ask whether or not statements in combinatorial optimization, that a specific solution is optimal, are meaningful. I will talk about the meaningfulness of conclusions about the generalization of graph coloring called T -coloring. I will give a variety of examples of meaningful and meaningless conclusions about scheduling problems. This will lead to the question of when a greedy solution to an optimization problem is a meaningful optimal solution. I will talk in general about 1991 Mathematics Subject Classification. Primary 90.30; Secondary 05C, 05.55, 90C, 90.50, 90.70, 90.99, 92G. The author thanks the U.S. National Science Foundation for its support under grants SBR9709134 and INT96-05174 to Rutgers University. c ©0000 American Mathematical Society 1052-1798/00 $1.00 + $.25 per page