Lunch will be served.
A statement involving scales of measurement is called meaningless if its truth or falsity can depend on the particular versions of scales that are used in the statement. We will develop the mathematical foundations of a theory of measurement that will allow us to make the notion of meaningless statement precise. We will then give a variety of examples of meaningless statements. For example, we will show that the conclusion that a given solution to a problem is optimal might be meaningless and in particular we will describe such results for shortest path problems, for graph coloring problems arising from frequency assignments, and for scheduling problems. We will also discuss limitations (through functional equations) on the possible averaging functions which allow meaningful comparisons in different applications such as choosing new technologies, comparing the abilities of different groups of students, etc.