(This is the third chapter of the first volume of The Scientific Experience, by Herbert Goldstein, Jonathan L. Gross, Robert E. Pollack and Roger B. Blumberg. The Scientific Experience is a textbook originally written for the Columbia course "Theory and Practice of Science". The primary author of "Measurement" is Jonathan L. Gross, and it has been edited and prepared for the Web by Blumberg. It appears at MendelWeb, for non-commercial educational use only. Although you are welcome to download this text, please do not reproduce it without the permission of the authors.)
1. There are many different reasons why a prize might be awarded to a baseball team for performance in league competition. Identify the type of scale to which each of the following possible criteria belongs, and explain your answer.
2. Jones has an expensive foreign-made analog watch with a jeweled movement that gains at most 12 seconds a month. Every month, Jones resets his watch according to the time announced by the telephone company time service. Smith has an inexpensive digital watch that gains at most 2 seconds a year. However, Smith sets her watch five minutes fast, in order to avoid missing commuter trains. Whose watch is more precise? Whose watch is more accurate?
3. Consider a two-dimensional square of definite size. Among all possible transformations that can be made on this square, there are some that will leave it in a position indistinguishable from its original position (e.g. rotation in the plane by 90 degrees; a rotation of 180 degrees around the diagonal of the square, or a reflection in a two-sided mirror positioned positioned perpendicular to the square and passing through its middle). We call such transformations the symmetry transformations of the square, and say that the more symmetry transformations a system has, the higher its degree of symmetry. What sort of scale is being used in this sort of measurement? Under what conditions would it be sensible to compare the degrees of symmetry of two systems?
4. A conversation is taking place between two members of your college class, and they are talking about their academic standing when they were in high school. One student tells the other that she was 2nd in her graduating class, and the other student remarks "Oh, well, you're about three times as smart as I am; I was only 6th in my class." Explain what's wrong with the second student's reasoning. Would the second student be warranted in asserting instead that the first student had done better than he had in high school?
5. Write a recursive program for Fibonacci numbers. Use your program to calculate the 5th, 10th, 15th, 20th, 25th, and 30th Fibonacci numbers. Use a wrist watch to record the time it takes for each calculation.
6. The text of this chapter would seem to imply that in deciding whether or not a particular discipline is or is not scientific, reproducibility of measurements is what ultimately counts. Do you think this is a reasonable way to demarcate science from "non-science", and/or can you think of other measurement criteria which you think are more important than "reliability" in establishing knowledge claims? Do you think that the criterion of reliability rules out the possibility of scientific status for some disciplines in principle (e.g. economics, psychoanalysis, literary theory)? Please explain your answers carefully.
7. For the given four-state example, show that an increase from 30 representatives to 32 would not have caused an instance of the Alabama paradox, but that an increase from 30 to 34 would.
8. Either construct a two-state example of the Alabama paradox, or prove that it is impossible to do so.
9. Construct a three-state example of the Alabama paradox.
10. A "yes-and-no" voting system can be viewed as a reduction of a preference ranking system, in which all that matters is who each voter likes best and least. If voters favoring a second-most popular candidate cast their "no"-votes against the most popular candidate, instead of a candidate they actually like least, this can sometimes tilt the net tally so that the second-most popular candidate wins. In other words, a "yes-and-no" system can deteriorate into deliberate manipulation. Construct a 4-candidate example in which this phenomenon could occur.
 At M.I.T., literature, art, music, foreign languages, and history are collectively a single department, while there is finer differentiation among the sciences and the engineering disciplines).
Deciding which subjects or groups of subjects are comparable raises a number of difficult questions, and this issue is often left to experts. This obviously presents problems if these same experts have a stake in the outcome of a particular research program.
One of these other criteria, of course, would be explanation. As you read more about models, you might ask yourself how or whether successful models actually explain, or merely describe the phenomena being modeled. Do you think that explanation, as opposed to description, is very important in science, or is description + prediction the only thing we should be concerned about? Similarly, you might consider whether you think the predictive capability of certain physical theories is the defining characteristic of natural science; that is, consider whether or not you would classify a discipline as "pseudo-science" if the theories of that discipline were not predictive.
 This is not to say that scientists always comply with this requirement. Sometimes scientists temporarily conceal a few crucial details for a number of months, in the interest of keeping a lead over other researchers, and thereby advancing their own careers. Like investment counselors, some scientists have been known to protect what they perceive as their own self-interest.
 Named for the 14th century English philosopher, William of Ockham, this is the maxim that the number of assumptions introduced to explain something, or the number of entities postulated by a theory, must not be multiplied beyond necessity. Of course, what constitutes "necessity" is often far from clear, and the criterion of "simplicity" in the evaluation of models is notoriously difficult to specify.
 You may recognize the name as that which describes the sequence of numbers, each of which is the sum of the two previous numbers. Named for the 13th century Italian mathematician Leonardo Fibonacci ("Leonardo, son of Bonaccio"), these numbers have fascinated mathematicians and scientists alike. In this case, the program Fibonacci instructs the computer to simulate the growth of the variable p(n).