When people refer to “Goedel’s Theorem” (singular, not plural), they mean the incompleteness theorem that he proved and published in 1931. Kurt Goedel, the Austrian mathematician, actually proved quite a few other theorems, including a completeness theorem for first-order logic. But the incompleteness theorem is the one for which he is most famous.

To get some sense of the impact of Goedel’s Theorem on the mathematical community, consider how Herman Weyl, perhaps the greatest mathematician of the first half of the twentieth century, reacted to it. According to Weyl, up until Goedel’s Theorem, mathematicians were optimistic that all questions in mathematics could be definitively answered either one way or another (see his contribution to Oldenbourg’s Handbook of Philosophy).

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When you encounter a number, what sort of numerical information is it giving you? Is the number functioning merely as a label? Or is it counting how many? Does it denote a magnitude or merely indicate an ordering relation? All such questions address the issue of scale. When you see a number, it belongs to one of five scales:

Nominal scale. Here the numbers are merely serving as a label to distinguish different items. Think of social security numbers. They simply distinguish people in the United States, nothing more. The fact that one social security number is bigger than another tells us nothing about people’s respective ages, heights, or anything else about the people who possess them. Social security numbers belong to a nominal scale.

Ordinal scale. Here the numbers indicate an order but provide no magnitude information. For instance, imagine that prisoners in a prison get their numbers assigned by the order in which they entered the prison. The first prisoner gets number 1, the second number 2, and so on. The prisoners’ numbers thus tells us whether one prisoner entered prison before or after another, but tell us nothing about when or how much longer one prisoner entered prison before another. These prison numbers belong to an ordinal scale.

Interval scale. Here the numbers provide information of relative magnitudes. Consider street addresses in a city that is laid out as a grid, such as Chicago. A difference of 800 indicates one mile. Thus Western Avenue, which is 2400 west, is two miles from Pulaski Avenue, which is 4000 west (both these streets run north-south). But allowing 800 to indicate one mile is arbitrary, as is assigning Western Avenue to 2400 west. Instead of 800, city planners could have let 500 indicate a mile; and they could have let Western Avenue be 0 west or even 600 east. The functionality of Chicago street numbers is unchanged by both the addition and the multiplication of a constant, which leave interval information unchanged. These street numbers therefore belong to an interval scale.

Ratio scale. Here the numbers provide information about relative magnitudes, but the zero point is fixed. Consider distance from a given point. One can measure the distance in meters or feet or miles; all that’s required to make the conversion of units is multiplication by a constant. But the distance from a point to itself is always zero and doesn’t depend on the units of measurement. Distance information in this case is unchanged by the multiplication of a constant, which is the ratio of units. Distance therefore belongs to a ratio scale.

Absolute scale. Here the numbers can be only what they are. Such numbers are used to count the exact number of objects in some collection. How many marbles are in that jar? The answer admits only one number, that is, the exact number of marbles in the jar. When a number gives an exact count, it admits no variation. It is what it is and can be no other. In that case, the number belongs to an absolute scale.

REFERENCE: Warren S. Torgerson, Theory and Methods of Scaling. New York: Wiley, 1958.