A nominal number is a symbol of a number used for naming. Wikipedia defines it as a " a one-to-one and onto function from a set of objects being named to a set of numerals. . . it is a function because each object is assigned a single numeral, it is one-to-one because different objects are assigned different numerals, and it is onto because every numeral in the set at a given time has associated with it a single named object."
It seems to me that such symbols are not natural numbers and that addition would not be defined with respect to them, yet I routinely see people manipulating them algebraically. Are nominal numbers natural numbers? Are arithmetic operations such as addition and multiplication defined for them?
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$\begingroup$With help from Lord_Farin and others I have answered this question and wanted to post the answer here, in case anyone else should have a similar question.
Scientists study the world, and one way that they do it is by analogizing between the properties of the objects of study and some subset of the real numbers. The process of deciding how to model the real world in numbers is called "measurement theory" (at least in psychometrics) and the seminal paper on levels of measurement was by Stevens, defining nominal, ordinal, interval, and ratio data. After creating a numeric model, scientists then use the laws of mathematics to come to mathematical conclusions, and then map these back to the world to come to conclusions about the world. Error is possible in one of two places: in the choice of numeric model or in the mathematical manipulation, both of which will result in error in the mapping back of the numeric answer to come to a conclusion about the world.
Nominal numbers do not have the properties of integers and the operations that are defined for integers are not defined for nominal numbers. The only operations that would be defined on nominal numbers are from set theory, because they are simply numeric symbols -- names -- for objects that could as well be called a, b, and c.
So yes, treating nominal numbers as integers and doing simple addition is a math error -- it is meaningless from the point of view of mathematics. Or it may be a mistake in modeling, and when the person said that they chose to model the data as nominals they really meant something else, such as integers. But this then raises the question of whether the underlying data has the right properties to be modeled as integers; if not, the result of the math calculations will not map back to the world in any meaningful way.
Many thanks to Lord_Farin and a few others on math stack exchange for their patience as I worked this one out.
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The only context in which I've seen [algebraic manipulation of nominal numbers] requires to separate the nominal number from its associated object, after which it of course is just a natural number and hence allows algebraic manipulation.
by which I mean that although our numbers were obtained as a function $f: A \to \Bbb N$ for some suitable collection $A$, we separate the numbers from how they were obtained by pretending that we just have the numbers themselves, i.e. the set $f[A] \subseteq \Bbb N$ of numbers assigned to $A$.
So we "forget" about $f$ and deal with the numbers as usual. As I read it, this is also going on in your second reference. The professor distributes his "football numbers" in a more or less normally distributed way (by which I assume it is meant that the function $f: A \to \Bbb N$ under consideration is somehow chosen using a normal distribution). Then, the prof claims that these numbers (which have now been separated from their object counterpart) behave just like those obtained from an ordinary normal distribution sampling. (I'm tempted to say "Big whoop.")
On the other hand, in the first example, we deal with an ordinary variable: it just so happens that every possible input state corresponds to a unique output for the variable. But since we intend to use properties of the output numbers (namely, that $0 < 1$) it is not valid to label it a "nominal variable".
In conclusion, arithmetic is not defined on nominal variables. This holds more or less by definition (otherwise the variables wouldn't be nominal, since they would be used for purposes beyond identification). But of course, every nominal number can be dissociated from its corresponding object. Then, we're left with just a number. Evidently we can perform arithmetic with this.
I hope that clarifies the matter for you.
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