Categorical or Quantitative?

A colleague and I had a conversation about whether the following variables are categorical or quantitative.

1) Social security numbers

2) Phone numbers

3) Postal zip codes

We agreed that all three are in fact categorical, but couldn't agree on a good reason.

The definition of a categorical variable (at least here In statistics, a categorical variable is a variable that can take on one of a limited, and usually fixed, number of possible values, thus assigning each individual to a particular group or "category."

However, it seems a somewhat weak case can also be made that the variables are discrete valued random variables. I need one good reason to convince students of why these variables are not quantitative. Any ideas?

• In a famous commentary on Stevens, Frederic M. Lord (1953) gives an example in which a statistical analysis treats football jersey numbers very effectively as being quantitative (even though these are archetypically categorical). One of his points--and one echoed over many decades by John Tukey--is that many data are not inherently "categorical" or "quantitative." These concepts may be useful for training novices, but they are misleading or worse as guides to statistical analysis. – whuber Mar 22 '16 at 19:15
• Nick Cox's thoughtful answer to a closely related question at stats.stackexchange.com/questions/67551/… may be relevant to your conversation. – whuber Mar 23 '16 at 15:38

• That's a fine mathematical principle--but it isn't useful for data analysis, because it arbitrarily precludes effective solutions. Indeed, your immediate two comments seem contradictory: yes, $0.5+0.6=1.1$ makes mathematical sense, but it has no meaning in the context I posited. Likewise, summing two SSNs (interpreted as nine-digit base-10 representations of integers) makes mathematical sense but it has no meaning in most conceivable contexts. – whuber Mar 23 '16 at 15:35