I introduced this series by noting that there are different kinds of numbers – more precisely, different kinds of quantities represented by numbers – and that each allows you to draw different kinds of conclusions.
The most basic kind of quantity that I can think of, from which you can draw the least amount of conclusions, is what I will call a key. In their purest form, keys are rare, but an example that comes close is a person’s Social Security Number (SSN) in the USA. The Social Security Number is a 9-digit number, usually written with dashes, like 123-45-6789. There are precious few conclusions you are supposed to be able to draw from knowing somebody’s social security number. If my SSN is bigger than yours, it doesn’t mean that I applied for mine later than you did. Neither does it mean that I was born after you were, or that I paid more for mine than you did for yours. The SSN is not a pure key, in the sense that you can tell something from the first three digits about where the SSN was issued. In its pure form, a key simply serves to be unique. Two different people have different SSNs, two different SSNs mean two different people, the same SSNs mean the same person.
A key, in its pure form, has certain operations be meaningful. In essence, there are four:
(a) equality testing: comparing one key with another to see if they are identical or different
(b) linking: though you can’t tell anything from the key itself, the key must allow access to other data for the key to be meaningful. That other data may be private, it may be hidden, it may be hard to get at. But without that other data being there, the key is rather useless. Somebody, somewhere, needs to be able to link the SSN to a particular person, or perhaps confirming the link between an SSN and a particular person.
(c) issuing: at some point, a person is first issued their SSN. Whoever issues the SSN must make sure that the assigned SSN is unique.
(d) retiring: at some point, presumably after death, the SSN is de-linked from the person, and retired.
There are also operations of secondary importance, like rendering the SSN in readable form e.g. “123-45-6798” or copying an SSN from one place to another.
As important as listing the kind of operations that are meaningful on a key is listing some operations that are not meaningful on a key: testing two keys for “bigger”, adding two keys, subtracting two keys, squaring a key, adding “1” to a key, multiplying a key by 2.
In looking at these operations, both the ones that we consider meaningful and the ones we consider not meaningful, you may wonder why the SSN is represented by a number at all. SSNs don’t seem to use their number for most of the things we use numbers for: adding, subtracting, multiplying, dividing. Could a unique key instead be done as a unique musical chord, or as a unique pattern of coloring a chess board, or as a unique pattern of wide and narrow black bands on a white background like a UPC? I think there is little doubt that it could. So why use numbers?
Numbers are convenient for representing keys for at least two reasons. One is that a key such as a Social Security Number can be easily transmitted using any keyboard (or, in the old days, typewriter). Another is that we all know that our number system is capable of producing a large number of distinct combinations. It is not hard to see that a 9-digit Social Security Number allows for a billion (a thousand million) combinations, quite ample for at least all of the people currently living in the USA (about three hundred million). It is true that numbers aren’t the only way to easily generate large numbers of different combinations. But they are handy, convenient, and familiar.