## 11 November 2007

### Genealogical nomenclature

The nomenclature of "kth cousin l times removed" clearly was not invented by a mathematician.

A friend of mine (hi, Kate!), in an effort to explain the nomenclature, said that "Nth cousins share an (N-1)th-great grandparent." This makes sense with respect to the traditional nomenclature, but that "-1" is kind of awkward. But you can't make it go away; the rephrasing is "Nth cousins share a common ancestor (N+1) generations back".

Therefore siblings are zeroth cousins, since they share an ancestor one generation back (their parents). (For the sake of simplicity I'm ignoring half-siblings, who share one parent, and all other half-relations.)

And in the degenerate case, you are your own negative-first cousin; you share an ancestor with yourself zero generations back (yourself!)

The relevant number is the number of generations between you and the first common ancestor; thus what we call "first cousins", for example, should really be indexed by the number two. This would make the indexing start at zero, not minus one.

The "removed" nomenclature gets even weirder. If two people share a common ancestor, who is k generations above person A and k+l generations above person B, where l is positive, then they are (k-1)th cousins l times removed. So if my grandparent is your great-grandparent, then we have k = 2, l = 3, so we're first cousins once removed. (Incidentally, if my grandparent is your great-grandparent, then you must live in the future; as of 2007 no such person exists.) The nomenclature is symmetrical, which is surprising because the only other English-language kinship terms that are symmetrical involve people of the same generation. (And this is only true if you equate "brother" and "sister", or use the gender-neutral word "sibling".) If person U is your uncle, you're not his uncle; you're his niece or his nephew.

Now, I share an ancestor with my uncle (my mother's brother); his mother is my grandmother. (His father is my grandfather, too, but my grandfather's dead, and I didn't much like him when he was alive.) So we have k = l = 1, and so he's my 0th cousin once removed.

And my grandmother? She's my negative first cousin twice removed. (In general, your direct ancestors, n generations backed, are your negative first cousins n times removed.)

The simplest fix would seem to be shift all the cousin numbers up by one, so your siblings are now your first cousins, the people currently called "cousins" (people with the same grandparent) are second cousins, and so on. That way the indexing starts at zero, not negative one. The "removed" nomenclature seems a bit funny, but I suppose that in a lot of cases the important quantity is the difference in generation number between two people, so I'd actually keep it the way it is if I were reforming the system -- although perhaps de-symmetrizing it, so that one can immediately tell which of the two people involved is of the older generation.

.mau. said...

We also use the "kth cousin" nomenclature. However, in Italian legal terminology (I do not know it it applies to the States too) they talk about relatives of kth degree, and the relationship counts one for each direct ancestor or offspring. So, a second cousin is a relative of fourth degree (you go up two steps towards your grandparent and down two steps towards the cousin), an uncle-niece relationship is of third degree, and you are the (only) zeroth-degree relative of yourself.
From a mathematician's point of view it is more apt :-)

Aaron said...

Oh maaan! That's the first coherent explanation of cousin terminology that I've ever heard, and I definitely plan to remember it. :) If you're looking for a greater challenge, you should tackle the Hindi equivalent next... someone tried to explain to to me once, but it made my head explode.

.mau.—I like the "kth degree" terminology you mentioned, because it's exactly the same as (the logarithm of) the "relatedness coefficient" that biologists use to measure relatedness. And it's simple too! :)

Flooey said...

I actually think that the original nomenclature (indexing starting at negative one) is a good idea, in the context of English, because any relationship which would be a zeroth or negative-first cousin, or involve people who were zero times removed, has an existing word for it, so we never would use those values. Therefore, the cousin relationships are numbered with 1 representing the closest relationship for which there is no more specific word.

Your only negative-first cousin is yourself.

Any nth cousin that is zero times removed is a direct ancestor of yours, so they're a parent (first cousin), grandparent (second cousin), or great^(n-2) -grandparent.

Any cousin that is a zeroth cousin is the child of a direct ancestor of yours, so they're a sibling (zero times removed), an aunt or uncle (once removed), or a great^(n-1) -aunt or -uncle.

Mike Cassidy said...

The system is clumsy.
It helps to remember it is tracking the "primary" relationship. So your first cousin's children are your first cousin once removed but second cousins to your children. Your first cousin's grandchildren are your 1st cousins twice removed, your children's second cousins once removed and your grandchildren's third cousins.

Sometimes I think people like using cousins [ 18th cousin 20th removed] to feel we are all related; one big happy family.

There are several other numbering systems used to identify people's relationships. One used in writing genealogy uses a name and generation number.
An example: my great great grandfather was John1 Cassidy, his sons were: John2 [John1] , Patrick2 [John1]; Daniel3 [Patrick2, John1]. The problem with this system is that if I suddenly found an ancestry early then John1 everyone gets renumbered. That's a lot of work.

There are two other commonly used systems: legal and source. I like the one above because it supplies the names of the ancestors.

I also think the most interesting part of genealogy is locating your family in 1900 or 1870 and seeing who was living around them. Seeing the occupations, education level, how large families were, how many children died young and as infants, how many women support large families of children when their husband dies in their 40's. All of which has nothing to do with cousins, but for me is more interesting.

Jack said...

I have seen explanations of cousin terminology before but this is the first time I really got it. Thanks for the lucid exposition.

John Armstrong said...

It's often easiest to understand terminology when you push it back from where it originated. Here we push back to 0th cousins and -1st cousins.

I don't think I really felt the full tower of category theory until it was pointed out that an n-category is a category enriched over (n-1)-categories. Then a category (1-category) is enriched over sets, so a set is a 0-category. And then a set is a category whose morphisms just say whether two objects are equal or not (since they're all identities). That is, a (-1)-category is a Boolean truth-value!

But as for the topic of this post, all that is.. unrelated (massive groan)