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Stream: theory: mathematics

Topic: Cousins

John Baez (Aug 17 2025 at 08:14):

A study of the mathematics of cousins argues that the concepts of "0th cousin" and "(-1)st cousin" make sense. And a commenter says:

So could we use 'negative-2nd cousin' to refer to people who have a child together?

This reminds me of how James Dolan noticed that not only 0-categories but also (-1)-categories and (-2)-categories make sense. I don't see any relation, except that we tend to start counting a bit too late, after a pattern is already going.

Amar Hadzihasanovic (Aug 17 2025 at 14:07):

The commenter is @Oscar Cunningham who is also here!
In this case it seems that the suggestion is that the entire diagram has a "mirror image" which goes all the way into the negative numbers, with (-1, 0) as the centre of symmetry (rather than just a few extra steps as in the case of (-2) and (-1)-categories).

Amar Hadzihasanovic (Aug 17 2025 at 14:10):

Perhaps this suggests rather that we should have phrased relations in terms of "n-th sibling" = (n-1)th cousins.

Oscar Cunningham (Aug 17 2025 at 14:28):

Yes, my thought was just that an ancestor a negative number of generations up must instead be a descendant. But someone replied to me pointing out a qualitative difference: someone you share an ancestor with is a blood relative, whereas someone you share a descendent with isn't. So maybe the concepts aren't analogous after all, or maybe we should call such people 'coblood related'.

Mike Shulman (Aug 17 2025 at 15:23):

My family had noticed the generalization to 0-cousins and (-1)-cousins before, but not the extension beyond that. However, now that Oscar pointed it out, I think it makes perfect sense! It's also useful, because it incorporates existing named relationships (parents-in-law are (-2)-cousins-once-removed) as well as important existing relationships that don't have a name in English (my parents and my wife's parents are (-3)-cousins).

Mike Shulman (Aug 17 2025 at 15:25):

It's true that $k$-cousins are blood relatives for $k\ge -1$ and not for $k\le -2$, but I don't think that's a valid objection to the terminology. After all, an "uncle" is sometimes a blood relative and sometimes not, and we don't distinguish the two cases linguistically at all.

Amar Hadzihasanovic (Aug 17 2025 at 15:25):

If blood is symbolic for "genetic code", the asymmetry comes down to the fact that

• the overlaps of my blood with the blood of my immediate ancestors are both disjoint and jointly covering,
• the overlaps of my blood with the blood of my immediate descendants are neither necessarily disjoint nor jointly covering.

Mike Shulman (Aug 17 2025 at 15:27):

Kinship relations, as viewed from the periodic table of cousins at least, are not really about genes but just about varieties of proximity in a directed graph, which is of course forwards/backwards symmetric.

Amar Hadzihasanovic (Aug 17 2025 at 15:27):

But apart from this everything seems pleasantly symmetric.

Jacob Neumann (Aug 17 2025 at 18:44):

Hey, glad to see my little doodle sparked a conversation! I've had a few thoughts since posting:

For the one-sided cousin diagram (the kind posted, not including in-laws), I think labelling the columns -1, 0, 1, 2, ... makes sense. It corresponds to counting only the generations properly between the people in question and their most recent common ancestor: my sibling is my 0-cousin because there are zero generations between us and our mrca, our parents. Deciding that "there are -1 generations between me and my mrca with X" means "I am my mrca with X, i.e. X is my descendant" is a consistent choice.

However, for the proposed two-sided diagram (which I want to make), I do feel like Me=(0,0) is the logical choice and that there's no reason symmetry should be about -1. It seems more logical to relabel the "Me" column to 0, the sibling column to 1, the first cousin to 2, and so on. That way your "spouse" (more precisely: "co-parent" or "biological spouse", the other parent of your children) is your (-1,0) cousin, reflecting that they have the opposite relation to you as your (1,0) cousin, your sibling. Likewise, your (-2,0) cousins, the other grandparents of your grandchildren, have the opposite place as your "first cousins", your (2,0) cousins. The mismatch with the traditional numbering is unfortunate, but worth it in my opinion

Mike Shulman (Aug 18 2025 at 00:39):

In the homotopy type theory community we fought a long battle against Voevodsky's attempt to renumber "homotopy $n$-types" as "types of h-level $n+2$". His way is undoubtedly more logical, with the contractible types being of h-level 0 rather than "homotopy (-2)-types". But I always believed (and still do) that we had a hard enough job ahead of us selling the subject to algebraic topologists and category theorists without simultaneously trying to renumber all of their definitions.

Mike Shulman (Aug 18 2025 at 00:41):

Sometimes it's worth the trouble to try to change infelicitous terminology, but not at the expense of making a more substantive advancement.

Mike Shulman (Aug 18 2025 at 01:02):

By the way, regarding this:

your "spouse" (more precisely: "co-parent" or "biological spouse", the other parent of your children)

I think there's a case to be made for using "(-2)nd cousin" more broadly for "spouse", at least colloquially. From a logico-mathematical point of view, past and future should be symmetrical. Your 1st cousin is still your 1st cousin even after your shared grandparents are no longer alive, so logically your (-2)nd cousin should already be your (-2)nd cousin even before your shared child is alive. Of course as a practical matter we are always speaking at some point in time and we don't know for sure what will happen in the future. But it doesn't seem unreasonable to me to use the observable (and more socially relevant) relation "spouse" as a proxy for the unobservable "present or future co-parent". In cases when precision is required one could add an adjective like "biological".

Oscar Cunningham (Aug 18 2025 at 05:51):

It might be more convenient to write 'usin' instead of '(-3)rd cousin'.

Mike Shulman (Aug 18 2025 at 05:58):

That's even worse than a "ntinuous functor".

Amar Hadzihasanovic (Aug 18 2025 at 07:06):

I agree that, logically, someone with whom I will one day have a child together should already now be my (-2)nd cousin, but I disagree that this implies that we should be justified in using the term for a spouse, since there is already an epistemic aspect in the “justified usage” of the positive half of the diagram: people can discover that they share an ancestor, and only then they are justified in referring to each other as siblings or cousins.
This has in fact been from time immemorable a common plot point in narratives: X and Y discover the previously unknown fact that they share a parent, and now they are justified in referring to each other as siblings (0th cousins).
The dual trope, where X and Y discover the previously unknown fact that they share a child together, is equally common. Of course it is more common for X and Y to discover the fact already the moment that the child is born. But I would suggest that the correct dualisation is that this is the moment in which X and Y discover that they are (-2)nd cousins and only after that moment they are justified in referring to each other as such.

Amar Hadzihasanovic (Aug 18 2025 at 07:19):

It is fun to consider what kinds of “dramatic revelations” are dual:

X discovers that he is not the biological father of the child of his spouse Y
is dual to
X discovers that he has been adopted, while his sister Y is the biological child of their parents

Amar Hadzihasanovic (Aug 18 2025 at 07:24):

And dual responses to the revelation:

X decides that it does not matter and that he is the father of Y's child in every way that matters
is dual to
X decides that it does not matter and that he and Y are siblings in every way that matters

whereas

X leaves Y and her child, remarries and has a child with Z
is dual to
X leaves his adoptive parents, finds his biological parents and discovers a biological sibling Z

Amar Hadzihasanovic (Aug 18 2025 at 07:27):

I bet one could find some pairs of famous stories/novels/films whose core narratives are dual under this symmetry :grinning_face_with_smiling_eyes:

Amar Hadzihasanovic (Aug 18 2025 at 07:32):

“Self-dual Oedipus” is the story of Oedipus, and in addition Jocasta discovers that she has unknowingly murdered her other child