It seemed simple enough at first, I figured I would start from Tarski's axioms of geometry (I like the first-order theories as a dumb person :3), weaken them or drop them appropriately to get appropriate abstractness of "gender" but I kept thinking about it and more and more it seemed less and less like geometry. Like what does it mean for a gender "point" to be between other gender points? Are genders congruent??

So I started thinking, maybe gender would be best represented as some kind of topological space. I had never even heard of topology before so I spent like 5 hrs reading about it, tearing apart the Metamath definitions, and I still only very vaguely kind of get it (my vibe is that the definitions of topology I read about as collections of open subsets of sets or the neighborhood stuff don't actually describe very well what people actually use topology for lol). Now I will go up, starting from the bottom, the traditional topological space hierarchy and explain why I can't seem to make it work for gender:

Inner product spaces I quickly skipped over cuz they seemed too restricting, not general enough

A normed vector space initially seemed kinda close to the original informal definition of gender given by Lilypad:

"I have thought about gender as an N-dimensional space, where N≥0. For every persons experience of gender, i think of it as the area where for ∀n∈N, n>0, with completely agender being that for ∀n∈N, n=0."

Is a very pretty way to think of it imo. I moved on from this for similar reasons to metric spaces, which I will describe

What about metric spaces? Metric spaces are even closer to Lilypad's informal definition and are the traditional formalism for Euclidean spaces (a metric space with 3 dimensions r3). I couldn't make Tarki's geometry work conceptually but maybe a metric space would work? Is just that I ran into the same problem, which is that dimensions of a space imply 2 binary directions. Like what mean is (sorry am not so fluent in math language yet), a single-dimensioned vector is either going in a direction x or going in the complementary opposite direction of x (-x or signed x or idk). In Euclidean space it is like a "positive" x or y or z direction vs its complementary "negative" x or y or z direction. In other words, 2 vectors that are equal except for 1 component are on the same line. The problem is, I can't think of any gender thing like this except for the masculine-feminine binary and I don't want to concede that gender is really just a bunch of different gender binaries deep down. The other problem, which I think is also implicit in the informal definition, is that metric spaces by definition have a concept of "distance", which is a function that takes 2 points and "returns" a real-valued distance > 0 for 2 distinct points, and I can't think of a way to formalize the metaphorical distance between genders without breaking all of the axioms that make a metric space a metric space (Like I didn't mention distances have to obey the triangle inequality lol) and without a non-real-valued distance function

So then... I considered........ topological spaces which are so general that it's quite hard to get a grasp on and everyone is just making their own special useful topological spaces under a bunch of different names. To be more specific and vague at the same time, one definition is: a topology T of a set X is a collection of "open" subsets of that set that satisfies the following conditions: the empty set and X are in T, any arbitrary union (like adding the elements of 2 sets together into one big set) of elements of T is in T, the intersection (the set of only common elements between 2 sets) of any finite number of of elements of T is in T. This seems like the {good, bad, {}, {good, bad}} thing about topologies..... they're so fucking general that you can use them for so many things, yet so vague you're gonna be making your types of topologies to do anything cool with them. Like there are 2 kinds of topologies in general which are discrete topologies where you meet those requirements uhhh "enthusiastically" and do every possible union and as many intersections before it becomes infinite (idk where there that is ) or indiscrete topologies where "you" did the minimum possible to meet those requirements and your topology T of X is literally just { {}, X } lol. And all the interesting topologies are somewhere in-between those kinds of topologies. Oh, and that's just one definition, also the most general (most useless, I just want my gender topology smh). There is an equivalent definition, which is way more interesting for our purposes, based on a concept of "neighborhoods" which brings a concept of "closeness" into it (as in you can make topologies of sets based on how "close" the elements are to each other or something lol). I am getting tired of writing though, so go look that up for yourself if you want lol

A topological space is something like an ordered pair of a "base" set X and its topologies. So I guess what we are interested in for our "gender" topological space is the set of all "genders" and its topologies..... maybe. The basic idea here which I am clutching to and on the verge of dropping it into a vast sea of abstractness is uhhhh oh, yeah, I think I just lost it