Beyond the Binary

Do not at the outset of your career make the all too common error of mistaking names for things. Names are only conventional signs for identifying things. Things are the reality that counts. If a thing is despised, either because of ignorance or because it is despicable, you will not alter matters by changing its name.

—W. E. B. duBois

A common misconception about words is that they have definitions: look up the definition, and that tells you everything to know about that word ... right?

It can't actually work that way—not in principle. The problem—one of them, anyway—is that with a sufficiently active imagination, you can imagine edge cases that satisfy the definition, but aren't what you really mean by the word.

What's a woman? An adult human female. (Let's not play dumb about this today.) Okay, but then what does female mean? One common and perfectly serviceable definition: of the sex that produces larger gametes—ova, eggs.

That's one common and perfectly serviceable definition in the paltry, commonplace real world—but not in the world of the imagination! We could imagine the existence of a creature that looks and acts exactly like an adult human male down to the finest details, except that its (his?) gonads produce eggs, not sperm! So one might argue that this would be a female and presumably a woman, according to our definitions, yes?

But if you saw this person on the street or even slept in their bed, you wouldn't want to call them a woman, because everything about them that you can observe looks like that of an adult human male. If you're not a reproductive health lab tech and don't look at the photographs in biology textbooks, you'll never see the gametes someone's body produces. (You can see semen, but the individual spermatozoa are too small to look at without a microscope; people didn't even know that ova and sperm existed until the 17th century.) Does that mean this common definition of female isn't perfectly serviceable after all?

No, because humans whose gonads produce eggs but appear male in every other aspect, are something I just made up out of thin air for the purposes of this blog post; they don't exist in the real world. What this really shows is that the cognitive technology of "words" having "definitions" doesn't work in the world of the imagination, because the world of the imagination encompasses (at a minimum) all possible configurations of matter. Words are short messages that compress information about the world, but what it means for the world to contain compressible information is that some things in the world are more probable than others.

To see why, let's take a brief math detour and review some elementary information theory. Instead of the messy real world, take a restricted setting: the world of strings of 20 bits. Suppose you wanted to devise an efficient code to represent elements of this world with shorter strings, such that you could say (for example) 01100 (in the efficient code, using just 5 bits) and the people listening to you would know that what you actually saw in the world was (for example) 01100001110110000010.

If every length-20 bitstring in the world has equal probability, this can't be done: there are 2²⁰ (= 1,048,576) length-20 strings and only 2⁵ (= 32) length-5 codewords; there aren't enough codewords to go around to cover all the strings in this world. It's worse than that: if every length-20 bitstring in the world has equal probability, you can't have labels that compress information at all: if you said that the first 19 bits of something you saw in the world were 0110000111011000001, the people listening to you would be completely clueless as to whether the whole thing was 01100001110110000010 or 01100001110110000011. Just locating a book in the Jose Luis Borges's Library of Babel is mathematically equivalent to writing it yourself.

However, in the world of a non-uniform probability distribution over strings of 20 bits, compression—and therefore language—is possible. Say, if almost all the bitstrings you actually saw in the world were either all-zeros (00000000000000000000) or all-ones (11111111111111111111), with a very few exceptions that were still mostly one bit or the other (like 00010001000000000000 or 11101111111011011111), then you could devise an efficient encoding.

To be efficient, you'd want to reserve the shortest words for the most common cases: like 00 in the code to mean 00000000000000000000 in the world and 01 to mean 11111111111111111111. Then you could have slightly-longer words that encode all the various exceptions, like maybe the merely-eleven-bit encoding 10110101110 could represent 00100010000000000000 in the world (1 to indicate that this is one of the exceptions, a following 0 to indicate that most of the bits are 0, followed by the Elias self-delimiting integer codes for 3 (110) and 7 (101110) to indicate that the 3rd and 7th bits are actually 1).

Suppose that, even among the very few exceptions that aren't all-zeros or all-ones, the first bit is always in the majority and is never "flipped": you can have exceptions that "look like" 00000100000000000000 or 11011111111101111011, but never 10000000000000000000 or 01111111111111111111.

Then if you wanted an efficient encoding to talk about the two and only two clusters of bitstrings—the mostly-zeros (a majority of 00000000000000000000 plus a few exceptions with a few bits flipped) and the mostly-ones (a majority of 11111111111111111111 plus a few exceptions with a few bits flipped)—you might want to use the first bit as the "definition" for your codewords—even if most of the various probabilistic inferences that you wanted to make on the basis of cluster-membership concerned bits other than the first. The majoritarian first bit, even if you don't care about it in itself, is a simple membership test for the mostly-zeros/mostly-ones category system.

Unfortunately—deeply unfortunately—this is not a math blog. I wish this were a math blog—that I lived in a world where I could afford to do math blogging for the glory of our collective understanding of greater reality. ("Gender?" I would say, confused if not slightly disgusted, "I barely know her.") It would be a better way to live than being condemned to gender blogging in self-defense, hopelessly outgunned, outmanned, outnumbered, outplanned in a Total Culture War over the future of my neurotype-demographic. But since I do, somehow, go on living like this—having briefly explained the theory, let's get back to the dreary, how do you say?—application.

Defining sex in terms of gamete size or genitals or chromosomes is like the using the never-flipped first bit in our abstract example about the world of length-20 bitstrings. It's not that people directly care about gametes or chromosomes or even genitals in most everyday situations. (You're not trying to mate with most of the people you meet in everyday situations, and sex chromosomes weren't discovered until the 20th century.) It's that that these are discrete features that are causally entangled with everything else that differs between females and males—including many correlated statistical differences of various effect sizes, and differences that are harder to articulate or measure, and differences that haven't even been discovered yet (as gametes and chromosomes hadn't respectively been discovered yet in the 16th and 19th centuries) but can be theorized to exist because sex is a very robust abstraction that you need in order to understand the design of biological creatures.

Discrete features make for better word definitions than high-dimensional statistical regularities, even if most of the everyday inferential utility of using the word comes from the high-dimensional statistical correlates. A dictionary definition is just a helpful pointer to help people pick out "the same" natural abstraction in their own world-model.

(Gamete size is a particularly good definition for the natural category of sex because the concept of anisogamy generalizes across species that have different sex determination systems and sexual anatomy. In birds, the presence or absence of a W chromosome determines whether an animal is female, in contrast to the Y chromosome's determination of maleness in mammals, and some reptiles' sex is determined by the temperature of an lain egg while it develops. And let's not get started on the cloaca.)

But because our brains are good at using sex-category words to simultaneously encode predictions about both absolute discrete differences and high-dimensional statistical regularities of various effect sizes, without our being consciously aware of the cognitive work being done, it's easy to get confused by verbal gymnastics if you don't know the theory.

I sometimes regret that so many of my attempts to talk about trans issues end up focusing on psychological sex differences. I guess I'm used to it now, but at first, this was a weird position for me to be in! (For a long time, I really didn't want to believe in psychological sex differences.) But it keeps happening because it's a natural thing to disagree about: the anatomy of pre-op trans women is not really in dispute, so the sex realist's contextual reply to "Why do you care what genitals someone might or might not have under their clothes?" often ends up appealing to some psychological dimension or another, to which the trans advocate can counterreply, "Oh, you want to define gender based on psychology, then? But then the logic of your position forces you to conclude that butch lesbians aren't women! Reductio ad absurdum!"

This is a severe misreading of the sex-realist position. No one wants to define "gender" based on psychology. Mostly, definitions aren't the kind of thing you should have preferences about: you can't coerce reality into changing by choosing different definitions! Rather, there's already a multivariate distribution of bodies and minds in the world, and good definition choices help us coordinate the concepts in different people's heads into a shared map of that territory.

One of the many distinctions people sometimes want to make when thinking about the multivariate distribution of bodies and minds in the world, is that between the sexes. But sex is by no means the only way in which people differ! In many situations, you might want to categorize or describe people in many different ways, some more or less discrete versus continuous, or high- versus low-dimensional: age or race or religion or subculture or social class or intelligence or agreeableness.

It's possible that the categories that are salient in a particular culture ought to be revised in order to fit the world better: maybe we should talk about categories like "masculine people" (including both typical men, and butch lesbians) more often! But the typical trans advocate shell game of just replacing "sex" with "gender" and letting people choose their "gender" isn't going to fly, because sex actually exists and we have a need for language to talk about it—or maybe, the fact that we have a need for language to talk about it (the fact that the information we observe admits compression) is what it means for sex to "actually" "exist".

One of the standard gender-critical complaints about trans ideology is that it's sexist on account of basing its categories on regressive sex stereotypes. On the categories-as-compression view, we can see that this complaint has something to it: if you remove the discrete, hard-line differences like genitals and chromosomes from your definitions of female and male, there's nothing left for the words to attach to but mere statistical tendencies—that is, stereotypes.

Conversely, another classic gender-critical trope is that sex is just about genitals and chromosomes and gamete size. Any "thicker" concept of what it means to be a woman or man is sexist nonsense. With some trepidation, I also don't think that one's going to fly. It's hard to see why most gender-critical feminists would care so much about maintaining single-sex spaces, if sex were strictly a matter of genitals or (especially) chromosomes or gamete size; it would seem that they too want mere statistical tendencies to be part of the concept.

This is somewhat ideologically inconvenient for antisexists like I used to be, insofar as it entails biting the bullet on masculine women and feminine men being in some sense less "real" women and men, respectively. Are our very concepts not then reinforcing an oppressive caste system?

I don't think the situation is quite that bad, as long as the map–territory relationship stays mostly one-directional: the map describing the territory, rather than the territory being bulldozed to suit the map—outliers needing a slightly longer message length to describe, rather than being shot. In my antisexist youth, I don't think I would have wanted to concede even that much, but I couldn't then have explained how that would work mathematically—and I still can't. Let me know if you figure it out.