The Problem With Statements of the Form “X is Y”

Cats are mammals.

Gravity is the curvature of spacetime.

Both of these sentences are true. And they both use conjugations of the verb “to be”. But the usage of that verb is very different.

The first sentence is true in the sense that we as humans have defined a category called “mammals”, members of which have certain characteristics, and cats have the required characteristics to put them into that category.

The second sentence is true in the sense that “the curvature of spacetime” describes a thing that exists in the universe, to which we have given the name “gravity”. It’s a literal definition, not a taxonomical categorization.

The problem with these two different uses of “to be” is that people mistake the one for the other. They think that when scientists reclassified Pluto as a dwarf planet, it was because they’d learned something new about Pluto, as opposed to having simply revised the definition of a “planet”. But this is incorrect. A planet is whatever humans say a planet is, because the entire purpose of the category is to allow us to talk about “planets” without having to say “celestial bodies that have assumed a roughly spherical shape, orbit a star, and have cleared the space around their orbits” every time.

This isn’t to say that categories are meaningless, or can be defined in any way we please. Because a category is a convenient way of referring to a set of traits, those traits have to be common enough as a complete set to be worth referring to. We shouldn’t define a category to mean “a person with black hair and green eyes”, because those two traits aren’t any more likely to occur together than any other hair/eye color combination, and that set of traits doesn’t imply anything else useful about the person.

It is to say, however, that a category is not a fact about the universe, and that a category can and should be changed to better suit our purposes. Take, for example, the categories “male” and “female”. Most of us are accustomed to using these categories to imply a common combination of gender presentation, chromosomes, and sex characteristics. However, a lot of people have recently been changing these categories. In doing so, they aren’t violating any universal law that says “all masculine-presenting humans must have XY chromosomes”, because there is no such universal law. They are redefining a category that was defined in the first place by English-speaking humans and can be redefined by those same humans.

(Another recent change to categories: birds are reptiles.)

This same problem occurs the other way around: people mistake observations about the universe for human-made categories. I learned to say the phrase “gravity is the curvature of spacetime” when I was 9. It successfully impressed a lot of grownups, but the image in my head was a stretched-out tarp with a ball in the middle. It took nearly a decade for me to actually understand enough physics and math to grasp its real meaning. And when I did, my thought was basically “oh shit, gravity literally is the curvature of spacetime”.

When we talk about facts we understand, we’re not saying riddles or passwords. If somebody who doesn’t have the relevant background knowledge to understand us overhears, they might not realize that what we say is a literal observable fact, which they too can observe if they know where to look, but that doesn’t change it. The unfortunate reality is simply that the conversion from thought to language is not lossless. And until we invent telepathy, it probably never will be. Still, simply knowing the distinction between these two definitions of “X is Y” type sentences was helpful to me, and so I’m passing it on.

Language: A Cluster Analysis of Reality

Cluster analysis is the process of quantitatively grouping data in such a way that observations in the same group are more similar to each other than to those in other groups. This image should clear it up.

Whenever you do a cluster analysis, you do it on a specific set of variables: for example, I could cluster a set of customers against the two variables of satisfaction and brand loyalty. In that analysis, I might identify four clusters: (loyalty:high, satisfaction:low), (loyalty:low, satisfaction:low), (loyalty:high, satisfaction:high), and (loyalty:low, satisfaction:high). I might then label these four clusters to identify their characteristics for easy reference: “supporters”, “alienated”, “fans” and “roamers”, respectively.

What does that have to do with language?

Let’s take a word, “human”. If I define “human” as “featherless biped”, I’m effectively doing three things. One, I’m clustering an n-dimensional “reality-space”, which contains all the things in the universe graphed according to their properties, against the two variables ‘feathered’ and ‘bipedal’. Two, I’m pointing to the cluster of things which are (feathered:false, bipedal:true). Three, I’m labeling that cluster “human”.

This, the Aristotelian definition of “human”, isn’t very specific. It’s only clustering reality-space on two variables, so it ends up including some things that shouldn’t actually belong in the cluster, like apes and plucked chickens. Still, it’s good enough for most practical purposes, and assuming there aren’t any apes or plucked chickens around, it’ll help you to identify humans as separate from other things, like houses, vases, sandwiches, cats, colors, and mathematical theorems.

If we wanted to be more specific with our “human” definition, we could add a few more dimensions to our cluster analysis—add a few more attributes to our definition—and remove those outliers. For example, we might define “human” as “featherless bipedal mammals with red blood and 23 pairs of chromosomes, who reproduce sexually and use syntactical combinatorial language”. Now, we’re clustering reality-space against seven dimensions, instead of just two, and we get a more accurate analysis.

Despite this, we really can’t create a complete list of all the things that most real categories have in common. Our generalizations are leaky in some way, around the edges: our analyses aren’t perfect. (This is absolutely the case with every other cluster analysis, too.) There are always observations at the edges that might be in any number of clusters. Take a look at the graph above in this post. Those blue points at the top left edge, should they really be blue, or red or green instead? Are there really three clusters, or would it be more useful to say there are two, or four, or seven?

We make these decisions when we define words, too. Deciding which cluster to place an observation happens all the time with colors: is it red or orange, blue or green? Splitting one cluster into many happens when we need to split a word in order to convey more specific meaning: for example, “person” trisects into “human”, “alien”, and “AI”. Maybe you could split the “person” cluster even further than that. On the other end, you combine two categories into one when sub-cluster distinctions don’t matter for a certain purpose. The base-level category “table” substitutes more specific terms like “dining table” and “kotatsu” when the specifics don’t matter.

You can do a cluster analysis objectively wrong. There is math, and if the math says you’re wrong, you’re wrong. If your WCSS is so high that you have a cluster that you can’t label more distinctly than “everything else”, or if it’s so low you’ve segregated your clusters beyond the point of usefulness, then you’ve done it wrong.

Many people think “you can define a word any way you like”, but this doesn’t make sense. Words are cluster analyses of reality-space, and if cluster analyses can be wrong, words can also be wrong.

This post is a summary of / is based on Eliezer Yudkowsky’s essay sequence, “A Human’s Guide to Words“.