# Where to Draw the Boundaries?

**Followup to:** [Where to Draw the Boundary?](https://www.lesswrong.com/posts/d5NyJ2Lf6N22AD9PB/where-to-draw-the-boundary)

_Figuring where to cut reality in order to carve along the joints—figuring which things are similar to each other, which things are clustered together:_ this _is the problem worthy of a rationalist. It is what people_ should _be trying to do, when they set out in search of the floating essence of a word._

_Once upon a time it was thought that the word "fish" included dolphins ..._

The one comes to you and says:

> The list: `{salmon, guppies, sharks, dolphins, trout}` is just a list—you can't say that a list is _wrong_. You draw category boundaries in specific ways to capture tradeoffs you care about: sailors in the ancient world wanted a word to describe the swimming finned creatures that they saw in the sea, which included salmon, guppies, sharks—and dolphins. That grouping may not be the one favored by modern evolutionary biologists, but an alternative categorization system is not an error, and borders are not objectively true or false. You're not standing in defense of truth if you insist on a word, brought explicitly into question, being used with some particular meaning. So my definition of _fish_ cannot possibly be 'wrong,' as you claim. I can define a word any way I want—in accordance with my values!

So, there is a legitimate complaint here. It's true that sailors in the ancient world had a legitimate reason to want a word in their language whose [extension](https://www.lesswrong.com/posts/HsznWM9A7NiuGsp28/extensions-and-intensions) was `{salmon, guppies, sharks, dolphins, ...}`. (And modern scholars writing a translation for present-day English speakers might even translate that word as _fish_, because [most](https://www.lesswrong.com/posts/4mEsPHqcbRWxnaE5b/typicality-and-asymmetrical-similarity) members of that category are what we would call fish.) It indeed would not necessarily be helping the sailors to tell them that they need to exclude dolphins from the extension of that word, and instead include dolphins in the extension of their word for `{monkeys, squirrels, horses ...}`. Likewise, most modern biologists have little use for a word that groups dolphins and guppies together.

When rationalists say that definitions can be wrong, we don't mean that there's a _unique_ category boundary that is the True floating essence of a word, and that all other possible boundaries are wrong. We mean that in order for a proposed category boundary to _not_ be wrong, it needs to capture some statistical structure in reality, even if [reality is surprisingly detailed](http://johnsalvatier.org/blog/2017/reality-has-a-surprising-amount-of-detail) and there can be _more than one_ such structure.

The reason that the sailor's concept of _water-dwelling animals_ isn't necessarily wrong (at least within a particular domain of application) is because dolphins and fish actually _do_ have things in common [due to convergent evolution](http://www.brooklyn.cuny.edu/bc/ahp/LAD/C21/C21_Convergent.html), despite their differing ancestries. If we've been told that "dolphins" are water-dwellers, we can _correctly_ [predict](https://www.lesswrong.com/posts/a7n8GdKiAZRX86T5A/making-beliefs-pay-rent-in-anticipated-experiences) that they're likely to have fins and a hydrodynamic shape, even if we've never seen a dolphin ourselves. On the other hand, if we predict that dolphins probably lay eggs because 97% of known fish species are oviparous, we'd get the _wrong answer_.

A standard technique for understanding why some objects belong in the same "category" is to [(pretend that we can)](http://zackmdavis.net/blog/2013/05/dimensionality/) visualize objects as [existing in a very-high-dimensional configuration space](https://www.lesswrong.com/posts/WBw8dDkAWohFjWQSk/the-cluster-structure-of-thingspace), but this "Thingspace" isn't particularly well-defined: we want to map every property of an object to a dimension in our abstract space, but it's not clear how one would enumerate all possible "properties." But this isn't a major concern: we can form a space with _whatever_ properties or variables we happen to be interested in. Different choices of properties correspond to different [cross sections](https://en.wikipedia.org/wiki/Cross_section_(geometry)) of the grander Thingspace. Excluding properties from a collection would result in a "thinner", lower-dimensional [subspace](https://en.wikipedia.org/wiki/Linear_subspace) of the space defined by the original collection of properties, which would in turn be a subspace of grander Thingspace, just as a line is a subspace of a plane, and a plane is a subspace of three-dimensional space.

Concerning dolphins: there would be a cluster of water-dwelling animals in the subspace of dimensions that water-dwelling animals are similar on, and a cluster of mammals in the subspace of dimensions that mammals are similar on, and dolphins would belong to _both_ of them, just as the vector [1.1, 2.1, 9.1, 10.2] in the four-dimensional vector space ℝ⁴ is simultaneously close to [1, 2, 2, 1] in the subspace [spanned](https://en.wikipedia.org/wiki/Linear_span) by _x₁_ and _x₂_, _and_ close to [8, 9, 9, 10] in the subspace spanned by _x₃_ and _x₄_.

Humans are already functioning intelligences (well, sort of), so the categories that humans propose of their own accord won't be _maximally_ wrong: no one would try to propose a word for "configurations of matter that match any of these 29,122 five-megabyte descriptions but have no other particular properties in common." (Indeed, because we are [not-superexponentially-vast](https://www.lesswrong.com/posts/82eMd5KLiJ5Z6rTrr/superexponential-conceptspace-and-simple-words) minds that evolved to function in a simple, ordered universe, it actually takes some ingenuity to construct a category _that_ wrong.)

This leaves aspiring instructors of rationality in something of a predicament: in order to _teach_ people how categories can be more or (ahem) [less wrong](https://tvtropes.org/pmwiki/pmwiki.php/Main/TitleDrop), you need some sort of illustrative example, but since the most natural illustrative examples won't be _maximally_ wrong, some people might fail to appreciate the lesson, leaving one of your students to fill in the gap in your lecture series eleven years later.

The _pedagogical_ function of telling people to ["stop playing nitwit games and admit that dolphins don't belong on the fish list"](https://www.lesswrong.com/posts/d5NyJ2Lf6N22AD9PB/where-to-draw-the-boundary) is to point out that, without _denying_ the obvious similarities that motivated the initial categorization `{salmon, guppies, sharks, dolphins, trout, ...}`, there is _more structure_ in the world: to maximize the [(logarithm of the)](http://yudkowsky.net/rational/technical/) probability your world-model assigns to your observations of dolphins, you need to take into consideration the many aspects of reality in which the grouping `{monkeys, squirrels, dolphins, horses ...}` makes more sense. To the extent that relying on the initial category guess would result in a worse Bayes-score, we might say that that category is "wrong." It might have been "good enough" for the purposes of the sailors of yore, but as humanity has learned more, as our model of Thingspace has expanded with more dimensions and more details, we can see the ways in which the original map failed to carve reality at the joints.

-------

The one replies:

> But reality doesn't come with its joints pre-labeled. Questions about how to draw category boundaries are best understood as questions about values or priorities rather than about the actual content of the actual world. I can call dolphins "fish" and go on to make just as accurate predictions about dolphins as you can. Everything we identify as a joint is only a joint because we care about it.

No. Everything we identify as a joint is a joint not "because we care about it", but because it _helps us think about_ the things we care about.

_Which_ dimensions of Thingspace you bother paying attention to might depend on your values, and the clusters returned by your brain's [similarity-detection](https://www.lesswrong.com/posts/jMTbQj9XB5ah2maup/similarity-clusters) algorithms might "split" or "collapse" according to which subspace you're looking at. But in order for your map to be _useful_ in the service of your values, it needs to reflect the statistical structure of things in the territory—which depends on the territory, not your values.

There is an _important difference_ between "not including mountains on a map because it's a political map that doesn't show any mountains" and "not including Mt. Everest on a geographic map, because my sister died trying to climb Everest and seeing it on the map would make me feel sad."

There is an _important difference_ between "identifying this pill as not being 'poison' allows me to [focus my uncertainty](https://www.lesswrong.com/posts/GJ4ZQm7crTzTM6xDW/focus-your-uncertainty) about what I'll observe after administering the pill to a human (even if [most possible minds](https://www.lesswrong.com/posts/tnWRXkcDi5Tw9rzXw/the-design-space-of-minds-in-general) have never seen a 'human' and would never waste cycles imagining administering the pill to one)" and "identifying this pill as not being 'poison', because if I publicly called it 'poison', then the manufacturer of the pill might sue me."

There is an _important difference_ between having a utility function defined over a statistical model's _performance_ against specific real-world data (even if another mind with different values would be interested in different data), and having a utility function defined over features of _the model itself_.

Remember how [appealing to the dictionary](https://www.lesswrong.com/posts/9ZooAqfh2TC9SBDvq/the-argument-from-common-usage) is irrational when the _actual_ motivation for an argument is about [whether to infer a property on the basis of category-membership](https://www.lesswrong.com/posts/4FcxgdvdQP45D6Skg/disguised-queries)? But at _least_ the dictionary has the virtue of documenting typical usage of our shared communication signals: you can at least see how "You're defecting from common usage" might _feel_ like a sensible thing to say, even if one's [true rejection](https://www.lesswrong.com/posts/TGux5Fhcd7GmTfNGC/is-that-your-true-rejection) lies elsewhere. In contrast, this motion of appealing to _personal values_ (!?!) is _so_ deranged that Yudkowsky apparently didn't even realize in 2008 that he might need to warn us against it!

You _can't_ change the categories your mind _actually_ uses and still perform as well on prediction tasks—although you can change your [_verbally reported_](https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-the-teacher-s-password) categories, much as how one can verbally report "believing" in an [invisible, inaudible, flour-permeable dragon](https://www.lesswrong.com/posts/CqyJzDZWvGhhFJ7dY/belief-in-belief) in one's garage without having any false anticipations-of-experience about the garage.

This may be easier to see with a [simple](https://www.lesswrong.com/posts/HnPEpu5eQWkbyAJCT/the-simple-math-of-everything) _numerical_ example.

Suppose we have some entities that exist in the three-dimensional vector space ℝ³. There's one cluster of entities centered at [1, 2, 3], and we call those entities Foos, and there's another cluster of entities centered at [2, 4, 6], which we call Quuxes.

The one comes and says, "Well, I'm going redefine the meaning of 'Foo' such that it also includes the things near [2, 4, 6] as well as the Foos-with-respect-to-the-old-definition, and you can't say my new definition is wrong, because if I observe [2, \_, \_] (where the underscores represent yet-unobserved variables), I'm going to categorize that entity as a Foo but still predict that the unobserved variables are 4 and 6, _so there_."

But if the one were _actually using_ the new concept of Foo _internally_ and not just _saying the words_ "categorize it as a Foo", they _wouldn't_ predict 4 and 6! They'd predict 3 and 4.5, because those are the average values of a generic Foo-with-respect-to-the-new-definition in the 2nd and 3rd coordinates (because (2+4)/2 = 6/2 = 3 and (3+6)/2 = 9/2 = 4.5). (The already-observed 2 in the first coordinate isn't average, but by [conditional independence](https://www.lesswrong.com/posts/gDWvLicHhcMfGmwaK/conditional-independence-and-naive-bayes), that only affects our prediction of the other two variables _by means_ of its effect on our "prediction" of category-membership.) The cluster-structure knowledge that "entities for which x₁≈2, also tend to have x₂≈4 and x₃≈6" needs to be represented _somewhere_ in the one's mind _in order to get the right answer_. And given that that knowledge needs to be represented, it might also be useful to have a _word_ for "the things near [2, 4, 6]" in order to efficiently share that knowledge with others.

Of course, there isn't going to be a _unique_ way to encode the knowledge into natural language: there's no reason the word/symbol "Foo" needs to represent "the stuff near [1, 2, 3]" rather than "both the stuff near [1, 2, 3] and also the stuff near [2, 4, 6]". And you might very well indeed want a [short word](https://www.lesswrong.com/posts/soQX8yXLbKy7cFvy8/entropy-and-short-codes) like "Foo" that encompasses both clusters, for example, if you want to contrast them to another cluster much farther away, or if you're mostly interested in x₁ and the difference between x₁≈1 and x₁≈2 doesn't seem large enough to notice.

But if speakers of particular language were _already_ using "Foo" to specifically talk about the stuff near [1, 2, 3], then you can't swap in a new definition of "Foo" without _changing the truth values_ of sentences involving the word "Foo." Or rather: sentences involving Foo-with-respect-to-the-old-definition [are _different_ propositions](https://www.lesswrong.com/posts/shoMpaoZypfkXv84Y/variable-question-fallacies) from sentences involving Foo-with-respect-to-the-new-definition, even if they get written down using the same symbols in the same order.

Naturally, all this becomes much more complicated as we move away from the simplest idealized examples.

For example, if the points are more evenly distributed in configuration space rather than belonging to cleanly-distinguishable clusters, then essentialist "X is a Y" cognitive algorithms perform less well, and we get [Sorites paradox](https://plato.stanford.edu/entries/sorites-paradox/)-like situations, where we know _roughly_ what we mean by a word, but are confronted with real-world (not merely hypothetical) edge cases that we're not sure how to classify.

Or it might not be obvious which dimensions of Thingspace are most relevant.

Or there might be social or psychological forces anchoring word usages on identifiable [Schelling points](https://www.lesswrong.com/posts/yJfBzcDL9fBHJfZ6P/nash-equilibria-and-schelling-points) that are easy for different people to _agree_ upon, even at the cost of some statistical "fit."

We could go on listing more such complications, where we seem to be faced with somewhat arbitrary choices about how to describe the world in language. But the fundamental thing is this: _the map is not the territory_. Arbitrariness in the map (what color should Texas be?) doesn't correspond to arbitrariness in the territory. Where the structure of human natural language doesn't fit the structure in reality—where we're not sure whether to say that a sufficiently small collection of sand "is a heap", because we don't know how to specify the positions of the individual grains of sand, or compute that the collection has a Standard Heap-ness Coefficient of 0.64—that's just a _bug_ in our human power of [vibratory telepathy](https://www.lesswrong.com/posts/SXK87NgEPszhWkvQm/mundane-magic). You can exploit the bug to confuse humans, but that doesn't change reality.

Sometimes we might _wish_ that something to belonged to a category that it doesn't (with respect to the category boundaries that we would ordinarily use), so it's tempting to avert our attention from this painful reality with [appeal-to-arbitrariness](https://www.lesswrong.com/posts/wqmmv6NraYv4Xoeyj/conversation-halters) language-lawyering, selectively applying our philosophy-of-language skills to pretend that we can define a word any way we want with no consequences. ("I'm not late!—well, okay, we agree that I arrived half an hour after the scheduled start time, but whether I was _late_ depends on how you choose to draw the category boundaries of 'late', which is subjective.")

For this reason it is said that [knowing about philosophy of language can hurt people](https://www.lesswrong.com/posts/AdYdLP2sRqPMoe8fb/knowing-about-biases-can-hurt-people). Those who know that words don't have intrinsic definitions, but _don't_ know (or have seemingly forgotten) about the [three or six dozen optimality criteria](https://www.lesswrong.com/posts/FaJaCgqBKphrDzDSj/37-ways-that-words-can-be-wrong) governing the use of words, can easily fashion themselves a Fully General Counterargument against _any_ claim of the form "X is a Y"—

> Y doesn't unambiguously refer to the thing you're trying to point at. There's no Platonic essence of Y-ness: once we know any particular fact about X we want to know, there's no question left to ask. Clearly, you don't understand how words work, therefore I don't need to consider whether there are any non-ontologically-confused reasons for someone to say "X is a Y."

[Isolated demands for rigor](https://slatestarcodex.com/2014/08/14/beware-isolated-demands-for-rigor/) are great for winning arguments against humans who aren't as philosophically sophisticated as you, but the evolved systems of perception and language by which humans process and communicate information about reality, _predate_ the Sequences. Every claim that X is a Y is an expression of [_cognitive work_](https://www.lesswrong.com/posts/QkX2bAkwG2EpGvNug/the-second-law-of-thermodynamics-and-engines-of-cognition) that cannot simply be dismissed just because most claimants doesn't know _how_ they work. Platonic essences are just the limiting case as the overlap between clusters in Thingspace goes to zero.

You should _never_ say, "The choice of word is arbitrary; therefore I can say whatever I want"—which amounts to, "The choice of category is arbitrary, therefore I can believe whatever I want." If the choice were _really_ arbitrary, you would be satisfied with the choice being _made_ arbitrarily: by flipping a coin, or calling a random number generator. (It doesn't matter which.) Whatever criterion your brain is using to decide which word or belief you want, _is_ your non-arbitrary reason.

If what you want isn't currently true in reality, maybe there's some action you could take to make it _become_ true. To search for that action, you're going to need accurate beliefs about what reality is _currently_ like. To enlist the help of others in your planning, you're going to need precise terminology to _communicate_ accurate beliefs about what reality is currently like. Even when—_especially_ when—the current reality is inconvenient.

Even when [it hurts](https://www.lesswrong.com/posts/dHQkDNMhj692ayx78/avoiding-your-belief-s-real-weak-points).

(Oh, and if you're actually trying to optimize other people's models of the world, rather than the world itself—you could just _lie_, rather than playing clever category-gerrymandering mind games. It would be a lot simpler!)

-----

[Imagine that you've had a peculiar job in a peculiar factory](https://www.lesswrong.com/posts/4FcxgdvdQP45D6Skg/disguised-queries) for a long time. After many mind-numbing years of sorting bleggs and rubes all day and enduring being trolled by Susan the Senior Sorter and her evil sense of humor, you finally work up the courage to ask Bob the Big Boss for a promotion.

"Sure," Bob says. "Starting tomorrow, you're our new Vice President of Sorting!"

"Wow, this is amazing," you say. "I don't know what to ask first! What will my new responsibilities be?"

"Oh, your responsibilities will be the same: sort bleggs and rubes every Monday through Friday from 9 _a.m._ to 5 _p.m._"

You frown. "Okay. But Vice Presidents get paid a lot, right? What will my salary be?"

"Still $9.50 hourly wages, just like now."

You grimace. "O–_kay_. But Vice Presidents get more authority, right? Will I be someone's boss?"

"No, you'll still report to Susan, just like now."

You snort. "A Vice President, reporting to a mere Senior Sorter?"

"Oh, no," says Bob. "Susan is _also_ getting promoted—to _Senior_ Vice President of Sorting!"

You lose it. "Bob, this is _bullshit_. When you said I was getting promoted to Vice President, that created a bunch of probabilistic expectations in my mind: you made me _anticipate_ getting new challenges, more money, and more authority, and then you reveal that you're just slapping an inflated title on the same old dead-end job. It's like handing me a blegg, and then saying that it's a rube that just happens to be blue, furry, and egg-shaped ... or telling me you have a dragon in your garage, except that it's an invisible, silent dragon that doesn't breathe. You may _think_ you're being kind to me asking me to believe in an unfalsifiable promotion, but when you [replace the symbol with the substance](https://www.lesswrong.com/posts/GKfPL6LQFgB49FEnv/replace-the-symbol-with-the-substance), it's actually just cruel. _Stop fucking with my head!_ ... sir."

Bob looks offended. "This promotion isn't _unfalsifiable_," he says. "It _says_, 'Vice President of Sorting' right here on the employee roster. That's an sensory experience that you can make falsifiable predictions about. I'll even get you business cards that say, 'Vice President of Sorting.' That's another falsifiable prediction. Using language in a way _you_ dislike is not lying. The propositions you claim false—about new job tasks, increased pay and authority—is not what the title is meant to convey, and this is known to everyone involved; it is not a secret."

-------

Bob _kind of_ has a point. It's tempting to argue that things like titles and names are part of the map, not the territory. Unless the name is written down. Or spoken aloud (instantiated in sound waves). Or _thought about_ (instantiated in neurons). The map is _part_ of the territory: insisting that the title isn't part of the "job" and therefore violates the maxim that meaningful beliefs must have testable consequences, doesn't quite work. Observing the title on the employee roster indeed tightly constrains your anticipated experience of the title on the business card. So, that's a non-gerrymandered, predictively useful category ... right? What is there for a rationalist to complain about?

To see the problem, we must turn to information theory.

Let's imagine that an abstract Job has four binary properties that can either be `high` or `low`—task complexity, pay, authority, and prestige of title—forming a four-dimensional Jobspace. Suppose that two-thirds of Jobs have `{complexity: low, pay: low, authority: low, title: low}` (which we'll write more briefly as [low, low, low, low]) and the remaining one-third have `{complexity: high, pay: high, authority: high, title: high}` (which we'll write as [high, high, high, high]).

Task complexity and authority are hard to perceive outside of the company, and pay is only negotiated after an offer is made, so people deciding to seek a Job can only make decisions based the Job's title: but that's fine, because in the scenario described, you can infer any of the other properties from the title with certainty. Because the properties are either _all_ low or _all_ high, the [joint entropy](https://en.wikipedia.org/wiki/Joint_entropy) of title and any other property is going to have the same value as either of the individual property entropies, namely ⅔ log₂ 3/2 + ⅓ log₂ 3 ≈ 0.918 bits.

But since H(pay) = H(title) = H(pay, title), then the [mutual information](https://www.lesswrong.com/posts/yLcuygFfMfrfK8KjF/mutual-information-and-density-in-thingspace) I(pay; title) has the same value, because I(pay; title) = H(pay) + H(title) − H(pay, title) by definition.

Then suppose a _lot_ of companies get Bob's bright idea: half of the Jobs that used to occupy the point [low, low, low, low] in Jobspace, get their title coordinate changed to high. So now one-third of the Jobs are at [low, low, low, low], another third are at [low, low, low, high], and the remaining third are at [high, high, high, high]. What happens to the mutual information I(pay; title)?

I(pay; title) = H(pay) + H(title) − H(pay, title)  
= (⅔ log 3/2 + ⅓ log 3) + (⅔ log 3/2 + ⅓ log 3) − 3(⅓ log 3)  
= 4/3 log 3/2 + 2/3 log 3 − log 3 ≈ 0.2516 bits.  

It went down! Bob and his analogues, having observed that employees and Job-seekers prefer Jobs with high-prestige titles, _thought_ they were being benevolent by making more Jobs have the desired titles. And perhaps they have helped [savvy employees who can arbitrage the gap between the new and old worlds](https://www.lesswrong.com/posts/8XDZjfThxDxLvKWiM/excerpts-from-a-larger-discussion-about-simulacra) by being able to put "Vice President" on their resumés when searching for a new Job.

But from the perspective of people who wanted to use titles as an easily-communicable correlate of the other features of a Job, all that's actually been accomplished is _making language less useful_.

-------

In view of the preceding discussion, to ["37 Ways That Words Can Be Wrong"](https://www.lesswrong.com/posts/FaJaCgqBKphrDzDSj/37-ways-that-words-can-be-wrong), we might wish to append, "38. Your definition draws a boundary around a cluster in an inappropriately 'thin' subspace of Thingspace that excludes relevant variables, resulting in [fallacies of compression](https://www.lesswrong.com/posts/y5MxoeacRKKM3KQth/fallacies-of-compression)."

Miyamoto Musashi [is quoted](http://yudkowsky.net/rational/virtues/):

> The primary thing when you take a sword in your hands is your intention to cut the enemy, whatever the means. Whenever you parry, hit, spring, strike or touch the enemy's cutting sword, you must cut the enemy in the same movement. It is essential to attain this. If you think only of hitting, springing, striking or touching the enemy, you will not be able actually to cut him.

Similarly, the primary thing when you take a word in your lips is your intention to reflect the territory, whatever the means. Whenever you categorize, label, name, define, or draw boundaries, you must cut through to the correct answer in the same movement. If you think only of categorizing, labeling, naming, defining, or drawing boundaries, you will not be able actually to reflect the territory.

Do not ask whether there's a rule of rationality saying that you shouldn't call dolphins fish. Ask whether dolphins are fish.

And if you speak overmuch of the Way you will not attain it.

_(Thanks to Alicorn, Sarah Constantin, Ben Hoffman, Zvi Mowshowitz, Jessica Taylor, and Michael Vassar for feedback.)_