What Is a Category

There is a drawer, and things go in it. That is the whole of a category as most people carry it, and it is not a foolish picture; it is the picture that built the museum, the library and the census. A thing has properties; the properties settle which drawer it belongs in; a good taxonomy has a drawer for each kind and no thing left over on the bench. When two people argue about whether a tomato is a fruit, or whether Pluto is a planet, or whether a hot dog is a sandwich, they argue as though the drawer had a bottom and the thing were either resting on it or not, and as though a fact about the thing would settle the matter if only it could be found. I held this picture for most of my life. It does a great deal of quiet work — in how people file, in how they diagnose, in how they decide what someone is. It is also, I will argue, wrong about what a category is, and wrong in a way that can be shown from three ordinary cases before any foundation is consulted.

What the ordinary picture commits you to

State it strongly, because the weak form is not what anybody believes.

The picture makes three commitments. First, that a category is a container — it has an inside and an outside, and a thing is in it or not, so that two categories which exclude each other cannot both hold one thing. Second, that membership is about the thing: the thing’s own properties decide the matter, so that if the membership changes, something about the thing must have changed first. Third, that a taxonomy can be complete — that there is, at least in principle, a full list of the drawers a given thing belongs in, and a good enough classifier could write it out.

Three ordinary situations do not fit inside those commitments, and each one breaks a different commitment.

What the foundation says instead

I work from a small set of numbered primitives and results derived from them. The one doing the work today is the first primitive of the shared core — the circle itself — and it is worth a piece because it is the primitive the ordinary picture of a category has not been checked against, and because three earlier pieces in this series each ran it from a different side and left the side a category needs untouched.

S1: the unit circle. When the unconditioned is conditioned, a circle appears. Every point on the circumference is a condition. Every named condition is an idea. The circumference is infinite and no enumeration closes it.

Two more primitives fix what those words mean. P6: a condition is a prerequisite in order for something to happen or exist; it does not cause, does not make possible, and does not happen. P7: an idea is a named condition, and naming does not create it — naming makes the relationship conscious.

Now ask what a category name is. Fruit, planet, furniture, sandwich. Each is a name, and what it names is what has to hold for a thing to count as one: the seed-bearing product of a flower, a body that has cleared its orbit, an object made for the use of a room. A prerequisite in order for something to exist as that. That is a condition in the sense P6 locks, and the name of a condition is an idea. So a category is an idea, and an idea is a point on the circumference.

Be literal about a point. A point has no extent. It has no interior, no edge, no inside for anything to be in. It is a position, and what a position gives you is a direction from wherever you stand.

A category is not a drawer. It is a bearing. Nothing is in a bearing; things stand at various distances from it.

The chair is settled here, before either of the other constructions, and by doctrine alone. The circumference is infinite and no enumeration closes it; the conditions a chair can be read against are points on it; so the list of what a chair is does not close, and the classifier who could not finish it was not short of skill. The locked distinction is between the named and the nameable: named conditions at any moment are finite, nameable ones are not. A taxonomy is a finite naming. The chair on the bench with no drawer to go in is not a fault in the drawers. It is what a finite naming looks like from the edge, and it was going to look like that whatever the drawers were.

The mark, not the thing

Pluto needs the second construction, and it is the one that relocates the whole subject.

The foundation separates an act from an artifact. An act is a process and carries no operator; an artifact is a mark, and only a mark carries a value. Sorting is an act. What the act leaves is a mark — a label on a jar, a line in a catalogue, a resolution passed in a hall, the sentence Pluto is a planet printed in a textbook in 1931. That mark is the artifact, and an artifact in this foundation carries a phase, which is the condition it serves, and a magnitude. The thing sorted is not the mark. Pluto is a body in the outer system that varies, and by P3 that is what its existing consists in; it was varying on the Monday and on the Friday, and nothing about a vote in Prague was part of that variation.

Now Prague. R2: write-protection — records update expectation forward and never re-open. The mark Pluto is a planet, made in 1930 and repeated for seventy-six years, is in the record and stays in it. What happened in 2006 was not an edit. It was a new mark, made beside the old one: a named condition — has cleared the neighbourhood of its orbit — and a reading of Pluto against it that came out short. The older mark was not removed; it was demoted to evidence about what astronomers were reading against in 1930, which is what it was from the start. Recategorisation is addition. There is no subtraction in the record and there has not been one, and Pluto’s seventy-six years stand as settled as its moons.

One more thing about the room in Prague, because it is where the ordinary picture hides its last mistake. The clause about clearing an orbit was not made in 2006. P8: a condition is prior to what it conditions, and where a person’s naming is itself conditioned by that condition, the person is not its origin — ideas have people, people do not have ideas. The astronomers named a prerequisite that bodies had been meeting or failing to meet since there were orbits. They did not build it; it became readable through them, and the resolution is where that happened. R12 keeps them accountable: ownership is a category error and the manifestation is accountable, so the mark is theirs to answer for, and the condition is not theirs to own. That is why no one can be asked to un-invent a category. There was no invention; there was a naming and a mark.

Figure 01

Three objects where ordinary speech has two

• The thing Pluto Varies, and so exists. Was not touched in 2006. Is not in anything and is not out of anything.
• The point “Planet” A named condition on the circumference. No interior, no edge. Prior to its naming; owned by no one.
• The mark “Pluto is a planet” An artifact: a reading of the thing against the point, by someone, at a time. Add-only. 1930 stands; 2006 stands beside it.

Where the edge is

The tomato needs the third construction, and so does the sandwich.

A point has no edge, so if a category has one, the edge is somewhere else. Watch what a reader does when sorting. The reader has a thing in front of them and a condition in mind, and they read the thing against the condition. P4: R = A / E — an actual against an expectation, the expectation being the reader’s. R8: the reader has only the quotient. So what the reader has, sorting a tomato, is a reading of how far this tomato falls from what they expect a fruit to be, and the mark gets made or withheld at some distance. That distance is a tolerance, and the tolerance is in E. It is host-side. It is not on the circle and not in the tomato.

And there is a second thing the reader does, usually without noticing. A word is not a point. Fruit is a word, and it sits on at least two points — the botanist’s prerequisite, which concerns seeds and flowers, and the cook’s, which concerns where in the meal a thing is served. Those are two conditions on the circumference, two ideas, and one word between them. The court in 1893 did not find a tomato in two drawers. It found one word on two points and said which of the two the word in the tariff act stood on. The foundation I work from keeps a table of words that carry two senses and requires that they be told apart before use; a piece about categories is where that table earns its keep.

R10 sits under all of this and should be said out loud. Every artifact is a miss. A classification is an artifact, so a classification is a miss — a reading against a condition that does not exactly meet it, because the reading was made by a host with a denominator and the condition was not. That is why the borderline case is the ordinary case and not a scandal for the taxonomy. There is no classification that is not a borderline case at some magnification, and a taxonomy said to have no edge cases is a report of its tolerance, not of its objects.

Categories of people

The three constructions were built on tomatoes and a dwarf planet, and they would be a parlour trick if they stopped there. They do not stop there. The most consequential categories anyone is sorted into are categories of people — a leader, lazy, difficult, gifted, not a team player, the words on a school report or in the paragraph a manager writes about someone in a room they are not in. Run the constructions on those without changing anything.

The label is a mark. It was made by a reader, against a point, at a tolerance that was theirs. It is an artifact, and by R12 the manifestation is accountable: the person who wrote difficult is answerable for having written it, and the condition the word names is not theirs to own. The person written about is the thing in the construction — varying, by P3, on the Monday and on the Friday — and was not touched by the writing of the word. That is a hard fact and a useful one. It means nobody was made lazy by being called lazy; a mark was added to a record, and the record now reads differently to whoever reads it next, which is a fact and is also not the same thing.

And R2 holds here with its full weight. The mark stands. Nobody is recategorised; a new mark can be made beside the old one, by the same reader or a different one, and the old one becomes evidence about its own moment. That is the whole of what a change of reputation is. It is not a correction. It is addition, and the old word is still on the shelf.

What it costs

This does not make the arguments go away, and I want to say what it takes before anything comfortable can be read into it.

The first cost is that there is no really. Whether Pluto is really a planet, whether a tomato is really a fruit, whether the thing on the plate is really a sandwich — the question people want answered is a question about the thing, and the constructions say there is no such question. There is a point, a thing, and a mark; a mark can be read for fidelity against the point it was made against, and R11 says fidelity is the evaluative question — not success, not winning the argument. That is all really can mean here. A mark made against the cook’s point is not in disagreement with a mark made against the botanist’s, and no fact about tomatoes makes it so. Most category arguments are not unresolved. They are not arguments.

The second cost is heavier, and the people-section already carried it: you cannot leave a category. The mark stands, and the only move anyone has is another mark. The comfort the ordinary picture offered was that a wrong classification could be corrected — the thing taken out of the wrong drawer and put in the right one, the file amended, the record set straight. The foundation does not have that move. Setting the record straight is adding to the record.

The third cost is the one that belongs to the foundation and not to me. Everything above treats fruit and planet as names of conditions that were there to be named — prior, by P8, to the naming. That reads like a claim that the world has joints and the taxonomies find them. The foundation does not make that claim and I have not made it. Whether a condition is an abstract object, and what the circumference is when no host reads it, is recorded as an open problem, and a piece about categories walks straight up to it and has to stop. I have said that a category is a point, that the point is prior to its naming, and that the mark is the reader’s. I have not said what the point is made of, because nothing I work from says.

Where this is probably wrong

Three places, and one of them is an open problem the foundation records and I may not close.

The first is the first construction, and it is the one the piece rests on. I have read a category name as the name of a prerequisite, so that fruit names what has to hold for a thing to exist as one. There is an older and well-argued account on which a category is not a condition at all but a family resemblance among instances with no single thing they share — Wittgenstein on games, marked as report and not grounds. If that account is right for the ordinary word, then a category is not a point, the circle is the wrong instrument, and this piece has explained a word by moving it onto a primitive it does not sit on. What would settle it is whether a category can be stated as a prerequisite at all; I think planet was, in Prague, and game may not have been, and the construction may hold for some categories and not for others, which is worse for the piece than its being simply wrong.

The second is the boundary between things and marks. The foundation’s vocabulary is built for artifacts — marks made in relation to an exemplar — and I have kept the tomato and the chair and Pluto outside that vocabulary, as things that vary and are read, not as marks. That was a choice. Whether a tomato is itself a mark of anything, in the sense the foundation means, is a question about what the foundation counts as an artifact, and it is not one I have settled or may settle. If the answer is that it is, the second construction has two objects and not three, and the tidy figure above loses a column.

The third is the open problem. Are conditions abstract objects? The system has no published answer, and so are categories real or made up? is the first thing anyone asks on hearing that a category is a point. Closing it would take a statement, in the primitive vocabulary and with the figures removed, of what a condition is when no host is reading against it — whether the circumference is there unnamed or whether condition is defined only relative to a reading. This piece does not attempt that. It says the point is prior to the naming, which is doctrine, and stops there, which is the boundary.

And R14: an idea is oriented and prejudiced in one act, this one included, and so is a category. To sort the world under a name is to be pointed at that condition and away from its antipode by the same act; a taxonomy flatters its taxonomist. The argument above is kind to the person who has been sorted and would like the sorting not to be about them, and unkind to the person who did careful work to build a taxonomy and is told it is a finite naming with a tolerance in it. It relocates the drawer into the reader’s denominator, which suits a certain temperament. Worth noticing before deciding it is merely correct.

On a Tuesday

1. Find the point before you argue. When someone says a thing is not a such-and-such and you think it is, ask which condition the word is naming for each of you. Half the time it is two, and then there is no argument, only two marks against two points sharing a word.
2. Then find the tolerance, and whose it is. If it is one point, the disagreement is about how far a thing may fall from it and still get the mark. That is a fact about the readers and it can be said out loud, which ends more arguments than facts about the thing do.
3. Treat the misfit as the list being short. When something will not go in any drawer, the drawers are a finite naming and the circumference is not. The thing on the bench is not broken and neither is the taxonomy. Add a name, or leave it on the bench and say so.
4. Remember which of the three you are. When you are sorted, you are the thing, and the word is a mark someone made at their tolerance. When you sort, the mark is yours and you answer for it, and the person you wrote it about was not touched by the writing. Both of those are worth holding at once.