Two kinds of claim

This site has published two statistics about the same fifteen claims and they point in opposite directions.

Nothing here is as old as it sounds reports that for eight of the fifteen, the date in circulation is earlier than the oldest surviving thing that attests it — by a median of two hundred and one years, with the largest single gap at nine hundred and eighty.

The name is not the date reports that the mean gap between a result in this field and the name it is known by is twenty-two years, and that it runs in one direction: the result comes first and the name comes later.

Both are correct and they have never been reconciled, because the reconciliation needs a column the record did not have. The claims are not one population. They are two, they behave differently, and a statistic over the whole record describes neither.

The column, and that it is a judgement #

The record carries a year for the popular date, a year and a source for the oldest attestation, a count of how many independent sources there are, and what kind of thing the oldest one is. Every one of those is a fact somebody can check.

The new column is not. It says what sort of thing each claim is about, in one of three words:

• technology — paper itself: made here, arrived there. Archaeology.
• practice — a thing people did. A fold, a ceremony, a toy, a lesson.
• result — a thing somebody proved, published or designed.

That is a judgement and it is written down separately for exactly that reason. It is a closed set of three, so a new entry cannot quietly invent a fourth and land outside every figure, and the classification of every row is visible beside the evidence rather than folded into a conclusion.

Fifteen rows: three technology, six practice, six result.

Every practice runs ahead. All of them. #

The six claims about things people did:

Not one runs the other way. Not most of them, not nearly all of them — every single claim in the record about a practice is dated earlier than the oldest thing that attests it, with a median of three hundred and forty-seven years and a spread of nine hundred and seventy-nine.

The Fröbel row at +1 is the one that keeps the claim from being vacuous, and it is instructive. It is a practice with a founder, a curriculum and a printed book, which is as close as a practice gets to being a publication — and it is the only practice in the record whose overrun is smaller than a lifetime.

And most results run behind #

The six claims about something proved, published or designed:

Five of the six are negative — the evidence is older than the date in circulation — with a median of −14 and a spread of 131. That is the twenty-two-year mean in the other essay, read as a distribution instead of as a mean.

The exception is the fold-and-cut theorem at +76, and it is the exception that explains the rule. Its popular date is 1922, which is when Houdini put a five-pointed star into a book of paper magic; the theorem that any straight-line drawing can be released by a single cut is of 1998. So the row is a practice date attached to a result claim, and it lands in the practice’s range because that is what it is a date of.

The spreads differ by a factor of seven #

The medians are the headline and the spreads are the more useful number.

Practices scatter over 979 years. From +1 to +980, with no clustering anywhere in between.

Results scatter over 131. From −55 to +76, and if the misfiled fold-and-cut row is set aside, from −55 to −7 — a range of forty-eight years, which is a working life.

That factor is the mechanism stated as a measurement. A result has a paper with a date on it. Somebody wrote it, somebody printed it, and the date of the printing is the date, so a claim about a result can be wrong by the length of a delay in recognition and not by more. A practice has no first day. Nobody folded the first paper crane; a habit accumulates, and the date attached to it afterwards is a guess dressed as a fact. The patterns nobody owns is the same silence read from the shelf rather than from the record.

The median over everything is one year #

The number that makes the partition necessary rather than merely tidy is the one nobody had computed.

The median gap over all fifteen claims is one year.

Not two centuries, not minus twenty-two: one. It sits almost exactly between the two populations’ medians of +347 and −14, and it is a fact about neither of them. A reader given only that number would conclude that this subject’s chronology is in excellent shape, which is the opposite of what every row of the table says.

That is what a median over two populations does. The published figure of two hundred and one years is a median over the eight positive gaps — which is not the same statistic and is chosen precisely because the whole-record median is uninformative — and the published mean of twenty-two years is over the results. Neither essay was wrong; both were quietly working around a partition nobody had written down.

Why the two behave differently #

The mechanism is not subtle and it is worth stating in one paragraph so that the numbers are not the only argument.

A result is an event with a document. Somebody proves a thing, writes it down, and the writing has a date printed on it. What can go wrong afterwards is recognition: the field can take twenty years to notice, and the name it eventually attaches belongs to whoever it noticed. So the date in circulation is the date of the noticing, the date in the record is the date of the doing, and the gap is a delay in one direction bounded by a career.

A practice is not an event. Nobody folded the first paper crane; a habit accumulates in a population, and the question “when did this start” has no answer of the kind the question presupposes. What gets recorded is the moment the practice became worth writing about — codified into a manual, printed in a book, drawn in an engraving — which is late by construction, because a practice has to be established before anybody bothers.

So the two gaps have different causes and different bounds. The result’s overrun is a delay in attention. The practice’s is a delay in literacy, and the second has no natural size.

Where the technology sits #

The three technology claims — when paper was made in China, when it reached Japan, when it was made in Europe — are the third group and they are the least interesting, which is worth saying rather than hiding.

They run −94, 0 and +110, with a median of zero and a spread of 204. So they sit between the two other groups in every respect, which is what would be expected: paper-making is a practice with a physical output, so it leaves artefacts the way a result leaves papers, and the artefacts can be dated.

Three rows is too few to say anything with, and the essay does not. What they contribute is a demonstration that the classification is not a two-way split dressed up as a three-way one: there is a middle group, it behaves like a middle group, and it does not need the story to be tidier than it is.

What the partition is not #

Three things, and the first is the standing caution of the whole field.

It is not a sample. The record has fifteen rows because fifteen claims are common enough to be worth checking, and nothing computed from it generalises beyond it. A median over six practices is a fact about those six.

It is not a claim that practices are young. The gap is between a date and its evidence, and a practice with a nine-hundred-year overrun may well be nine hundred years old — what is missing is anything that says so. The difference between this is when it happened and this is the oldest surviving evidence that it happened is the whole of this field.

And it is not gated. Every other claim on this site is re-derived on every build and a wrong one stops it. A date is checked once, by hand, against a record that is itself a survivor, and the classification added here is a judgement on top of that — which is the one thing this field can never gate. What can be checked is that the vocabulary is closed, that every row has one of the three words, and that the arithmetic over them is what it says — and those are checked on every build.

What a fourth column would have to be #

The partition suggests an obvious refinement and it is worth saying why it is not made.

The natural fourth category is a claim about an object — a specific model, a specific pattern, a specific fold — as against a practice, which is a habit, and a result, which is a statement. The pajarita is arguably that: it is one folded bird rather than a way of behaving.

Two reasons not to. The record has fifteen rows and cutting it into four groups leaves cells of two and three, which support no statement at all. And the distinction does not track the behaviour: the pajarita’s overrun is +293, which is comfortably inside the practice range and nowhere near the results’.

That is the test a taxonomy has to pass to be worth having. Three words separate the record into groups that behave differently; a fourth would separate it into groups that behave the same, and a category that predicts nothing is a category that costs a column and earns nothing.

What the partition predicts #

A classification earns its place by predicting something it was not built from, so here is the prediction and the check.

The rows were sorted into three groups by what each claim is about, using no numbers at all. If the grouping were arbitrary, the gaps within each group would scatter like the whole record’s.

They do not. The six practices occupy a contiguous band from +1 to +980 with nothing else in it, and the six results occupy a contiguous band from −55 to +76 with one intruder. The two bands overlap in exactly one row, and that row is the fold-and-cut theorem, whose popular date is a magician’s performance rather than a publication.

So the partition made from the claims’ subject matter reproduces the partition a statistician would have drawn from the numbers, with one identifiable exception whose exceptionality has an explanation that does not appeal to the numbers. That is as close to a check as a column of judgements can get.

How unlikely the separation is #

Calling the previous section as close to a check as a column of judgements can get invites a number, and there is an exact one available. It assumes no distribution and fits no model, because the only thing it reads is rank.

Set the three technology rows aside and take the twelve claims that are either a practice or a result. Sorted by gap, largest first, the six practices take the first five places and the seventh; the six results take the sixth place and the last five. Count the pairs that are out of order — a result whose gap is larger than a practice’s — and there is exactly one, the fold-and-cut theorem’s +76 standing above Fröbel’s +1.

Now ask what that would look like by chance. Had the three-word column been assigned by coin rather than by reading the claims, the six practices would be a random six of the twelve, and there are $\binom{12}{6} = 924$ ways to choose them. Exactly one of those puts the practices in the top six with no pair out of order. Exactly one more leaves a single pair out of order. So a partition at least this clean arises by chance twice in nine hundred and twenty-four attempts — about one time in four hundred and sixty.

That is not a large sample enjoying a small probability. Twelve rows are twelve rows, and a rank test is the only thing twelve rows will support. What it establishes is narrow and worth having: the separation is not something a reader would have got by sorting these claims into any three groups they liked. It is a property of these particular three words, and it was available to be computed the moment the column existed.

The test that must not be run #

There is an obvious way to make that number better, and it is the reason to state the rule before the temptation.

The single out-of-order pair is the fold-and-cut row, and this essay has already argued — on grounds that have nothing to do with the size of any gap — that its popular date belongs to a practice. Move it into the practices and the separation becomes perfect: one arrangement in nine hundred and twenty-four rather than two.

That number must not be quoted, and the reason is not modesty. The row would have been reclassified because it was the exception, and a taxonomy adjusted until its own test passes has stopped being a taxonomy and become a curve fitted to six points. The record is kept under a rule that settles this in advance: the classification is written from what the claim is about, before anybody looks at the gap, and it stands whatever the gap turns out to be. So the fold-and-cut theorem is a result, it is filed as a result, and the inversion it causes is left in the arithmetic where a reader can see it.

What the argument about that row does earn is a smaller and different claim. The exception has an explanation that was not reverse-engineered from the numbers: a magician’s performance is not a publication, so 1922 is not a date of the kind the results column is made of. That statement would have been just as true had the gap come out the other way, which is the test of whether an explanation is doing work or decorating a result.

What a sixteenth row would do #

The record gains a row whenever a claim turns out to be repeated often enough to be worth checking, and it is worth knowing in advance what that does to all of this.

A new practice with a long overrun changes nothing; it lands among the others and the rank test barely moves. A new result attested a decade before its popular date, likewise. The informative row is the one that lands in the wrong band — a practice attested within a few years of the date in circulation, or a result running four centuries ahead of anything that shows it — and the honest thing to do with such a row is to record it, classify it by its subject matter as before, and let it move the numbers. Not to look for a reason it is special.

The partition survives one exception that has an argument attached to it. It would not survive three, and it should not: three would mean the three words are not tracking what the gaps do, and the column would have earned its removal in the same way it earned its place.

What to do with a date #

The practical reading is short and it is the reason to bother.

When a date attaches to a practice, treat it as a lower bound on the evidence and nothing else. The record says the practice is at least as old as the oldest attestation and gives no upper bound at all; the number in circulation is a guess whose median error in this record is three and a half centuries.

When a date attaches to a result, look for the paper. The overrun is small, it usually runs the other way, and the thing being dated is nearly always the name rather than the work — which is the failure mode the eponym essay is entirely about.

And when a date attaches to a result by way of an anecdote, it is a practice date. The fold-and-cut row is the whole demonstration: a magician’s trick in 1922 and a theorem in 1998, quoted as one number.