When two of them are first attested together

Two traditions and a merge separates three lineages that are usually told as one: ceremonial wrapping, recreational folding and the kindergarten syllabus, with three purposes, three social locations and three earliest sources. Told as one continuous tradition, the oldest date in any of them becomes the age of all three.

Every one of those dates is contested. The record this collection keeps carries, for each claim, the year it is popularly given and the year of the oldest surviving thing that attests it, and the gaps are enormous — nine hundred and eighty years on recreational folding in Japan, eight hundred and ninety-seven on the thousand cranes.

That is the field’s difficulty and it is usually where the argument stops. There is a quantity in the same record that is not contested at all, and it is the one the merge story is actually about.

A joint date is a maximum #

The construction is trivial and that is the point.

Two claims are both attested from the later of their two earliest sources. If ceremonial wrapping is attested from 1600 and recreational folding from 1680, then the earliest year at which the record can put both of them in the world is 1680.

A joint date is a maximum of two source years, and a maximum inherits the better-attested member. So a joint claim about a well-attested lineage and a badly-attested one is as good as the better of the two, not as bad as the worse — which is the opposite of how error usually compounds and is why the quantity is worth having.

It also has nothing in it that anybody disputes. The individual dates are argued about because the argument is over how much older a practice is than its earliest record; a joint date is a statement about records and not about practices, so the argument does not reach it.

The numbers #

Take the five claims in this record that are about practice — ceremonial wrapping, recreational folding, the thousand cranes, the pajarita and the kindergarten syllabus.

Their popular dates run from 700 to 1838, a spread of over a millennium. Their earliest surviving sources run from 1600 to 1838, a spread of two hundred and thirty-eight years.

All five are jointly attested from 1838, which is the kindergarten syllabus’s own source and the latest of the five. And the year at which the same five claims are popularly dated together is 1837, one year earlier.

So the joint date is contested by a single year, against nine hundred and eighty for the worst individual one. The record dates the moment at which all five lineages could first have met to within a year, and dates none of their beginnings to within a century.

Reading the pairs rather than the whole set gives the intermediate structure. Wrapping and recreational folding are jointly attested from 1680; add the pajarita and it is 1793; add the cranes and it is 1797; add the syllabus and it is 1838. The record’s own account of when these things were in the world together is a short sequence of dates from the seventeenth century to the nineteenth, and every one of them is a printed book or a surviving picture rather than an inference.

What this does to the merge story #

The merge is the event the rung below this one is about: the lineages were independent until the late nineteenth century, and they met when kindergartens reached Japan in the 1870s. That essay’s own closing note is that the traffic ran in both directions and that the better-documented flow went from a German syllabus into Japanese schools.

The joint dates say something the individual ones cannot. The merge is the best-dated event in the field, not because more was written about it but because a joint date is a maximum and the syllabus is the best-attested of the five claims.

That inverts the usual difficulty. A field whose beginnings are all obscure has, in its own record, one event that is dated tightly — and it is the event that made the subject possible, since the mathematics this collection is about descends from the third lineage rather than the first two.

It also gives the record’s shape a use. Nothing here dates the merge to the 1870s; the record has no entry for it. What the arithmetic supplies is a bracket: all five lineages are jointly on the record from 1838, so any claim about them interacting before then is a claim about at most four of them, and the record can say which four.

What the bracket forbids #

A bracket is only interesting if it rules something out, and this one rules out a specific and common form of statement.

The story that runs “paper folding began in Japan and spread outward” — which nothing here is as old as it sounds takes apart one row at a time — is a claim about at least two lineages at one time. Before 1680 the record attests exactly one of the five — ceremonial wrapping, from 1600 — so any statement about two of them before 1680 is a statement the record cannot support at all, whatever is true.

Before 1793 it attests two. Before 1797, three. Before 1838, four.

So there is a computable answer to “how many of these lineages can the record put in the world at a given year”, and it is a step function with four steps in it, all of them in the last four centuries. That is a much sharper instrument than the individual gaps, and it is derived from the same fifteen rows.

Its value is that it converts an argument about how old a practice is into an arithmetic about what survives, which is the only kind of question this field can settle.

Adding a sixth claim moves nothing #

The construction has a property that is worth testing rather than asserting: a joint date over a set is decided by the set’s latest source, so adding a claim to the set changes the answer only if the new claim is later than everything already there.

Add the arrival of paper in Japan to the five. Its popular date is 610 and its earliest surviving source is the Nihon Shoki at 720 — the oldest source in the whole record, and a chronicle compiled a century after the event it describes.

The joint date of all six is still 1838. Every pairwise date involving the new claim is decided by the other member, because 720 is earlier than every one of them.

A claim with an ancient source contributes nothing to a joint date, which is a strange-sounding sentence and is exactly right. The bracket is a statement about when the last of a set arrives on the record, so the oldest members are irrelevant to it and the newest one decides everything.

That is the property that makes the construction robust and it is also what concentrates all its risk in one place. Every joint date in this set is the kindergarten syllabus’s date, and every claim about how firm the bracket is, is a claim about that one entry.

Why the individual gaps are so large and this is not #

Two of the five claims run enormously ahead of their evidence — nine hundred and eighty years and eight hundred and ninety-seven — and it is worth saying why, because the reason is what makes the joint construction work.

A practice leaves a record when somebody has a reason to write it down. Recreational folding in Japan is attested from a poem of 1680 mentioning folded butterflies — the largest gap in the whole record — and a poem mentions a thing in passing because the thing is already ordinary. So the source is late by however long it took the practice to become worth mentioning in passing, and that interval is unknowable.

The kindergarten syllabus is different in kind. It was published as a curriculum, deliberately, by somebody who intended it to be taken up — so the record of it is contemporary with the thing itself, and the one-year gap is a rounding rather than a dispute.

The record is good exactly where somebody meant to make a record. A practice that grew has a late source and a practice that was promulgated has a contemporary one, and this field’s largest gaps are all in the first category.

Which explains the arithmetic’s shape. The joint date inherits the latest member, the latest member is the promulgated one, and the promulgated one is the well-attested one. The construction is firm because the thing that arrived last is the thing somebody wrote down on purpose — and that is a coincidence of this particular set rather than a property of the method.

What kind of source each one is #

The record grades its sources as well as dating them, and the grades change how the bracket should be read even though they do not change the arithmetic.

The closed set runs, in descending order of what a source can bear: an artefact — a surviving folded object or a contemporary picture of one; a manuscript; a printed book with a publication date; a secondary account with no surviving primary behind it; and an inference, which is not attested at all.

Among the five practice claims, ceremonial wrapping rests on manuscripts, the pajarita on an artefact, and recreational folding, the cranes and the syllabus on printed books. None of the five is a secondary account or an inference, which is worth saying because several of the record’s technology claims are.

So the bracket is built from sources at the good end of the scale, and its latest member — the one that decides everything — is a printed book of known date. That is about as firm as this field’s evidence gets.

It also gives the bracket a second reading. A joint date is the year from which the record can support a statement about a set of claims; a joint date built from artefacts and printed books is a year from which somebody could have checked such a statement, which is a stronger thing than a year from which it happened to be true.

The one thing the arithmetic cannot fix #

The joint date inherits the better-attested member, and that is a strength and a trap.

A joint date is firm because it rests on the later source, so it is as good as that source and no better. If the kindergarten syllabus’s 1838 attestation were wrong, every joint date involving it would move — and every one of them involves it, since it is the latest.

So the whole bracket rests on a single entry. That is not a small dependency and the record makes it visible: the syllabus rests on four independent surviving sources, which is the best-attested claim in the whole set, and its popular date and its source date differ by one year.

Which is fortunate rather than robust. A field whose tightest date rests on its best-attested claim is in a good position; the same construction on a set whose latest source were a single manuscript would produce an equally tight-looking number resting on nothing.

That is worth stating because a joint date carries no warning about which member it inherited. A maximum looks the same whatever it is a maximum of, and reading one requires knowing how good its latest member is — which is the next rung of this ladder.

Which theorem was checked and how #

Every joint date is required to lie inside its own bracket. It must be at least as late as both of its members’ sources and no later than the whole set’s, which catches a maximum computed the wrong way round.

The individual claims must be widely contested. The worst gap must exceed two hundred years, or there is nothing for the joint date to be firmer than and the essay has no subject.

The joint date must be tighter than the worst single one, which is the finding, and it is stated as a condition on the figure existing rather than left for a reader to compare.

And every row goes through the record’s own checker, which refuses an entry with no dated source, an unnamed source, a source of a kind outside the closed set, or a source count that is not a count — the machinery a record is not a proof is about.

Where the model stops #

Five claims is five. The record carries fifteen entries and ten of them are about technology or about results; the five here are the ones about practice, which is the set the merge story is about.

A joint date is about the record and not about the world. Two practices might have coexisted for centuries before either was written down, and the arithmetic says nothing whatever about that — it says when the surviving evidence can first put both of them in the world.

The popular dates are what is generally circulated and are not a measurement of anybody’s belief. They are carried in the record as a stated quantity and the essay uses their maximum in one comparison, which is the weakest use it makes of them.

And nothing here dates the merge. The record has no entry for the merge itself, so the bracket is a constraint on statements about it rather than a date for it, and the difference is the whole of this field’s discipline.

What the picture cannot show #

The rows are joint dates and cannot show what happened at them. A year at which two lineages are both attested is a year at which the record could support a statement about both, and whether anybody made one, or whether the two had any contact, is not in it.

Nor can it show what the record is missing. The whole construction is an arithmetic over surviving sources, and the sources that did not survive would move every date in it — presumably earlier, since a lost source is a source that would have attested something sooner.

The most conspicuous absence is a fifth lineage nobody has found. The set is five because the record has five entries about practice, and a tradition attested nowhere is a tradition this arithmetic cannot see. A bracket computed over what survives is a bracket on statements, not on history.

The idealisation, named #

The record is treated as a set of claims each with one earliest source and one popular date, and the joint date is the maximum of the source years.

The strongest simplification in that is one source per claim. A claim attested by four sources has an earliest one, and the record carries only that; a claim whose second-earliest source is a century later is in a different position from one whose sources cluster, and the arithmetic cannot tell them apart.

That matters for the bracket because a joint date rests entirely on its latest member’s earliest source. If that source is one of four clustered around the same decade, the date is robust; if it is one of four spread over two centuries, the date is a single document’s date and the other three are decoration.

Which is the question the next rung asks of the whole record, and it is asked because this rung’s arithmetic depends on the answer.

Where the ladder goes next #

The bracket rests on how good the sources are, and the record states how many survive for each claim without anybody having asked what the number does.

Five of the fifteen entries rest on exactly one surviving source. One lost source and the story changes takes them away one at a time and finds the record’s most-quoted statistic — how far ahead of its evidence a popular date typically runs — falling by nearly half, while the median hardly moves. The effect is not a tendency spread through the record; it is two particular documents, and both of them are single-witness.

Sideways from here, the joint-date construction belongs beside the record’s account of what it cannot prove, which is where this field’s own limits are stated as a table rather than as a preamble. The joint date is the one quantity in that table’s neighbourhood that gets better rather than worse as the claims get shakier, and it is the only one.

The habit worth carrying is about combining uncertain quantities. A maximum inherits its best member and a minimum inherits its worst, so a derived quantity can be firmer than anything it is derived from — and looking for the combination that inherits upward is often the difference between a field that can say nothing and one that can say something.