The Talemi numerals, and the saint who did not invent them

It was a bright afternoon in Venice, and I had spent most of it in an antiquarian shop going along the shelves in no particular order — old printed books for the most part, and among them a case of manuscript material sold by the leaf.

I came on them there: a few leaves plainly older than anything shelved with them. Being a cautious sort of reader I assumed at first that I had misjudged the hand, which is easy enough to do if you would like it to be old. The assumption held less well the longer I stood there, and by the time I reached the sums I had stopped asking it to.

My Latin is basic and my Greek is worse, so what I could establish standing at the case was only the kind of thing I was holding: a puzzle, with numbers in it, set in a language that was neither of the two I was struggling with.

What held me was not the age of the leaves, though that was part of it. It was the flat fact that the thing on the page worked. Six statements in an unknown tongue, six sums beside them in Greek and in Latin, and between the two a relation that either holds or does not. Whatever else about the leaves was open to doubt — and nearly everything was — the arithmetic on them was true in an absolute sense, and would go on being true whether or not I ever established where they had come from, or who had written them, or in what language.

I did not buy them. The price was a long way past what I had come out with, and the dealer, who had shown no interest in me at all until the moment I stopped turning pages, became attentive in the way that makes a camera impossible. So I copied them out by hand instead, as faithfully as I could manage, on the understanding that I would study the copy properly later. This is later.

Everything that follows rests on that copy and on nothing else. It is the first thing to hold against anything surprising below, and one surprise in particular turns out to need it.

The leaves credit the puzzle to Eutychios of Mokissos, a holy man of Justinian’s Cappadocia, patron — one is tempted to suppose — of inconvenient notation and recreational arithmetic. The provenance is doubtful at nearly every link in the chain, and the one link that holds turns out to matter. The puzzle is excellent.

The manuscript

The page is headed DE QVAESTIONE SANCTI EVTYCHII, on the question of Saint Eutychius, and opens with the scribe explaining at some length how little he is prepared to vouch for:

Temporibus Iustiniani imperatoris fuit in Cappadocia vir quidam sanctus, Eutychius nomine, qui apud Mocissum habitabat. Hic hanc quaestionem discipulis suis proposuisse fertur.

Ex codice Graeco antiquo, qui ad manus meas pervenit, haec transfero. Verbum de verbo in Latinum reddo, quantum Latinitas pati potest, nihil de sententia sponte mutans.

In the time of the emperor Justinian there was in Cappadocia a certain holy man, Eutychius by name, who lived at Mokissos. He is said to have set this question to his disciples; the scribe is working from an old Greek codex that came into his hands; he renders it word for word so far as Latin will bear, altering nothing of the sense on purpose. Four hedges, and the only sentence that carries no hedge at all is the one giving a place and a date.

Which is convenient, because a place and a date can be checked. The place holds up well: Mokissos was a real town in Cappadocia, rebuilt by Justinian and raised to the rank of a metropolis, and both its history and its site are reasonably well established.¹ Eutychios is another matter. On him the sources are silent.

The scribe then reports a note in the margin of the Greek original, in an older hand:

Ταῦτα ὁ ὅσιος Εὐτύχιος ὁ ἀπὸ Μωκισσοῦ ἐξ ἑτέρας γλώσσης εἰς τὴν Ἑλληνικὴν γλῶσσαν μετήνεγκεν.

These things the blessed Eutychios of Mokissos translated out of another tongue into Greek. The epithet is ὅσιος rather than ἅγιος — the word used of a monastic, an ascetic, rather than of a martyr — which fits a man remembered for setting puzzles to disciples. And it means the saint is not the author either: he is one more copyist in the chain. Then the sentence that turns a text into a puzzle:

Cuius autem gentis sive linguae verba primum fuerint, neque auctor nominat neque ego invenire potui.

Of what people or language the words first were, the author does not say and I could not find out. The language is not merely lost; it was never named — though a damaged gloss may preserve a corner of the name. I call it Talemi below.²

One more editorial note, and it is the one that makes the page worth photographing:

Numeros, qui in exemplari Graecis litteris notati sunt, sicut scriptos inveni servavi; iuxta eos autem eosdem Latinis notis posui. Voces vero peregrinas similiter primum Graecis litteris, deinde Latinis litteris reddidi.

He has kept the Greek numerals as he found them and set Roman ones beside them, and done the same for the foreign words. Everything below therefore appears twice, which is a considerable gift: it lets a reader check the transcription against itself.

The six computations

The same six in the unnamed language

The scribe is careful to warn that the order has been changed — ordo autem computationum mutatus est — so the reader may not simply read down the two lists in parallel.

What is asked

Four tasks, in the manuscript’s own order. First, which of the six sentences answers to which computation. Second, and this is the real one:

Deinde ex ipsis vocibus inveniendum est qua ratione homines illius linguae numeros componant.

From the words themselves it must be discovered by what method the people of that language compose their numbers. Third, four further words are given for the reader to evaluate:

And last, four numbers to be put back into the language:

Explicit quaestio sancti Eutychii.

The hand that wrote it down

Below the explicit, the scribe signs off without signing:

Scriptum Patavii, anno ab incarnatione Domini MCLXXVII, imperante Friderico augusto et Alexandro papa sedente, a fratre qui sub regula beati Benedicti militat. Qui legis, ora pro scriptore.

Written at Padua, in the year 1177 from the Incarnation, under the emperor Frederick and with Pope Alexander on the throne, by a brother serving under the Rule of Saint Benedict; you who read, pray for the writer. A place, a year, two men he dates by, an order, and no name.

He does not say which house he belonged to, but the place and the year narrow it, and a little reading narrows it further. A Benedictine at Padua in 1177 points to the abbey of Santa Giustina, which had been the Benedictine house of the city for two centuries by then.³ If that is where he was writing, the least likely part of his story turns ordinary: an old Greek codex — coming out of Byzantium, in all probability — reaching his hands there in one form or another.

The puzzle in modern notation

Three things have to be changed before any of this can be worked on, and none of them is mathematics. They are worth doing carefully, because two of the three are places where a transcription can quietly destroy the puzzle.

The numerals. Greek alphabetic numerals run α to θ for 1 to 9, ι to ϙ for the tens, and take a keraia — the mark ʹ — to show they are numbers rather than letters. So ιηʹ is $10+8=18$ and νθʹ is $50+9=59$. One of them is a trap for the unwary: μϛʹ uses stigma, ϛ, the obsolete letter that holds the place of 6, so μϛʹ is 46 and not something involving a sigma. The Roman numerals agree throughout, and they are worth a second look: the scribe writes XXXXV, XXXXVII, LVIIII, never XLV or LIX. His own numerals are purely additive. That he did not modernise them is a small sign he was not modernising anything else either — and it is quietly funny, given that the system he is transmitting turns out to subtract.⁴

The transliteration. Greek ου spells u, and β at this date spells v, so βοτου is votu. The one that matters is ιου, which the scribe Latinises as iu: Greek has no letter for the glide, and writes it with iota. It is a single syllable, and I will write it yu from here on, so that every root in the lexicon is one syllable spelled one way. Nothing hangs on the choice; it is the same sound the codex writes ιου and the scribe writes iu.

The numbers themselves. With both of those done, the first table is six equations over $\mathbb{N}$,

10 + 8  = 18
15 + 7  = 22
22 + 13 = 35
34 + 11 = 45
27 + 20 = 47
41 + 18 = 59

and the second is six equations over words, in an order the scribe has told us is not the same:

A)  petu   + katu = kasasu
B)  sutu   + ri   = petu
C)  sunami + votu = kasana
D)  penami + yu   = susana
E)  lonasu + tetu = kasalo
F)  pe     + vo   = tetu

Now the task can be stated. Sixteen distinct words appear. We are looking for a valuation $\mathrm{val}$ assigning each an integer, together with a matching $\sigma$ pairing the lettered sentences with the numbered equations, such that every sentence is true under $\mathrm{val}$ and agrees with the equation $\sigma$ sends it to.

Put that baldly and the problem is hopeless, because it is also trivial: the six equations constrain sixteen unknowns, and one can satisfy them in enormously many ways by simply declaring sixteen numbers. What rules those out is a condition the manuscript states in a single word. Qua ratione numeros componant — by what method they compose numbers. We are not to find sixteen values. We are to find a way of building values, so that $\mathrm{val}$ is not a table but a function of the shape of the word.

That is what turns the problem into a well-posed one, and it is the whole reason the last two tasks are there. Four words the reader has never seen must come out right, and four numbers never yet spelled must be spellable. A table of sixteen entries can do neither. In modern terms the manuscript has handed us a training set of six equations and a held-out test set of eight items, and it is asking for a grammar and a semantics rather than a lookup.

So the object to be recovered is a triple: an alphabet of roots, a grammar saying how roots combine into words, and a valuation defined by recursion on that grammar. The rest of this post finds all three, and then checks them.

Form before meaning

The first useful move is not to guess at meanings. It is to look at shapes.

Four Talemi words are a single syllable — pe, vo, ri, yu — and every other word in the corpus has two or three. The natural hypothesis is that one syllable is one root, and that the longer words are built out of them.

If that is right, then equation F is special: it is the only one whose two addends are both bare roots. Among the decimal equations, only $10+8=18$ has both addends small enough to be plausible as bare roots at all; the others add 15, 22, 34, 27 or 41 to something. So F translates $10+8=18$.

This is the one genuinely abductive step in the reconstruction. Everything after it is forced — forced, that is, under the syllable hypothesis, and it is worth saying out loud how much that hypothesis is carrying. It holds that one syllable is one element of the numeral, that a bare root is a digit, that a recurring fragment means the same arithmetic thing every time it recurs, and that a word like tetu comes apart at all rather than being a single name for eighteen. Grant those and there is nothing further to choose. Refuse them and the six equations admit readings this post never looks at. The check on the guess, and on the hypothesis carrying it, is that all six equations close and the eight items the manuscript holds back fall out of the same machinery.

Now look at B, whose result petu begins with the pe of F and ends with the tu of F’s own result tetu. Its second addend ri is a bare root, so B must be $15+7=22$. Reading off:

pe = 10      petu = 22
vo = 8       tetu = 18
ri = 7       sutu = 15

The ending that means twelve

Compare pe with petu. Ten against twenty-two: the ending tu contributes exactly twelve. That immediately predicts su $=15-12=3$ and te $=18-12=6$, and it tells us what kind of system this is.

That last point is the transferable one. In a puzzle of this kind, the base announces itself as a difference, long before any word for it turns up.

The remaining four equations

With petu $=22$ in hand, equation A is determined: the only decimal equation with 22 as an addend is $22+13=35$, so katu $=13$ and therefore ka $=1$.

Equation D has a bare root, yu, as its second addend. Of the equations still unassigned, only $34+11=45$ has an addend small enough to be one, so yu $=11$ and penami $=34$. Since pe is ten and $34-10=24=2\cdot 12$, the word segments as pe-na-mi, which asks us to read mi as 2 and the medial na as an instruction to add twelve times what follows:

$$\begin{array}{}\text{(}\text{1}\text{)}& \mathrm{val}(D\text{-na-}N)=d+12\mathrm{val}(N).\end{array}$$

Two words later this stops being a hypothesis and starts being a prediction. Equation C opens with sunami, which $\text{(1)}$ reads as $3+12\cdot 2=27$ — and 27 is exactly what C needs.

The results of A, D and E are a different shape. A gives kasasu $=35$, and $35=3\cdot 12-1$ with ka $=1$ and su $=3$ already known. So sa is the mirror of na:

$$\begin{array}{}\text{(}\text{2}\text{)}& \mathrm{val}(D\text{-sa-}N)=12\mathrm{val}(N)-d.\end{array}$$

Both remaining results fall out. D’s susana is $12\cdot 4-3=45$, which fixes na $=4$; E’s kasalo is $12\cdot 5-1=59$, which fixes lo $=5$. Everything else in the corpus now checks itself — votu $=20$, lonasu $=41$, kasana $=47$ — and the correspondence is:

F)  10 + 8  = 18
B)  15 + 7  = 22
A)  22 + 13 = 35
D)  34 + 11 = 45
C)  27 + 20 = 47
E)  41 + 18 = 59

Note that na leads a double life: the digit 4, and the additive linker. I will come back to that, because it looks like trouble and turns out not to be.

The manuscript’s own test

That is the first task answered and the second one guessed at. The third and fourth are what decide whether the guess was worth making, because nothing in them was used to arrive at it.

The four words read off without further comment. Each segments into roots and linkers exactly one way, and $\text{(1)}$ and $\text{(2)}$ do the rest — with the manuscript’s iutu written yutu, as agreed:

yutu   = yu-tu        11 + 12       = 23
minasu = mi-na-su      2 + 12 · 3   = 38
susalo = su-sa-lo     12 · 5 - 3    = 57
rinana = ri-na-na      7 + 12 · 4   = 55

rinana is the nicest of the four, because it puts the two lives of na next to each other — linker then digit — and still reads only one way.

The four numbers go back the other way, and here something happens that the manuscript gives no sign of noticing:

17  =  5 + 12         lotu
29  =  5 + 12 · 2     lonami
46  = 10 + 12 · 3     penasu
    or 12 · 4 - 2     misana
71  = 11 + 12 · 5     yunalo
    or 12 · 6 - 1     kasate

Two of the four have two answers apiece, of the same length, with nothing to choose between them. A puzzle that asks the reader to express a number has quietly assumed there is one way to do it, and for 46 and 71 there is not. That is the first crack in the system, it appears in the manuscript’s own exercises, and the rest of this post is largely about how wide it goes.

The reconstructed system

The six equations fix a fragment. What remains is to say what the language is — including the root the corpus never wrote, and the recursion it never needed to.

The hole in the middle

Collect the roots the corpus has given us, and something is wrong with the list:

ka = 1
mi = 2
su = 3
na = 4
lo = 5
te = 6
ri = 7
vo = 8
?? = 9
pe = 10
yu = 11

Every digit from 1 to 11 turns up somewhere in the corpus except nine. Not “nine is rare” — nine is absent, and there is no arithmetic reason for it. Nor is there a zero, though that is a different kind of absence and I will come back to it.

The gap is hard to explain away, because everything else about the puzzle is carefully made. The six equations are ordered so that each new root is forced by the ones before it. The two tasks aim squarely at the machinery those equations establish — reading in one direction, writing in the other — and if one of them stumbles into an ambiguity the author seems not to have seen, it still stumbles into it at exactly the interesting place. This is not the work of somebody who would lose a digit.

So I think something is missing from what I have, and I can see three places it could have gone. The Greek codex may already have been incomplete when it reached Padua. The scribe may have dropped a line he did not understand — he was, after all, copying words in a language nobody had been able to name. Or — and I cannot rule this out either — I did, copying by hand at speed in a shop where I was not welcome to linger.

Of the three I think the middle one is the least likely, which is an odd thing to say about the only person in the chain who was doing the job properly, and is precisely why. A man who keeps XXXXV rather than tidying it to XLV, who hedges four times in two sentences, and who tells you outright that he has reordered the computations, is not a man who quietly loses a line. Note also what his promise actually says: he alters nothing of the sense sponte, on purpose. It is a careful word, and it covers nothing accidental — but a scriptorium is the one place in this story with a discipline against accidents.

Which leaves the codex and me. I would rather it were me, because that is the only one of the two that can still be fixed.

It is worth being plain about what this means. Something came away from the text somewhere on its way to this page, and the part that is gone is not a decoration but a digit — one of the eleven pieces the mechanism needs in order to run at all. What I have is a fragment of a fragment, and I cannot tell you at which pair of hands it became one.

Three more things are missing along with the ninth root, and once they are listed together they look less like four gaps than like one:

• nothing says how to say an exact multiple of twelve;
• nothing says whether the two linkers are interchangeable where both fit;
• nothing says whether the $N$ in $\text{(1)}$ may be more than a single root.

Completing the system

From here I stop reading and start proposing. What follows is a reconstruction — the smallest set of assumptions I can find that turns the fragment into a numeral system — and none of it is attested. I mark each assumption where it arrives.

A root for nine. The inventory needs one, and nothing tells me what it sounded like. I need a name to state theorems about, so I will write it ze, chosen to fit the shape of the others and meaning nothing more than “the ninth root, whatever it was”. Every claim below that mentions ze is a claim about the slot, not about the syllable.

A zero remainder. This much is forced; whether the language had an overt root for it is not. An exact multiple of twelve needs a zero in the units place: twenty-four is $0+12\cdot 2$, and no other digit will do, because $\text{(1)}$ with $d\ge 1$ gives at least $1+12k$ and $\text{(2)}$ would need a digit of twelve. So the units slot has to be fillable by something that contributes nothing. For the formal model I give that slot the conventional root o, and then exact multiples of twelve go onate $=72$, onari $=84$, onaze $=108$.

Both linkers live. The corpus uses na and sa both, and gives no sign that either is preferred where the two would fit. I assume they are simply alternatives, which is the weaker assumption — and the manuscript’s own exercises have already shown what it costs, since 46 is both penasu and misana and 71 is both yunalo and kasate. What a grammar permits and what a speaker would prefer are separate questions, and the second needs usage data that six equations cannot supply.

Unbounded recursion. Here parsimony decides. If the $N$ in $\text{(1)}$ must be a bare root, the language stops at $11+12\cdot 11=143$ and the rule needs an extra clause to say so. If $N$ may be any numeral, no clause is needed and the language goes on forever. The corpus reaches 59 and so cannot tell the two apart, but one of them requires a restriction that nothing supports, and the other is what you get by writing the rule down and leaving it alone. So depth is unbounded, and 1731 is a word:

$$\text{sunaonatu}=3+12\cdot 144.$$

Which is Horner’s method,⁵ $a_0+12(a_1+12(a_2+\cdots ))$, spelled out in morphology — with the subtractive linker allowing a local digit to be negative.

Everything after this point is about the reconstructed system. Where it turns out to behave badly — and it does, in a way I did not expect — the fault may be the language’s or may be mine, and I will say which cases I think are which.

The grammar

One subtlety deserves to be stated before the grammar rather than after it. Thirteen is katu, never kanaka. By $\text{(1)}$ those would be the same number, $1+12\cdot 1$, and only one of them is Talemi. So tu is not a construction of its own: it is the fused realisation of the linker na followed by a ka that ends the word. Writing $D\text{tu}$ as a separate production would generate both forms and get the language wrong.

That leaves two productions:

with $\text{(1)}$ and $\text{(2)}$ as the semantics, and the fusion applied to the finished word.

That fusion is narrower than “na ka surfaces as tu” makes it sound, in two ways worth pinning down. It reaches the bottom rung only, where ka is the whole remainder rather than a digit with more word behind it: thirteen padded out to kanakanao keeps its internal na ka intact and does not collapse to katunao. And where the digit on that bottom rung is o, the o goes unsaid as well, so twelve is tu and not otu, and $145$ is kanatu and not kanaotu. Stated on the syllable string, the two rules are: word-final na ka becomes tu, and then word-final o tu becomes tu.

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Pick a word · click a syllable to stop there

Writing it down so a machine can check it

Everything from here on is a theorem in a small Lean library, which you can read or download at the end. The statements and their proofs are given here in words; what the library adds is that a machine has checked every step of them, and the tactic scripts are of no interest to anybody who is not writing tactic scripts.

Three things have to be kept apart, and the prose solution above runs them together: a numeral as a tree, the number that tree denotes, and the syllables it is said with. Most of the benefit of formalising is the discipline of not letting those merge.

The tree is the grammar read as an inductive definition: a numeral is a bare digit, or a digit joined by na to a numeral, or a digit joined by sa to a numeral. The recursive occurrence — “a numeral”, not “a digit” — is the unbounded depth assumed above.

The number is the fold: $\text{(1)}$ and $\text{(2)}$, with $\mathrm{val}$ of a bare digit being the digit. Its values are integers and not naturals, which is already a finding rather than a design choice. The grammar builds kasao $=12\cdot 0-1$, so insisting on $\mathbb{N}$ would make $\mathrm{val}$ a partial function and every later theorem uglier.

The syllables are a second fold, $\mathrm{say}$, which sends a numeral to a sequence of syllables rather than to a string — the theorem worth proving is about syllables, and a string would have thrown away the boundaries. It has four clauses where $\mathrm{val}$ has three:

The second clause claims $d\text{-na-}\mathrm{ka}$ before the third can, and that is the whole of the fusion: kanaka is not discouraged, it is unsayable, because nothing in $\mathrm{say}$ can emit it. Note where the fusion lives — in the speller, where it cannot reach a value.

The double life of na costs nothing

The digit 4 and the additive linker are the same syllable. Call that an accidental homonymy and move on, and you have conceded more than you need to: it introduces no ambiguity whatsoever. Every root is one syllable, so the syllable sequence is fixed; in that sequence digit positions and linker positions alternate; a reader going left to right never faces a choice.

The way to say that so it can be checked is to write the reader as a function in its own right, without looking at the speller, and then prove the two agree. Call it $\mathrm{read}$; it takes a sequence of syllables to a numeral or to nothing, and it decides by looking at the front of the sequence. A lone tu is twelve. A lone root is that bare digit. A root followed by tu is that digit plus twelve. A root, then na or sa, then anything that reads, is the corresponding compound. Nothing else reads at all.

The load-bearing step is the third one. The reader is never in doubt about whether a na in front of it is the digit 4 or the linker, because it is not deciding by syllable but by position, and the positions cannot drift: every root is exactly one syllable, so digits sit at even offsets and linkers at odd ones, all the way down. That is a stronger claim than the informal solution makes, and it is the sort of claim that is very easy to believe wrongly.

The corpus as a test suite

A grammar that compiles is not yet a grammar worth having; $\text{False}\to \text{anything}$ compiles too. So every attested word is written out as a tree and checked twice over: that $\mathrm{say}$ sends each tree to the string the manuscript actually prints, and that $\mathrm{val}$ sends it to the number the manuscript actually counts.

Both halves are needed. Without the spelling check the arithmetic would say only that some arithmetic is consistent — not that it is the arithmetic of these words.

The manuscript’s own test set is checked the same way, including the part the manuscript got wrong: that misana and penasu are distinct words of equal length both worth 46, and likewise kasate and yunalo for 71. It is a small satisfaction to have the flaw in a twelfth-century exercise recorded as a theorem.

What formalising it exposes

Syntax is not semantics. kasatu and yunayu are different trees that $\mathrm{val}$ sends to the same 143, and penasu and misana both to 46. The redundancy is a property of the language, not a defect of the reconstruction.

Surface is not structure. The fusion of na ka into tu lives in $\mathrm{say}$ alone. The tree behind katu still holds a rung with two parts, and $\mathrm{val}$ never learns that they were said as one syllable.

The grammar is permissive, and says so. kasao is a well-formed word denoting $-1$. Whether Talemi speakers would utter it is a question about speakers, of whom we have none; if the answer is no, the fix is a well-formedness predicate, and it belongs in the type, not in a footnote.

And counting spellings is the wrong question. My first instinct was to ask whether every number has finitely many Talemi forms. It does not, and not for an interesting reason: a zero remainder stacks without limit, so ka, kanao, kanaonao, kanaonaonao are all the number 1, and the same trick pads anything.

The question worth asking is about forms without padding, or about a canonical shortest spelling. Which turns out to be the thread that unravels the rest of the system.

What the system does

The library is not an ornament on the reconstruction. It is where the reconstruction starts answering questions the manuscript did not ask.

Can Talemi name every number?

The corpus gives twenty-odd words and stops. That leaves open whether Talemi is a notation for arithmetic at all, or a list of number names that happens to run out — and the way to settle it is a construction rather than an existence argument. The construction is long division, read backwards: a number below twelve is its own root, and anything else is its remainder on division by twelve, then na, then the word for the quotient. Write $\mathrm{enc}(n)$ for the result. The recursion terminates because dividing by twelve shrinks anything that is at least twelve.

So 144 is onatu, 1728 is onaonatu, and 20735 is yunayunayunayu. None of those needed a manuscript entry — the grammar generates them, which is the only credential a numeral system can offer.

But the construction has a property worth more than the theorem it proves. Look at what it never writes. Long division produces remainders between zero and eleven, which is precisely the range the additive linker covers, so sa is never reached — not as an optimisation, but because there is never an occasion for it.

What the subtractive linker is for

If na alone names every natural number, then sa adds nothing above zero. It does add something, and the formal statement says exactly what.

Read left to right that says no additive word is negative; right to left, that $\mathrm{enc}$ reaches everything that is not. And below zero the system does not give out either — a second construction bottoms out at $d\text{-sa-o}$, which is $12\cdot 0-d$, and reaches the rest of the integers from there.

Minus one is kasao, minus thirteen is yunamisao, minus a hundred is vonazesao, and every one of them contains sa, necessarily.

That is a sharper answer than the manuscript could give to the question of why the language has two linkers, and it is slightly deflating. Across the whole manuscript sa appears five times, and every one of those five words is a positive number — kasasu is 35 and could equally have been yunami. The one job only sa can do is a job the monk never asks it to do.

Nor does it save breath

The remaining defence of the subtractive linker is economy. Roman numerals subtract for exactly that reason: IX spends two characters where VIIII spends five, and that is the whole of its justification. kasasu reads as “one short of three twelves”, and if that were a syllable cheaper than “eleven and two twelves” then sa would have earned its place after all.

It is not, and the theorem is general.

The proof turns on a single function. Let $\mathrm{cap}(k)$ be the largest value any word of $k$ syllables can denote:

$$\mathrm{cap}(k+3)=11+12\mathrm{cap}(k+1).$$

The recurrence steps by two rather than by one, and that is the fusion showing up in the arithmetic. A rung normally costs two syllables, a digit and a linker; the bottom rung costs one when it fuses into tu. So one syllable already reaches tu $=12$, two reach yutu $=23$, and from there each further pair of syllables multiplies the reach by twelve and adds eleven for the new digit. The first few values are $-1$, $12$, $23$, $155$, $287$, $1871$ — the $-1$ recording that no word is empty.

The theorem is then two inductions that meet in the middle.

So subtraction in Talemi is not the subtraction of Roman numerals. It reaches no new number and shortens no old one.

Where uniqueness comes from

What subtraction does do is create ties. kasasu and yunami are both 35, both three syllables, and neither can be improved on.

How typical is that? Counting requires enumerating, and an enumeration is worth nothing until it is known to be exactly right, so both halves are proved before anything is counted: that the enumeration misses no numeral of the given value, and that it invents none.

The enumeration is smaller than it looks, for a reason worth pausing on. At each rung the value fixes the digit. If $v=d+12w$ with then $d$ must be and $w$ must be $\lfloor v/12\rfloor$, with nothing left to choose, and the subtractive rung is pinned the same way. A Talemi numeral is therefore not a free choice of digits at all. It is a sequence of choices between na and sa, one per rung — a binary tree, and a shallow one, since the ceiling bounds a shortest spelling below 1872 at five syllables and so at three rungs.

Three runs — 0 to 23, 145 to 155, 277 to 287 — and not a scatter. Blocks want an explanation, and the ceiling turns out to be it. Those runs end at 23, 155 and 287, which are $\mathrm{cap}(2)$, $\mathrm{cap}(3)$ and $\mathrm{cap}(4)$: exactly the most that two, three and four syllables can say.

The reason is one line of the ceiling argument, read the other way round.

Eleven short, every time — and for every word that subtracts anywhere, not merely for those that subtract last. That is what makes the conclusion go through: in the eleven values immediately under each ceiling there is no subtractive rival to be had at any depth, so only purely additive words are left in contention, and among those the value fixes every digit. One of them is shortest, and it is alone. Everywhere else there is slack, the two constructions come out level, and the tie stands. Uniqueness in Talemi is not a property of a number so much as of how tightly it is wedged under a ceiling.

It keeps going, too, and counting rather than merely asking yes or no makes the shape clearer. Below three hundred the number of shortest spellings is not scattered either: it is 1 up to 23, then 2 as far as 144, then 1 again through 155, then 2 to 276, then 1 through 287, then 4. Constant between ceilings, and dropping to 1 over the eleven values before each of them.

Plotted against $n$ the stretches look nothing alike — the three-syllable band is 132 numbers wide and the five-syllable band is 1584 — but that is the wrong variable to plot against. Measured downwards from the ceiling they are the same stretch. Start at $\mathrm{cap}(k)$ for any $k$ from three up and walk down: 1 for eleven steps, then 2 for a hundred and twenty-one, then 3 for eleven more, then 4. All that changes with the length is how far you can walk before the band runs out underneath you.

What is proved is the near end of that. The library checks the uniqueness pattern below four hundred and the exact counts below three hundred, and both are theorems. That the same profile recurs under every ceiling is something I have computed and not proved, which is why it is a paragraph here and not a box.

The two stragglers are 0 and 1, bare roots that nothing three syllables long can tie.

Which leaves the reconstruction’s elegant feature looking thin. Above zero, signed digits earn nothing whatever — not a number, not a syllable, not a canonical form — and the one place the language does name a number uniquely is the place where its signed digits cannot reach.

The reconstruction

o  = 0
ka = 1
mi = 2
su = 3
na = 4
lo = 5
te = 6
ri = 7
vo = 8
ze = 9
pe = 10
yu = 11

Two constructions, $\text{(1)}$ and $\text{(2)}$, recursive in $N$, with na ka fused to tu. A signed-digit duodecimal system written as morphology, unambiguous despite appearances, able to name every integer, and — above zero — redundant twice over, since the subtractive half of it neither reaches a number the additive half misses nor says any number faster.

Two of those roots I supplied myself, and the recursion and the free choice of linker are assumptions rather than readings. What the corpus attests is a fragment; what is proved above is a system, and the joins between them are marked. Whether Eutychios would recognise the result is a matter for palaeography, and whether my ninth root is anywhere near the right syllable is a matter for somebody with better Greek and access to those leaves.

I have not been back for them. The price will not have moved in my favour, and I am not sure I would trust my own excitement a second time. What I have instead is a transcription made standing at a shelf, and twelve hundred lines of Lean that a machine rechecks from nothing in about fifteen seconds every time this page is built.

I am aware that this is the wrong way round. The manuscript is the real object, and the formalisation is a commentary on a copy of a translation of a copy. But the commentary is the part that is whole. Every piece the system needs is in it, including the pieces I had to supply, and it cannot quietly lose one the way a codex can: take a line out and the whole thing refuses to build, at once and in the open.

So I have stopped thinking of those files as a check on the reconstruction and started thinking of them as where the reconstruction lives. The leaves are in a case in Venice, priced beyond me, and the sums on them are still true. The system they belong to is here — complete, with its ninth root borrowed and its zero argued for. The arithmetic checks out, and this time that is not a figure of speech.

One line in those leaves was addressed to me, and I was slow to notice it. The scribe gave a place, a year, two men to date himself by and an order, and no name; then he closed with the only request the document makes of anyone. Qui legis, ora pro scriptore. You who read, pray for the writer. It is the single sentence in the whole manuscript I did not have to reconstruct, hedge, or supply a missing root for. It arrived whole, it was meant for whoever got this far, and it asks for the one thing I was in a position to give. So I gave it. He left no name to thank and asked for nothing else, and after months of taking from him it was not much to have been asked for.

Behind him stands the holy man at Mokissos, on whom the sources are silent and who is only said to have set the question at all. If he did, he set it to his disciples, which means he set it to be understood. And it is worth understanding. Strange as it looks on the page, once it gives way it is very simple: count in twelves, and let a digit be owed as readily as held. That is the whole of it — a small, clean idea, and a durable one. It survived a Greek codex, a translation by a man who said outright that he could not follow it, the loss of at least one line, and eight centuries of being worth nobody’s attention.

Would he be proud that I solved it? I have thought about that more than the question can really bear. I do not think he would have been very interested in me, and I find I do not mind. He set it so that someone would come to see what he had seen, and someone did — late, and by a route he could not have pictured, and needing a great deal more help than his disciples were given. The seeing is the same seeing. A man in Cappadocia arranged twelve syllables so that a mind meeting them fifteen hundred years later would end up standing where he had stood. I got there. Of everything written here, that is the part I am glad of.

The Lean sources

Every proof written out above is also a proof in a small Lean library, checked from its axioms each time this site is built. The prose and the library follow the same arguments in the same order, so a paragraph above should correspond to a recognisable stretch of a file below. The library imports no mathematics library: integers, induction and reflexivity are the whole toolkit, which also means it can be read by someone who does not know one.

The six files, in dependency order:

• Talemi.lean — the library root, and a map of the rest
• Numerals.lean — trees, values, syllables, and the reading-back theorem
• Range.lean — which numbers the system can name
• Shortest.lean — the ceiling argument
• Enumeration.lean — enumerating spellings, and the count below four hundred
• Examples.lean — the corpus, and every word quoted in this post

The boxes above are prose, so here is what they are called in the files, for anyone who wants to check that I have quoted them faithfully. Reading back is parse_toks and toks_injective; the infinity of spellings is padded_injective; the argument that the reconstruction had no choice about a zero remainder is multiple_needs_zero_head; the reach of the system is eval_encode for the naturals, eval_encodeInt for the integers, and additive_range for the fragment without sa; the ceiling argument is encode_shortest and sub_le_cap_sub_eleven, with the extension to a buried sa in usesSub_le_cap_sub_eleven; the tie at thirty-five is both_shortest_35; the count below four hundred is unique_shortest_below_400, sharpened from a yes-or-no answer to the counts themselves by tie_counts_below_300.

The names in the prose map across too: $\mathrm{val}$ is eval, $\mathrm{say}$ is toks, $\mathrm{read}$ is parse, and $\mathrm{enc}$ is encode.

Footnotes

1. Procopius of Caesarea, De aedificiis V.4.15–18 (the Loeb text and translation): a decayed Cappadocian fortress pulled down, rebuilt on ground too steep to assault, given churches and hospices and baths, and raised to a metropolis. On where that actually was, see Friedrich Hild, “Jerphanion und die Probleme der historischen Geographie Kappadokiens. Neue Forschungen und deren Ergebnisse”, Mélanges de l’École française de Rome. Moyen-Âge 110/2 (1998), 941–951, on Persée. Hild follows Honigmann against Jerphanion in placing Mokissos at Viranşehir near Helvadere, high on the slopes of Hasan Dağı, where the city ruins are sixth-century and the terrain answers to Procopius’s description. Procopius gives the name as Μωκησός where the leaves have Μωκισσός; both forms circulate. None of this literature knows anything of a Eutychios there, and looking for him is not helped by the fact that Justinian’s reign already has a well-known Eutychius — the patriarch of Constantinople — against whom every search promptly silts up. ↩

2. The name of the language has not survived. The Greek text designates it only as “another tongue”, ἑτέρα γλῶσσα. A damaged gloss standing beside the text does, however, yield the sequence …λημ…, which might be reconstructed as part of an otherwise unattested ethnonym — Ταλημοί. I use Talemi throughout as a working label, and it should be read as exactly that: a reconstruction, and not an attested autonym. Nothing below depends on it. If the gloss turns out to say something else, every theorem stands and only the word for it changes. ↩

3. Francesco G. B. Trolese, S. Giustina di Padova nel quadro del monachesimo italiano. Studi di storia e cultura monastica, ed. G. Carraro, R. Frison Segafredo and C. Marcon, Rome: Istituto Storico Italiano per il Medio Evo, 2014 (Italia Sacra, new series, 1); the volume, and its presentation in Padua. A caution on wording that I nearly got wrong: it is the abbey of Santa Giustina in 1177, not the congregation. The Congregation of Santa Giustina is a fifteenth-century reform movement, founded by Ludovico Barbo in 1419, and calling the house by that name in the twelfth century backdates it by some two hundred and fifty years. As for how a Greek book reaches such a house: Padua is some forty kilometres from Venice, which traded with Constantinople throughout the period, and Greek manuscripts came west along that route in quantity. ↩

4. The system is the Milesian (Ionic) alphabetic numerals: twenty-seven signs in three nines, so a number is a sum of letter-values rather than a place-value string. The account to start from is T. L. Heath, A History of Greek Mathematics, vol. 1 (Oxford: Clarendon Press, 1921), 31–36 (the 1921 text). Heath is also the place to catch the stigma. The sixth sign is the old digamma; by the seventh or eighth century it had come to be written in a form that scribes then called stigma from its likeness to the σ–τ ligature. A reader who looks for a sigma in μϛʹ has therefore made the mistake twice — once of alphabet, and once of century. The free-standing keraia that marks a letter as a number is itself a later habit; earlier practice was often an overbar. The scribe’s Roman spellings (XXXXV, LVIIII) are the ordinary medieval ones, not an archaism: subtractive XL and IX are the forms that print later made look obligatory. The modern handbook for both systems is Stephen Chrisomalis, Numerical Notation: A Comparative History (Cambridge: Cambridge University Press, 2010). ↩

5. Horner’s method is the ordinary way of evaluating a polynomial without computing any powers: $a_0+a_{1}x+a_2{x}^2+a_3{x}^3$ is rewritten as $a_0+x(a_1+x(a_2+x{a}_3))$, which costs one multiplication per coefficient instead of growing quadratically. The name comes from W. G. Horner, “A new method of solving numerical equations of all orders, by continuous approximation”, Philosophical Transactions of the Royal Society of London 109 (1819), 308–335, doi:10.1098/rstl.1819.0023 — a paper about finding roots, in which the nested evaluation is the inner step rather than the point. Like most things named after somebody it is older than its name: Newton was using the nested form a century earlier, and it appears in thirteenth-century China in the work of Qin Jiushao. Donald Knuth gives the method and its history in The Art of Computer Programming, vol. 2, §4.6.4; for the Chinese side, the standard study is Ulrich Libbrecht, Chinese Mathematics in the Thirteenth Century: The Shu-shu Chiu-chang of Ch’in Chiu-shao (MIT Press, 1973).

   What makes it worth naming here is that the nesting is not an optimisation applied to the language from outside; it is the shape of the language. Each na is one set of brackets. So sunaonatu is $3+12(0+12(0+12\cdot 1))=1731$, and its roots spell the base-twelve digits 3, 0, 0, 1 of that number, least significant first. The subtractive linker is the one departure from the textbook scheme, since it lets a coefficient come out negative. ↩

6. The equivalence is the standard one between right-linear grammars and finite automata: read a production $N\to w{N}^{\prime }$ as “emit $w$ and go to the state for $N^{\prime }$”, and $N\to w$ as “emit $w$ and stop”. What makes it available here is that Digit is a nonterminal with only terminal right-hand sides, so substituting it out leaves no recursion except at the right end — and recursion at the right end is iteration, which is exactly what a loop in an automaton is. The class is fixed by the shape of the productions rather than by how recursive the definition looks: a grammar with the nonterminal in the middle, Numeral → a Numeral b, is not right-linear, and grammars of that shape can generate languages no automaton recognises. The classification is Chomsky’s: N. Chomsky, “On certain formal properties of grammars”, Information and Control 2/2 (1959), 137–167, doi:10.1016/S0019-9958(59)90362-6; the finite-state case had been worked out the year before in N. Chomsky and G. A. Miller, “Finite state languages”, Information and Control 1/2 (1958), 91–112, doi:10.1016/S0019-9958(58)90082-2. Both are open archive at Elsevier. ↩

7. A transduction a finite-state transducer can perform is called rational, and rational transductions carry regular languages to regular languages — so a fusion applied to the output of a regular grammar cannot push the result out of the class, however context-sensitive the rule reads on the page. The argument is Nivat’s: every rational transduction factors as $\tau (X)=\psi ({\phi }^{-1}(X)\cap K)$ for morphisms $\phi ,\psi$ and a regular $K$, and inverse morphism, intersection with a regular set, and morphic image each preserve regularity. Jean Berstel, Transductions and Context-Free Languages (Stuttgart: Teubner, 1979), gives Nivat’s theorem as III.4.1 and the closure as III.4.2; the book is available from the author. Berstel writes “rational” where this post writes “regular”; over a free monoid the two words name the same family. ↩