bourbaki

Bourbaki's Éléments de mathématique as Markdown, with permanent tags, translations, and worked exercises.

The source PDFs go in, Markdown comes out. Every numbered statement gets a four-character tag that never changes, so you can cite a result and have the citation still work after the text is re-extracted, re-split, or renumbered. That idea is lifted straight from the Stacks Project.

This repo is the corpus only. The code that builds it lives in tamnd/bourbaki-solver.

What is in scope

All twelve Books of the Éléments, in the English translation where one was printed and in the French original everywhere else. 44 volumes, 15338 pages, of which 15 volumes and 6648 pages are English and 29 volumes and 8690 pages are French.

Book	English	French	Volumes	Pages
Theory of Sets	I to IV	I to IV	2	767
Algebra	I to VIII	I to X	8	3690
General Topology	I to X	I to X	4	1511
Functions of a Real Variable	I to VII	I to VII	2	683
Topological Vector Spaces	I to V	I to V	2	740
Integration	I to IX	I to IX	7	1732
Commutative Algebra	I to VII	I to X	5	1733
Variétés différentielles et analytiques	none held	1	1	190
Lie Groups and Lie Algebras	I to IX	I to VII, IX	8	2182
Théories spectrales	none held	I to V	2	925
Topologie algébrique	none held	I to IV	1	512
Elements of the History of Mathematics	1	1	2	673

The French is not a fallback for the English. Three Books were never translated, Algèbre chapters IX and X and Algèbre commutative chapters VIII, IX and X exist in French only, and where both printings are held the French is the original a disputed English sentence gets checked against. So both are in scope and both carry the same tags.

None held in a column means one of two things, and the table cannot tell them apart. Théories spectrales, Topologie algébrique and Variétés différentielles et analytiques have no English printing because none was ever made. Théorie des ensembles has a French printing, the original one, and this corpus holds only the English translation of it.

One more gap worth stating plainly. Groupes et algèbres de Lie, Chapitres 7 et 8 is a partial file: 61 pages that open at Chapitre VII and stop in the middle of the exercises around printed page 65, so the English Chapters 7-9 is the only complete copy of that material.

bourbaki books list prints the volume by volume detail, and manifests/books.yaml carries it as data, with the SHA-256 and page count of every file so that a swapped or re-downloaded PDF is caught rather than silently extracted.

The PDFs are not here and will not be. They are copyright Springer and N. Bourbaki. This corpus is for personal study.

Three extraction paths, not one

What a volume costs to read depends entirely on what its own text layer is worth, and the library splits three ways. bourbaki books add measures this rather than guessing it, by sampling the images and the text of a band of body pages, and records the answer as text_layer in the manifest.

Text layer	Volumes	What it means
native	6	Born digital. pdftotext -layout gives real text and real mathematics.
ocr	34	A scan somebody has already run OCR over. Good enough to read a running head off, useless for mathematics.
none	4	A scan with no text at all. Even the page map has to come out of vision OCR.

The six native volumes are Algebra, Chapter 8 and Lie Groups and Lie Algebras, Chapters 7-9 in English, and Algèbre, Chapitre 8, Théories spectrales, Chapitres 1 et 2, Théories spectrales, Chapitres 3 à 5 and Topologie algébrique, Chapitres 1 à 4 in French. They are cheap and they go first. The four with no text at all are Théorie des ensembles at 349 pages, Algèbre, Chapitre 10 at 222 pages, General Topology, Chapters 5-10 at 372 pages and Commutative Algebra, Chapters 1-7 at 642 pages, and they are the most expensive volumes in the library. The other 34 are the ordinary case.

Their OCR layer renders a pair of braces in Theory of Sets as R! x, y I, so nothing in it can be trusted as mathematics, but it is legible enough to build the page map from before a single page goes to vision OCR, which saves the expensive pass on every one of them.

Running heads differ per volume

Worth knowing before you touch the page-map code, because each volume prints its locators differently:

1998, Ch. I     I                    ALGEBRAIC STRUCTURES
                                     p-GROUPS                         §6.5
2003, Ch. IV    A. IV. 2             POLYNOMIALS AND RATIONAL FRACTIONS   §1
                No. 2                POLYNOMIALS                      A.IV.3
2023, Ch. VIII  No 4                 POLYNOMIALS WITH ...              A VIII.13

Bourbaki cross-references are page based, as in VIII, p. 3, Proposition 3, so the page map is not decoration. It is the key the reference resolver joins on. The 1998 volume does not print a page locator in its running head at all, which is why its page map is anchored and interpolated rather than read off directly.

Tags

#### Proposition 6 {#alg-viii-s1-prop-6 .statement tag=0A3F}

tags/tags is append only, one line per statement:

0A3F,alg-viii-s1-prop-6

A tag is never reused and never edited. If a label has to change, the tag follows the statement and the old label goes to tags/aliases. Every edition uses the same tag for the same statement, the French original included, so a tag is the one identifier that works across all of them and is what lets a translation be checked against the French rather than only against the English it was made from.

tags/runs says where one assignment stopped and the next began, one line per merge. Tags are handed out in reading order, so the tags of a file climb on the run that assigned them, and they do not climb across runs: a statement added to the middle of a § a month later takes a tag above everything under it. The boundaries are what let the audit tell that correct edit from a heading somebody pasted the wrong tag on to.

Layout

content/en/<book>/<CH>/NN_sN_<slug>.md         one file per §
content/en/<book>/<CH>/exercises/sN/NN.md      one file per exercise
content/fr/...                                 same tree, extracted from the French printing
content/{vi,zh,ja}/...                         same tree, translated
content/solutions/<lang>/<book>/<CH>/sN/NN.md  verified solutions
tags/                                          permanent tag index
manifests/                                     books, TOC, page maps, refs, glossary, errata, editions
figures/                                       cropped diagrams, small, committed
reports/                                       audit, usage, coverage, scorecards
imports/<book>/chapter_<n>/<n>.<m>.md          read off share links, not yet checked

content/fr is not a translation and is never generated from content/en. It is what the French volume prints, extracted the same way the English is, which is why it carries no translated_from and is exempt from the translation rules.

Because the two printings sit side by side, the places they disagree get noticed, and there are two manifests for what is found. manifests/errata.yaml is for a printing that is wrong, and it carries the words the page has and the words to read instead. manifests/editions.yaml is for the other case, where the printings differ and both are right: chapter VIII § 2 has twenty exercises in the 2023 English and nineteen in the 2012 French, the nineteen are the same nineteen, and the twentieth is one the later printing added. A difference like that looks exactly like a page nobody read, so the reading that settles it is written down once instead of being done again by whoever counts next.

pdf/, images/ and work/ are gitignored. Nothing large or copyrighted is committed.

imports/ is deliberately outside content/. It holds sections read off public ChatGPT share links, which cost an HTTP GET rather than 150 seconds of a browser per page, and which nobody has yet held against the printed book. The audit runs over content/, so anything dropped in there would pass every rule in it by default. See imports/README.md.

Coverage

Book	Chapter	Sections	Statements	Exercises	Tagged	Pages
Commutative Algebra	I	4 of 4	65	42	0	106
Commutative Algebra	II	5 of 5	183	84	0	218
Commutative Algebra	III	5 of 5	143	39	0	230
Commutative Algebra	IV	3 of 3	69	50	0	98
Commutative Algebra	V	3 of 3	122	60	0	150
Commutative Algebra	VI	10 of 10	151	71	0	211
Commutative Algebra	VII	4 of 4	130	85	0	302
Commutative Algebra	VIII	0 of 7	0	0	0	108
Commutative Algebra	IX	0 of 5	0	0	0	85
Commutative Algebra	X	0 of 10	0	0	0	179
Algebra	I	10 of 10	293	148	0	357
Algebra	II	12 of 12	384	118	0	446
Algebra	III	12 of 12	273	97	0	542
Algebra	IV	6 of 6	157	53	0	202
Algebra	V	17 of 17	425	156	0	382
Algebra	VI	2 of 2	83	75	0	92
Algebra	VII	5 of 5	146	64	0	162
Algebra	VIII	25 of 25	709	317	1024	964
Algebra	IX	0 of 10	0	0	0	207
Algebra	X	0 of 9	0	0	0	216
Theory of Sets	I	6 of 6	16	35	51	101
Theory of Sets	II	6 of 6	109	41	150	117
Theory of Sets	III	7 of 7	222	120	342	226
Theory of Sets	IV	3 of 3	29	15	43	255
Topological Vector Spaces	I	3 of 3	64	35	0	58
Topological Vector Spaces	II	8 of 8	203	162	0	195
Topological Vector Spaces	III	6 of 6	123	75	0	101
Topological Vector Spaces	IV	6 of 6	111	100	0	152
Topological Vector Spaces	V	4 of 4	132	37	0	181
Functions of a Real Variable	I	4 of 4	81	51	0	100
Functions of a Real Variable	II	3 of 3	54	29	0	78
Functions of a Real Variable	III	2 of 2	10	48	0	141
Functions of a Real Variable	IV	2 of 2	54	28	0	94
Functions of a Real Variable	V	5 of 5	107	28	0	113
Functions of a Real Variable	VI	3 of 3	25	7	0	67
Functions of a Real Variable	VII	2 of 2	14	12	0	73
Elements of the History of Mathematics	1	26 of 26	0	0	0	665
Integration	I	0 of 1	0	0	0	17
Integration	II	2 of 2	29	21	0	44
Integration	III	4 of 4	106	35	0	128
Integration	IV	7 of 7	285	94	0	332
Integration	V	8 of 8	206	108	0	290
Integration	VI	4 of 4	98	57	0	173
Integration	VII	5 of 5	120	48	0	207
Integration	VIII	5 of 5	109	59	0	188
Integration	IX	7 of 7	184	56	0	259
Lie Groups and Lie Algebras	I	7 of 7	166	125	0	250
Lie Groups and Lie Algebras	II	9 of 9	121	66	0	186
Lie Groups and Lie Algebras	III	11 of 11	393	108	0	481
Lie Groups and Lie Algebras	IV	3 of 3	66	45	0	108
Lie Groups and Lie Algebras	V	7 of 7	139	44	0	179
Lie Groups and Lie Algebras	VI	4 of 4	128	47	0	291
Lie Groups and Lie Algebras	VII	7 of 7	139	66	205	127
Lie Groups and Lie Algebras	VIII	13 of 13	299	172	471	211
Lie Groups and Lie Algebras	IX	11 of 11	244	112	356	284
Topologie algébrique	I	0 of 6	0	0	0	150
Topologie algébrique	II	0 of 5	0	0	0	78
Topologie algébrique	III	0 of 5	0	0	0	110
Topologie algébrique	IV	0 of 6	0	0	0	158
General Topology	I	11 of 11	302	152	0	279
General Topology	II	4 of 4	110	41	0	94
General Topology	III	7 of 7	164	113	0	198
General Topology	IV	8 of 8	106	91	0	204
General Topology	V	4 of 4	24	8	0	45
General Topology	VI	3 of 3	37	36	0	66
General Topology	VII	3 of 3	38	24	0	61
General Topology	VIII	4 of 4	25	22	0	71
General Topology	IX	7 of 8	191	48	0	266
General Topology	X	4 of 4	106	43	0	173
Théories spectrales	I	0 of 8	0	0	0	197
Théories spectrales	II	0 of 3	0	0	0	136
Théories spectrales	III	0 of 6	0	0	0	143
Théories spectrales	IV	0 of 5	0	0	0	228
Théories spectrales	V	0 of 4	0	0	0	194
Variétés différentielles et analytiques	1	0 of 8	0	0	0	95

388 of 487 sections are in the corpus, 80 per cent. 8622 statements and 4123 exercises, 2642 of them carrying a permanent tag.

What is translated

The table above is what has been read out of the printed books. This one is what has been carried into another language, which is the other half of the work and is where most of the machine time goes.

A section or an exercise is counted as translated when a file for it exists in that language, nothing more. It is not a claim about quality. The translation rules in the audit are what says whether a file that exists is any good, and reports/audit.md is where their findings are. Plenty of what is counted here was written by a cut down model in a hurry and is queued to be read again by a better one.

The source is the English, and the English is two directories. content/en is the Springer translation where one was printed. content/en-mt is this project's reading of the French, for the chapters no English was ever printed for, which is most of Algebra IX and X, Commutative Algebra VIII to X and Integration V to IX. They are kept apart so a reader can tell which is which, but a Vietnamese file is made from whichever of the two exists, so for this table they are one source.

The French is counted and not compared against the English. A file name carries a slug of its own title, so 02_s2_relevement_des_ideaux_premiers.md and 02_s2_the_lift_of_prime_ideals.md are one section under two names. Matching the two trees by path once reported 289 French sections with no English when the real answer was 41.

Book	Sections	Exercises	Vietnamese sections	Vietnamese exercises	Done	From machine English
Theory of Sets	30	211	30	211	100%	0
Algebra	129	1223	129	1028	86%	217
General Topology	80	681	78	578	86%	105
Functions of a Real Variable	34	241	34	203	86%	38
Topological Vector Spaces	34	457	34	409	90%	48
Integration	59	504	59	478	95%	26
Commutative Algebra	69	842	69	711	86%	436
Variétés différentielles et analytiques	10	0	10	0	100%	10, all of it
Lie Groups and Lie Algebras	85	798	85	785	99%	14
Théories spectrales	36	374	36	371	99%	410, all of it
Topologie algébrique	28	196	28	164	86%	224, all of it
Elements of the History of Mathematics	27	0	27	0	100%	1
All	621	5527	619	4938	90%	1529

The source column is the English, which is 621 sections and 5527 exercises: 4619 files in content/en where Springer printed an English translation and 1529 in content/en-mt where this project read the French instead. The French originals are 600 sections and 5377 exercises in content/fr, and they are counted here rather than compared, because a file name carries a slug of its own title and matching the two trees by path calls every honestly translated title a missing section.

Vietnamese has 619 of the 621 sections and 4938 of the 5527 exercises. Sections here means every file that is not an exercise, so the introductions, the notes to the reader and the historical notes are counted with the §§.

The last column is the part of a Book that was never printed in English, so the only English of it is this project's own reading of the French. A translation made from one of those is a translation of a translation, and that is 938 of the 5557 files in Vietnamese. Where the column says all of it the whole Book is in that position, and a hundred per cent in the Done column for such a Book is not the same claim as a hundred per cent for one Springer translated.

Building it

Registering a volume probes it and writes what it measured, so this is run once per file and then the manifest is the record:

export BOURBAKI_CORPUS=$PWD
bourbaki books add "pdf/en/Algebra Chapter 8 (2023, Springer Nature).pdf" \
  --id alg-viii --book alg --lang en --chapters VIII
bourbaki books add "pdf/fr/Algèbre_ Chapitre 8 (2012, Springer).pdf" \
  --id alg-viii-fr --book alg --lang fr --chapters VIII
bourbaki books list
bourbaki books verify

Then per volume. A native volume is three commands and a few minutes:

bourbaki pagemap build --book alg-viii
bourbaki extract       --book alg-viii
bourbaki assemble      --book alg-viii

A scan is four, and the middle two are the ones that take days:

bourbaki pagemap build --book alg-i-iii
bourbaki render        --book alg-i-iii --dpi 300
bourbaki ocr           --book alg-i-iii
bourbaki assemble      --book alg-i-iii

A volume with no text layer at all cannot have its page map read off the file, so on those three pagemap build runs after render and ocr rather than before. That is the third path and it is why Commutative Algebra, General Topology Chapters 5-10 and Algèbre chapitre 10 are the last volumes anybody should pick up.

The rest is corpus wide and runs after any volume changes:

bourbaki tags assign && bourbaki tags merge && bourbaki tags verify
bourbaki refs build
bourbaki report readme --write
bourbaki audit --report reports/audit.md

assemble writes the section files, the exercise files and the two manifests in one pass, and sweeps whatever an earlier split left behind. assemble --check is the same run with the writing taken out and a diff put in its place, which is what CI runs.

OCR runs against a small fleet of hosts over SSH. Round trips are slow, roughly 150 seconds a call, so every long stage is resumable and safe to interrupt.

Where the numbers are

reports/ is generated and checked by CI, so it is the state of the corpus rather than a snapshot somebody remembered to update. audit.md lists every finding by file and line. extraction-quality.md says how much of each volume has been read and how much of that passes the rules. refs-unresolved.md and out-of-corpus.md are the references that resolve to nothing and the ones that point outside the corpus.

The audit is 73 rules in nine groups: 12 structure, 9 tags, 13 mathematics, 6 figures, 3 references, 16 translation, 6 solutions, 1 publication and 7 hygiene. 59 of them are hard, which means a finding fails the build, and 14 are soft.

what-it-cost.md and reproduction.md are the two written by hand. The first says what the transport cost, which checks caught what, and what is still wrong, including the numbers that do not flatter the project. The second is the record of cloning both repositories fresh and running the pipeline from the PDFs to see whether it produces what is committed: 2775 of 2868 pages of the six deterministic volumes came back byte for byte, and everything derived from the pages came back identical.

Licence

Transcriptions, translations and solutions here are derived from copyrighted material and exist for personal study only. Éléments de mathématique is copyright N. Bourbaki and its publishers. No source PDF is distributed.