Mathematical Writing — Chapter-by-Chapter Outline

Authors: Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts First published: 1989 (based on Stanford CS 209 course, autumn quarter 1987; Stanford Technical Report STAN-CS-88-1193 circulated in 1988) Edition covered: First and only edition. Published by the Mathematical Association of America as MAA Notes, Series Number 14 (ISBN 0-88385-063-X). 115 pages. The book is a compiled transcript of the 31 lectures given by Knuth plus handouts distributed during the course; it is organized into 43 numbered sections (§1–§43) rather than conventional chapters. This outline treats each section as a chapter.

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Central thesis

Good mathematical and technical writing is a learnable craft — not a gift — governed by a set of specific, teachable principles about sentence construction, notation, structure, typography, and the relationship between author and reader. The book argues that writers in mathematics and computer science face problems peculiar to their disciplines (managing notation, presenting algorithms, handling symbols alongside prose, surviving copy editors and referees) that are rarely taught and must be addressed deliberately.

The overarching claim is that the reader's experience must come first. Every decision — which pronoun to use, when to display a formula, how to name a variable, what to say in a cover letter to a journal — should be made by asking: what does the reader know, and what does the reader need to know next? This principle subsumes all the individual rules.

What does the reader know so far? What does the reader expect next and why?

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Section 1 — Minicourse on Technical Writing

Central question

What are the most important, immediately actionable rules a mathematical writer should internalize before writing a single sentence?

Main argument

Separating symbols and words. Knuth opens with the rule that symbols in different formulas must be separated by words. "Consider Sq, q < p" is bad; "Consider Sq, where q < p" is good. Sentences must never begin with a symbol: "xn − a has n distinct zeroes" is wrong; "The polynomial xn − a has n distinct zeroes" is right. The symbols "…", "⇒", "∀", "∃" should be replaced by their word equivalents except in works on logic.

Theorem-statement hygiene. The statement just before a theorem should be a complete sentence or end with a colon. Theorem statements should be self-contained, not dependent on prior text. Every variable must be defined, at least informally, on first introduction.

The first-person and tense. Use "we" to mean "you and me together" — author and reader thinking jointly — rather than as a formal substitute for "I". Use present tense for timeless mathematical facts; reserve sequential past/future for lengthy papers where temporal orientation helps.

Rhythm and flow. Sentences have rhythm. Read what you have written at normal speed; change wording that does not flow. Vary sentence structure and word choice to avoid monotony; use parallelism where parallel ideas are expressed. Avoid "sticky" words (unusual or polysyllabic terms) in consecutive sentences.

Notation economy. Do not use the same notation for two different things. Avoid excessive subscripts: if a set X = {x₁, …, xₙ} is going to require subsets, the subscripted elements will cascade into subscripted subscripts. Give reference numbers only to the most important displayed formulas. Display important formulas on lines by themselves, never letting sentence flow depend on mid-line formula breaks.

Opening sentences. The worst way to start is "An x is y" — e.g., "An important method for internal sorting is quicksort." Better: "Quicksort is an important method for internal sorting, because…." The opening paragraph should be your best; a smooth beginning carries readers past later lapses.

Punctuation with formulas. Periods go inside parentheses only when the entire sentence resides inside. Commas and periods go inside quotation marks by convention, but brackets and parentheses obey strict logic. Colons should not precede a formula that merely completes the sentence.

Small numbers and capitalization. Spell out small numbers used as adjectives; use numerals for numbers as objects. Capitalize "Theorem 1," "Lemma 2," "Algorithm 3."

Key ideas

• Symbols in different formulas must be separated by words; never start a sentence with a symbol.
• Replace logical symbols (∀, ∃, ⇒) with words except in pure logic papers.
• Theorem statements should be self-contained; state all needed hypotheses locally.
• Use "we" to mean the author and reader thinking together; avoid both pure "I" and the passive voice.
• Keep notation consistent across a paper; avoid naming set elements unless subscripts are necessary.
• Read your prose at reading speed to detect rhythm problems; change wording that does not flow smoothly.
• The opening sentence is the most important sentence; never open with "An x is y."
• Vary sentence structure; but use tight parallel construction for parallel ideas.

Key takeaway

The first section provides Knuth's core checklist of 27 specific rules that cover notation, grammar, flow, and organization — the practical foundation on which all subsequent discussion rests.

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Section 2 — An Exercise on Technical Writing

Central question

What does "stylistically atrocious but mathematically correct" look like in practice, and how should it be rewritten?

Main argument

Knuth presents a sophomore term paper proof of the result "L(C, P) ⊆ Aₙ implies C, P ⊆ Aₙ" — a result about nonincreasing vectors in Nⁿ. The original proof is a series of unlabeled implications, symbols used as sentences, and proof steps with no verbal connective tissue. The rewrite introduces full sentences, defines every object formally on first use (giving Aₙ as the set of "nonincreasing" vectors and L(C, P) as the smallest closed set of a certain type), and frames the lemma as a self-contained statement. The rewrite demonstrates rules 1, 2, 4, 5, 11, 12, 13, 14, 15 from §1 in a single concrete example. Readers are asked to complete the proof beautifully.

Key ideas

• A mathematically correct proof can still be unreadable if all explanatory prose is stripped away.
• Defining the same object in two complementary ways (formal set-builder notation plus an English gloss like "nonincreasing") reinforces understanding.
• L(C, P) should be introduced as "the smallest subset of Nⁿ that contains C and is closed under addition of elements of P" — an English characterization alongside the symbolic definition.

Key takeaway

The exercise forces readers to apply §1's rules to a real piece of mathematics, making the transition from abstract rule to concrete practice.

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Section 3 — An Answer to the Exercise

Central question

What is the cleanest way to prove Lemma 1, and what does the proof reveal about proof strategy in general?

Main argument

The worked proof. Knuth presents his solution: assume L(C, P) ⊆ Aₙ; since C ⊆ L(C, P), we immediately get C ⊆ Aₙ. For P: suppose some (b₁, …, bₙ) ∈ P fails to be in Aₙ, i.e., bᵢ < bⱼ for some i < j. Taking k = cᵢ − cⱼ + 1 leads to a contradiction via inequality (3). The proof inserts "navigation" phrases — "Since C is nonempty, it contains…," "We want to show that this leads to a contradiction" — to keep the reader oriented.

The direct proof is better than proof by contradiction. The first proof uses contradiction. But the direct proof — "let (b₁, …, bₙ) be any element of P; we wish to prove bᵢ ≥ bⱼ…" — is cleaner, avoids stating c₁ ≥ … ≥ cₙ explicitly, and works for non-integer components. The lesson: proof by contradiction is natural when discovering an argument, but direct proofs are usually better exposition.

The role of the empty-set hypothesis. The condition C ≠ ∅ is necessary and must be stated; the proof fails if C is empty.

Key ideas

• Proof by contradiction is natural for discovery but often suboptimal for exposition; a direct proof is usually cleaner.
• Navigation phrases ("Since C is nonempty…", "We want to show a contradiction…") are essential guide-rails for the reader.
• The lemma's hypothesis C ≠ ∅ is an easy-to-miss technical necessity; surface such conditions explicitly.
• The alternative to the sequence of implications is a paragraph that reads as narrative.

Key takeaway

A correct proof is not automatically a good proof; exposition of a proof requires deliberate choices about structure, proof strategy, and reader guidance.

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Section 4 — Comments on Student Answers (1)

Central question

What patterns of error emerge when students first try to apply the §1 rules to mathematical prose?

Main argument

Excerpts from class, October 7. Knuth reviews anonymous student solutions. Key observations: (1) Not all formulas are equations — depending on content, call it a "relation," "definition," "statement," or "theorem." (2) Mathematical notation and programming notation differ; a star (∗) for multiplication is wrong in a math paper. (3) Distinguish "element of P" from "component of p" — precision of subordinate relationships matters. (4) Ellipses need commas before and after: write "(P₁, …, Pₙ)" not "(P₁...Pₙ)". (5) Line breaks in the middle of formulas are undesirable; TEX commands like \mid and \emptyset help. (6) Comments like "We demonstrate the second conclusion by contradiction" are useful navigation. (7) Numbering all displayed formulas is usually a bad idea; number the important ones only. (8) Parentheses around a quantity already delimited by context are extraneous: "let k be (cᵢ − cⱼ) + 1" should drop the parentheses.

Personal pronoun debates. Knuth prefers "we" for author-and-reader over "I," but the class discusses both views. For tense: either use present throughout, or write sequentially ("We saw this before; we will see this later").

Key ideas

• "Not all formulas are equations": precision in naming formula types matters.
• Computer science notation (array brackets, stars) is inappropriate in formal math papers.
• Ellipses in sequences require commas: "(P₁, …, Pₙ)" not "(P₁...Pₙ)".
• Navigational comments ("by contradiction," "we need only show") guide the reader through proof structure.
• Number only the most important displayed formulas.

Key takeaway

Common first-attempt errors fall into a few predictable categories: notation leakage from programming, missing connective tissue, and misused punctuation around symbols.

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Section 5 — Comments on Student Answers (2)

Central question

What form-level and content-level problems persist in student answers, and what principles govern their correction?

Main argument

Excerpts from class, October 9. Knuth continues reviewing homework. On form: overuse of colons (a colon after "We have:" before a formula that completes the sentence should be omitted); capitalization after a colon (yes if what follows is a full sentence, no otherwise); comma placement (too few commas can make sentences hard to parse; "Observe that if x is y then so is z because…" needs a comma before "because"). Excessive parentheses become distracting. On content: the use of "I" in scholarly writing (Knuth prefers "we"; van Leunen defends "I"); two approaches to tense — either present throughout, or sequential with "we saw / we will see"; Knuth's rewrite sequence (pencil first, type for flow, then TEX, then revisit).

Key ideas

• Colons should only appear before completions that do not continue the sentence mid-stream.
• Comma placement is not purely aesthetic; missing commas before "because" or "if" clauses cause parsing difficulty.
• Too many parenthetical asides within a sentence break reading flow.
• Two defensible tense strategies: uniform present tense, or explicit sequential narration.

Key takeaway

Form errors — misplaced colons, erratic commas, excessive parentheses — are distinct from content errors and must be addressed systematically.

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Section 6 — Preparing Books for Publication (1)

Central question

What does the book-production process actually look like, and how do authors collaborate with designers and copy editors?

Main argument

Excerpts from class, October 12. Knuth demonstrates Before/After pages from Concrete Mathematics (co-authored with Graham and Patashnik). The designer determined text width, fonts, chapter headings, and introduced "graffiti" margins — informal snappy comments from secondary voices, inspired by Approaching Stanford. The key point: writing for a book differs from writing a paper; the author learns what the book's style is only by writing enough of it, just as a novelist discovers characters by writing the story.

The revision sequence. Knuth's own process: first draft in pencil (hand speed matches thought speed); typing edits for flow; TEXing; further revisions. Nils Nilsson had five drafts of his non-monotonic reasoning chapter; Tony Hoare revised his CSP algorithm a dozen times over two years. "Any writing can be improved. But eventually you have to put something out the door."

Font choice. The Euler typeface by Hermann Zapf is used in Concrete Mathematics: it looks slightly handwritten, and its zero has a point at the apex because "when people write zeros, they never really close them." This creates a subtle distinction between mathematical numerals and English-language numerals.

Key ideas

• Book designers and copy editors are professionals whose expertise pays; authors who self-typeset must still engage them.
• The "graffiti margin" technique — informal commentary running alongside formal text — adds a secondary expository voice.
• The book's style emerges through the writing process itself; you discover what fits as you write more.
• Pencil composition often matches thinking speed better than keyboard composition.

Key takeaway

Publishing a book involves a team — author, designer, copy editor, typesetter — whose roles interact; understanding this process helps authors make better decisions about when and how to revise.

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Section 7 — Preparing Books for Publication (2)

Central question

What specific line-by-line editing decisions produce clearer mathematical exposition?

Main argument

Excerpts from class, October 14. Continuing with Before/After pages from Concrete Mathematics: (1) Reorder for the reader's expectations — if readers will wonder why a rule is true, prove it before stressing its significance. (2) Pronoun antecedents must always be unambiguous; "It is a big convenience" where "it" could refer to several things should be recast. (3) False equations displayed on their own line are dangerous — readers' eyes land on them and read them as gospel; embed false equations inside text. (4) Hyphens, en-dashes, minus signs, and em-dashes are four distinct marks with four distinct uses. (5) Fractions with a horizontal bar (the stacked form) should be replaced by slashed fractions in running text to avoid tiny numerals. (6) Exercises are the hardest writing to do unambiguously, since they have minimal context. (7) Use present tense for timeless facts; past tense is appropriate only for results that turned out to be wrong.

Key ideas

• Anticipate what the reader wonders at each point; answer questions in the order they arise.
• Never display false equations on their own line; they will be misread as true.
• Four typographic mark lengths: hyphen (word joins), en-dash (ranges "10–18"), minus sign (arithmetic "10 − 18"), em-dash (sentence interruption).
• Stacked fractions in running text produce tiny numerals; slashed fractions are generally cleaner.
• Exercises require extra clarity because they lack surrounding context.

Key takeaway

Fine-grained editing decisions — pronoun clarity, formula display, typographic mark choice — cumulatively determine whether a reader trusts and understands a text.

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Section 8 — Preparing Books for Publication (3)

Central question

What stylistic principles apply to formulas, variables, and the later stages of book production?

Main argument

Excerpts from class, October 16. When long formulas don't fit on a line, break them logically — before the equals sign in a displayed formula, after the equals sign in an inline formula. Moving a formula from embedded text to a displayed line sometimes enables cleaner logical division. Semicolons slow reading; periods speed it. Conjunctions at the start of a sentence are not evil. Variable names must not be misleading; systematic renaming is a text-editor's advantage. Copyright law now requires crediting previously published exercises; tracing theorem origins is difficult (Knuth spent four hours searching Lagrange's collected works for "Lagrange's inequality").

Key ideas

• Line-break position in formulas is not arbitrary: break before "=" in displayed context, after "=" in inline context.
• Periods speed reading; semicolons slow it — choose consciously.
• Copyright has changed: credit published exercises; trace theorem origins carefully.
• Consistent variable naming across co-authors requires deliberate coordination.

Key takeaway

Production-stage decisions about line breaks, punctuation, and citation are as much authorial decisions as early drafting choices.

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Section 9 — Handy Reference Books

Central question

What reference shelf does a serious mathematical writer need?

Main argument

Excerpts from class, October 16 (continued). Knuth presents seven books he keeps at his desk: (1) The Oxford English Dictionary (two-volume compact edition) — historical and etymological authority. (2) The OED Supplement (four volumes) — updates OED to current usage. (3) The American Heritage Dictionary — usage notes and Indo-European root appendix help with "compare to" vs. "compare with." (4) The Longman Dictionary of Contemporary English — current slang and street usage, with nuances like "mind-bending" vs. "mind-blowing" vs. "mind-boggling." (5) Webster's New Word Speller Divider — for those who spell poorly. (6) Roget's Thesaurus — Knuth owns two (home and office), uses them to recall forgotten words, avoid repetition, and name new technical terms. (7) Webster's Dictionary of English Usage (1989) — goes far beyond American Heritage usage notes; filled with examples.

American vs. British spellings. The rule: use your own country's conventions. Mix only when explicitly referring to the other convention (an American writing about the British Labour Party).

Key ideas

• The OED is the ultimate arbiter of historical usage; the Supplement covers newer words.
• Roget's Thesaurus serves three distinct purposes: recalling forgotten words, avoiding repetition, and naming new concepts.
• Usage dictionaries (American Heritage usage notes, Webster's Dictionary of English Usage) resolve questions that regular dictionaries leave open.
• American and British spelling conventions should not be mixed within a single text.

Key takeaway

A writer's reference shelf is a professional tool; maintaining it is part of craftsmanship.

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Section 10 — Presenting Algorithms

Central question

How should algorithms and computer programs be typeset and explained in mathematical and CS papers?

Main argument

Excerpts from class, October 19. Mathematical journals struggled to present programs until conventions were established. Algol 60's conventions — indentation, reserved words, mixed fonts — set the modern standard; these were developed by Myrtle Kellington at ACM with Peter Naur. Knuth needed fixed-width type for The Art of Computer Programming Vol. 1; publishers initially said it was impossible.

Knuth's own algorithm presentation. He presents algorithms as Theorems with labeled steps, using high-level English (not a specific programming language) with optional flow charts. Invariant relations appear as parenthetical remarks after 1968. He confesses the style "must just be a personal style that works for me — get a personal style that works for you." Recent papers use the pidgin Algol style of Aho, Hopcroft, and Ullman.

The program-as-literature thesis. "A computer program is a piece of literature." Treating programs literarily forces the programmer to get them right, because you have to think clearly about them. "I look forward to the day when a Pulitzer Prize will be given for the best computer program of the year."

Introduction of WEB/literate programming. The WEB system (first described in a British Computer Journal paper) lets one source file produce either a typesetting file or a Pascal source file. This is the segue into the next two sections.

User manuals preview. The section closes with a preview: user manuals are the "blight on the industry" — the next topic.

Key ideas

• Typographic conventions for programs originated with Algol 60; all modern conventions descend from these.
• Choose an algorithm presentation style that prioritizes the reader; stick to it consistently across a body of work.
• Algorithms in English (rather than a specific language) avoid alienating readers who don't know that language.
• Programs should be treated as literature — clear, motivated, readable by a thoughtful person.
• WEB allows a program to exist as a document first and executable code second.

Key takeaway

Presenting algorithms is a distinct and demanding genre with its own conventions; there is no universal correct style, but every choice must serve the reader's understanding.

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Section 11 — Literate Programming (1)

Central question

What is WEB, and how does the literate programming methodology change the relationship between code and explanation?

Main argument

Excerpts from class, October 21. Knuth shows WEB programs written by undergraduates. The central WEB philosophy: each module is introduced at the "psychologically right moment" — the natural order for reading and writing are the same. The TANGLE processor converts WEB programs to Pascal; WEAVE converts them to TeX. TEX: The Program has almost 1400 modules.

Module-naming conventions. Module titles should use imperative form: "Store the word in the dictionary" rather than "Stores the word…". Declarative modules (lists of declarations) use descriptor form: "Procedures for sorting." Capitalize only the first word of a module title (newspaper-headlines convention). Compound identifiers should use WEB's underscore shorthand (get_word rather than getWord).

Common student errors. Students who failed to change their thinking produced WEB programs that looked like ordinary Pascal with comments added. The correct approach breaks every complex task into tractable named pieces, each motivated by a paragraph of high-level explanation. Module lengths should be around a dozen lines of code. Variables declared in separate modules should still be close to their use; large procedures should have their local variables "distributed" as exposition proceeds.

Anthropomorphization. Knuth prefers "I" for the computer in user interfaces ("What file should I process?") and "we" for the author-and-reader voice in program comments; Dijkstra disagrees.

Key ideas

• WEB's central principle: introduce each module at the psychologically right moment for the reader.
• Module titles should be imperative (for active procedures) or descriptive (for declarations), capitalized like newspaper headlines.
• Each module should carry a paragraph of high-level explanation, not just a list of steps.
• About a dozen lines of code per module is the right target for tractability.
• The natural order for writing a program is also the natural order for reading it.

Key takeaway

Literate programming requires a fundamental shift in thinking: the program text is a document first and executable code second, organized for human readers rather than compilers.

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Section 12 — Literate Programming (2)

Central question

What advanced issues arise when writing real literate programs, and how do WEB's macro features help?

Main argument

Excerpts from class, October 23. A key WEB advantage: error-handling code can be placed in a separate module ("Check if the data is wrong") pointed to by a single line, keeping the main module focused on normal operation. Variable names should reflect function: "villains" for inconsistent data entries, enabling coherent exposition in later modules. WEB macros simulate two-parameter macros even though the base system allows only one parameter. The stringtype(#) macro example: `stringtype(#) ≡ packed array [1..#] of char` allows cleaner variable declarations.

Style details. Don't import conventions from old styles into a new one (e.g., adding end {while} comments to WEB is unnecessary because module structure already makes scope clear). Typewriter type for hexadecimal numbers provides "subliminal type-checking." Words in documentation must match variable names in code: using two terms for the same concept confuses the reader. Verbs name procedures; nouns name data.

Quotes vs. opening-quote/closing-quote. Computer typesetting distinguishes opening quotes (") from closing quotes ("); a typewriter's single quote character must be handled carefully to produce correct typographic output.

Key ideas

• Separate error-handling code into dedicated modules so the main module reads naturally.
• Variable names should be expressive enough to support coherent prose narration ("villains," "cut throats").
• Documentation and code must use the same vocabulary.
• Verbs name procedures; nouns name data — a principle that can guide naming decisions at design time.
• WEB macros extend Pascal's expressiveness for readable module titles and type declarations.

Key takeaway

WEB works best when the programmer designs module names and variable names with the eventual documentation in mind from the start.

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Section 13 — User Manuals

Central question

What distinguishes a good user manual from a bad one, and what do typical failures look like?

Main argument

Excerpts from class, October 26. Knuth presents four student-written manuals for the "Digiflash" dot-matrix display system, reviewed by his wife (the novice user test). Common failures: (1) jargon the author never notices ("Menu," "Scrolling" were mysterious to non-CS users); (2) formal definitions instead of task-based instructions; (3) failure to explain how to exit or recover from errors; (4) inconsistent capitalization (MESSAGES vs. MESSAGEs); (5) options with non-intuitive key assignments (OPTION-B for bold/unbold rather than OPTION-N for normal).

The best manual. The fourth team's manual — though not even typeset — had a good overview, step-by-step descriptions, friendly tone, short sentences, helpful redundancy, diagrams, and icons for directional scrolling. Its flaw: "up" meant "previous message," which was non-obvious. Best principle: "If you try to write for the novice, you will communicate with the experts — otherwise you communicate with nobody."

The AWK book. The AWK Programming Language (Aho, Weinberger, Kernighan) explicitly aimed to write for the absolute novice but failed because its authors had been in the field so long they couldn't recognize their own jargon. Still probably "the best book in Computer Science published this year."

Key ideas

• User manuals are written for people who don't share the author's vocabulary; test with a truly naive reader.
• Formal definitions are the wrong approach for novices; task-based instructions work better.
• Always explain how to recover from errors; this is the question users most often need answered.
• Key assignments should follow intuitive mnemonics.
• "If you try to write for the novice, you will communicate with the experts."

Key takeaway

User manual failure is almost always a failure of perspective: authors who know the system cannot recognize what their users don't know, so outside testing is essential.

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Section 14 — Galley Proofs

Central question

What happens at the galley-proof stage of journal publication, and how should an author handle it?

Main argument

Excerpts from class, October 26 (continued). Knuth shows actual galley proofs received from ACM with cryptic marginal notes: "Bad proof, Camera copy OK" means the copy was laser-printed rather than phototypeset. "Au" means "author" in copy-editing shorthand. The copy editor substituted "cleverer" for "more clever" (citing Fowler); Knuth noted that Scientific American had once changed "more common" to "commoner." The publisher at the time re-types all papers; only a few AI journals had adopted author TEX files. Proof-readers' symbols provide a standardized language for corrections. Publishers sometimes require more bibliographic detail (exact conference location and date); Knuth notes that nearly half the information in conference bibliographies is wrong, citing Berlekamp's finding from his coding-theory book.

Key ideas

• Galley proofs must be returned quickly; check not only prose but also technical correctness of mathematics.
• Copy-editor changes may be wrong (even grammatically motivated ones); authors have the right to revert.
• Bibliographic information degrades by copying; verify citations against primary sources.
• Many publication errors only appear at the page-proof stage (widow lines, misplaced subheadings).
• Standard proofreaders' marks provide efficient author-editor communication.

Key takeaway

Galley proofs are a final chance to correct errors introduced in typesetting; treat them seriously and return them promptly.

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Section 15 — Refereeing (1)

Central question

What is the referee's role, and what principles should guide refereeing well?

Main argument

Excerpts from class, October 30. The conventional view (Forscher's "Rules for Referees," 1965) treats refereeing as a legal duty to editors. Knuth disagrees: referees' primary responsibility is to authors and readers, not editors. Referees should act as teachers. A referee who says "incredibly poorly written" and nothing more teaches nothing; constructive criticism makes the next paper better. Knuth has contributed results anonymously to papers — "My thanks to the referee for Theorems 4, 5, and 6." Referees should also supply references to related work.

Calibrating referees. In the 1960s Knuth sent the same paper to multiple referees to calibrate their judgment. The paper was a badly written exposition of an interesting algorithm; the range of referee responses — from line-by-line complaints, to general structural critiques, to a recommendation for publication — was illuminating.

Choosing a journal. Look for the journal with the best referees, not the least critical editor. An author who genuinely wants to know if his work is a contribution should seek rigorous feedback.

The "Toilet Paper Problem." Knuth's famous paper on random toilet-paper selection was first submitted to the Monthly under editor Halmos. Halmos cautioned that "jokes are dangerous in our journal" and asked Knuth to reconsider scatological section titles; Knuth changed the section headings but kept the title. Reply: "Your toilet paper is accepted."

Key ideas

• Referees' primary duty is to authors and readers, not to editors.
• Act as a teacher: identify what the author needs to learn, not just what is wrong.
• Supply missing references; offer technical improvements anonymously if appropriate.
• Journals with rigorous referees serve authors better than journals with lenient editors.
• Humour in technical papers carries genuine risk; submit such work cautiously.

Key takeaway

Refereeing is a form of teaching and a cooperative institution for raising standards; it should be practiced accordingly, not treated as a bureaucratic gatekeeping function.

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Section 16 — Refereeing (2)

Central question

How does the refereeing process play out in practice, including its benefits and pathologies?

Main argument

Excerpts from class, November 2. Knuth submitted a Number Theory paper to the Journal of Number Theory under the pseudonym Ursula N. Owens (an Agatha Christie allusion — "U.N. Owen") to test whether his famous name biased referees. The referee found typographic errors, notation suggestions, and places where the presentation was misleading. More importantly, the referee significantly improved one theorem anonymously. Knuth shows referee reports on his own paper: two in favor, one against; they found missing references to earlier work in the field, caught subtle technical errors, and helpfully noted sections they enjoyed. Facile generalizations — mechanical manipulations of arguments that add no new insight (Pólya's term) — are not improvements. The refereeing system is "a cooperative effort — a correspondence between tens of thousands of world authorities."

Key ideas

• Submitting under a pseudonym reveals what a paper earns on its own merits, undistorted by the author's reputation.
• A good referee may improve the actual mathematical content, not just the exposition.
• Missing references to prior work is one of the most common and valuable things a referee can supply.
• Beware of facile generalizations — mechanical extensions that add no new insight.
• Refereeing is a cooperative global knowledge-building institution, not just gatekeeping.

Key takeaway

The referee relationship is collaborative: the best outcomes occur when referees see themselves as junior co-authors improving a manuscript rather than judges passing verdict.

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Section 17 — Hints for Referees

Central question

What concrete, numbered advice captures best practices for writing referee reports?

Main argument

This section is Knuth's actual "Hints for Referees" document, originally circulated in the 1960s and rediscovered just as the class needed it. The handout provides specific procedural and ethical guidance for referees: how to evaluate originality, how to give constructive criticism, what to do if the content is wrong vs. merely badly written, and how to supply missing references. The document was apparently revised by others after it left Knuth's hands; he no longer recognizes the style of all the comments. (The section in the PDF consists primarily of the handout itself, which is reproduced verbatim as a reference document.)

Key ideas

• Originality is the first criterion; is this a genuine advance?
• Distinguish bad content (worth only a brief negative) from bad writing around good content (worth detailed criticism).
• Constructive criticism outlasts a rejection; the author will write again.
• Anonymity is a privilege that should not be abused for tasteless personal attacks.

Key takeaway

Knuth's Hints for Referees document crystallizes the ethical and practical norms of the referee relationship into actionable guidelines.

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Section 18 — Illustrations (1)

Central question

How should mathematical writing use diagrams, charts, and figures effectively?

Main argument

Excerpts from class, November 4. Knuth begins with a biographical aside: his first publication was "The Potrzebie System of Weights and Measures" in MAD magazine (1957), for which he received $25. His second published illustration of note appears in The TEXbook. His RUNCIBLE paper (CACM, 1959) contained what he admits is "probably one of the worst spaghetti flow charts ever drawn," with an error (a misdirected arrow) and no mention of co-contributors.

Diagrams are good when kept small. A fold-out syntax chart for Algol 60 conveys a useful overview despite not being small. The line-breaking algorithm paper with Plass (Software — Practice & Experience, 1981) used a graph where each node on level p corresponds to a word after which the pth line might be broken, with edges labeled by demerits. This reduces line-breaking to finding a least-cost path — a well-understood search problem. The explanation is opaque without the diagram.

Tufte and Huff. Knuth recommends Tufte's The Visual Display of Quantitative Information and Huff's How to Lie with Statistics (two sacks drawn as volumes to exaggerate a 17% increase look like a 58% increase because visual volume scales as the cube).

Real data over sample data. Use a paragraph from Grimms' Fairy Tales as test data rather than data cooked up for the example; real data is more interesting and more credible.

Key ideas

• Diagrams reduce cognitive load for structural arguments (graphs, algorithms, sorting sequences); use them when a verbal account becomes opaque.
• Keep figures small if possible; large fold-outs are justified only when the overview they provide is impossible otherwise.
• Use real data (actual fairy-tale text, actual city distances) rather than contrived examples.
• The amount of work involved in preparing a paper is proportional to the cube of the number of illustrations.
• Tufte's book is the key reference for quantitative visualization principles.

Key takeaway

Illustrations earn their cost only when a structural idea genuinely cannot be conveyed without them; choosing and constructing them carefully is as much part of authorship as prose.

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Section 19 — Illustrations (2)

Central question

How does The Art of Computer Programming use tables, charts, and numerical data, and what can be learned from its choices?

Main argument

Excerpts from class, November 6. The class examines illustrations from TAOCP Vol. 3. Numerical tables are "sometimes thought to be unenlightening" but can show patterns — like numbers oscillating about 2 with period 2π — that cannot be captured graphically. A large fold-out illustration compares performances of sort-on-tape algorithms, revealing that certain patented sophisticated algorithms are slower than traditional ones due to unanticipated rewind times. TAOCP contains a running example of sorting 16 particular numbers by dozens of different algorithms, each leading to a different graphical representation across 25 pages (pp. 77, 82, 84, 97, 98, 106, 110, 113, 115, 124, 140, 143, 147, 151, 161, 165, 166, 172, 175, 205, 251, 253, 254, 359). Color is effective in talks but rarely in papers. Make sure final text is placed opposite the diagrams it references.

Key ideas

• Numerical tables are a legitimate visualization when oscillation and period patterns cannot be shown graphically.
• A single fold-out comparison can definitively settle performance questions that algorithmic analysis alone cannot.
• Running a single example through many different algorithms across a book provides cumulative intuition.
• Text describing a figure should appear opposite the figure, not pages away.

Key takeaway

Tables and charts are complementary tools: charts reveal structure and trends, tables reveal precise values and subtle patterns — skilled authors choose the right one.

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Section 20 — Homework: Subscripts and Superscripts

Central question

Can Gillman's already-simplified proof of Sierpiński's theorem be simplified further, eliminating subscripted subscripts and subscripted superscripts?

Main argument

This section presents the homework assignment itself. Gillman's version of Sierpiński's theorem involves a lemma establishing a bijection between reals and pairs of sequences (⟨nₖ⟩, ⟨tₖ⟩), followed by a theorem asserting the existence of functions fₙ: ℝ → ℝ such that for any uncountable S ⊆ ℝ, fₙ(S) = ℝ for all but finitely many n. The proof as given by Gillman already simplifies Sierpiński's original (which contained an error found after publication), but still has subscripted superscripts (nₖᵅ). The assignment: recast the proof so the same mathematical idea comes through with no double subscripts. Knuth notes: "The theorem implies the Continuum Hypothesis — it's not necessary to understand the proof completely to do this assignment."

Key ideas

• Double subscripts and superscripted subscripts are among the hardest notation problems; minimizing them is a concrete goal.
• Sierpiński's original proof contained an error (found post-publication) because its notation was so complex it confused even its author.
• The Continuum Hypothesis (neither provable nor disprovable in ZFC) is the mathematical context: Gödel (1938) showed consistency; Cohen (1963) showed independence.
• "Hold back" on enumerating set elements: say "S is countable" without immediately writing S = {s₁, s₂, …}.

Key takeaway

Notation is not just cosmetic: overly complex subscripting can cause mathematical errors; the assignment proves this by redesigning a historically error-prone proof.

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Section 21 — Homework: Solutions

Central question

What strategies — beyond Gillman's version — produce cleaner proofs of Sierpiński's theorem?

Main argument

Knuth presents four student solutions. Solution A (his own draft): replaces the pair of sequences ⟨nₖ⟩, ⟨tₖ⟩ with a set N and a sequence (α₁, α₂, …) — "hold back" on set enumeration avoids subscripted superscripts. Solution B (a student): recasts the lemma as a bijection to sequences of ordered pairs ⟨(nₗ, tₗ)⟩ₗ≥₁ — this factors the proof into two parts with a clean interface (a subgoal and a final step), and the word "counterpart" replaces notational correspondence. Solution B is easier to follow because it describes the construction as "essentially a diagonalization technique." Solution C: replaces subscripted superscripts with plain English words ("the integer sequence of α" and "the real sequence of α"). Solution D: presents seven illustrations (some in four colors) showing set membership relations; conceptually powerful but cannot be shown in full. Knuth notes: "A combination of the ideas from all these solutions would lead to a truly perspicacious proof."

Key ideas

• Holding back on set enumeration avoids subscripted subscripts.
• Factoring a proof into two parts with a clean interface (subgoal + conclusion) reduces what the reader must keep in mind simultaneously.
• Plain English words often substitute for notational conventions, making proofs more readable without sacrificing precision.
• "Essentially a diagonalization technique" is an orientation statement that repays its cost many times over.
• Illustrations can sometimes replace notation entirely — but require careful design.

Key takeaway

Multiple solutions to the same notational problem reveal that no single approach is universally best; combining techniques produces the clearest overall proof.

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Section 22 — Quotations

Central question

How should mathematical writers find, select, and use quotations in their work?

Main argument

Excerpts from class, November 9. Knuth confesses his "secret delight" in playing a library "as if it were a musical instrument." TEXbook has two quotations at the end of each of 27 chapters, 10 appendices, and a preface; METAFONT posed the same challenge. Strategy: (1) Bartlett's Familiar Quotations (PN 6000 in reference section) — Knuth found the Leonard Bacon quote on Technique under the heading "technique," which matched TEX/τέχνη (Greek for both "technique" and "art"). (2) The OED — lists memorable quotations alongside each word's history; excellent for finding quotations involving specific words. (3) Concordances of Shakespeare or Chaucer — find every use of a given word. (4) Computer search of online texts (Grimms' Fairy Tales and Wuthering Heights on SAIL). (5) Quote yourself in desperation; even ask someone famous to say something.

The "God is in the details" mystery. Knuth has been trying to trace this aphorism for years. It is attributed to Mies van der Rohe, but no one heard him originate it. The most likely trail: Aby Warburg used "Der liebe Gott steckt im Detail" as a seminar motto at Hamburg in 1925; Mies learned it in Germany. But its true origin in Flaubert or Nietzsche cannot be ruled out until their complete works are available electronically.

Key ideas

• Quotation dictionaries (Bartlett's) are indexed by subject, not just author — use the index to find quotations on a topic.
• The OED lists quotations as part of its citation history; excellent for word-specific searches.
• Computer full-text search of online literary texts enables quotation hunting by keyword.
• Quote out of context with grace: an economist's remark about "output" fits a chapter on computer output.
• The best quotations illuminate a technical point that only someone who understands it will appreciate.

Key takeaway

Finding the right quotation is a research skill requiring library literacy, serendipity, and creative out-of-context reading; its payoff in expository impact is disproportionate to the effort.

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Section 23 — Scientific American Saga (1)

Central question

What happens when a mathematical author's carefully crafted prose is edited by a general-audience magazine, and how should the author respond?

Main argument

Excerpts from class, November 11. In fall 1975, Scientific American's editor Dennis Flanagan invited Knuth to write on Algorithms for 600,000 readers at $500 (about 8 cents per word). Knuth submitted his manuscript saying "it will be interesting to see what you do to this, my masterpiece." Fourteen months later, he received an edited copy "edited for the general reader" that had rewritten every sentence. Knuth wrote to Martin Gardner that the editing "took perfectly good English and turned it into something worth no more than a B− on a high school term paper." Specific changes: all "we"s changed; some long sentences fragmented, short ones fused; commas removed; "which" changed to "that"; jargon removed; "most common" changed to "commonest"; errors introduced. The error introductions were, perversely, useful: they showed where Knuth's exposition had been insufficiently clear.

Key ideas

• General-audience editing often removes technical precision along with jargon; authors should prepare for this.
• Errors introduced by copy editors are diagnostic: they reveal where the original text was ambiguous enough to be misread.
• Seeing page proofs is essential for papers with significant mathematical typography; push for them.
• A 1.5-hour phone call with an editor can resolve many disputed changes more efficiently than correspondence.

Key takeaway

Author-copy-editor conflict is a normal part of publication; the author's goal is not to win every dispute but to produce the clearest final text, which sometimes means accepting changes and sometimes fighting them.

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Section 24 — Scientific American Saga (2)

Central question

How did the Scientific American editorial saga end, and what broader lessons does it teach about writing for general audiences?

Main argument

Excerpts from class, November 13. The page proofs for the April 1977 issue (Knuth had wanted an April slot ever since Gardner's April Fool hoax) required a 1.5-hour phone session. Simple errors: a "1" had become an "l," an "∅" had become a "φ" (typesetters unfamiliar with mathematics are creative with the empty set symbol). SA never displays equations — a policy that drives at least one scientist to refuse to read it. SA stuck by "commonest" over Knuth's "most common"; Knuth eventually conceded that the cooperative process removed jargon and made the paper more accessible to a general audience. The experience directly triggered TEX: "It was in April of 1977 that Don's travails with SA prompted him to investigate typesetting for himself; in May of that year he designed the first draft of TEX."

The Norwegian predecessor. The SA paper grew from a lecture Knuth gave to a non-technical Norwegian audience for Forskningsnytt. Ole Amble, a Norwegian numerical analyst, asked whether binary search and hashing could be combined — Knuth first said "obviously not," then realized what Amble meant, too late for TAOCP Vol. 3. This combination of methods made the conclusion of the SA paper.

Key ideas

• TEX was created directly in response to frustration with typesetting problems during the SA publication process.
• Typesetters unfamiliar with math misread special symbols systematically; page proofs are essential.
• General-audience writing is harder than technical writing; Martin Gardner attributes his success to not being a mathematician.
• A cooperation that begins in conflict can end with a genuinely improved paper.
• The "never display equations" policy of some general-audience outlets severely limits mathematical communication.

Key takeaway

The SA saga illustrates how friction between author precision and editorial accessibility, if navigated constructively, can produce writing that genuinely reaches a wider audience.

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Section 25 — Examples of Good Style

Central question

What do examples of excellent mathematical and technical writing look like in practice, and what makes them good?

Main argument

Excerpts from class, November 16. Knuth distributes handouts of good writing by each upcoming guest speaker as preparation for their lectures. Herb Wilf's "Editor's Corner" piece in the Monthly is a nice exposition of Riemann's Hypothesis. Wilf's paper "n coins in a fountain" illustrates the title-formula dilemma (Knuth would have capitalized N; Wilf eventually published "The Editor's Corner: n Coins in a Fountain"). Leslie Lamport's CACM paper "A simple approach to specifying concurrent systems" is the best technical paper Knuth has seen in the past year — unusual for its question-and-answer format (a dialogue form used before by Socrates, Galileo, Dantzig, Rényi). Knuth's own Surreal Numbers was written in six days as a dialogue in a Norwegian hotel room. Nils Nilsson and Genesereth's chapter on non-monotonic reasoning has an excellent blend of formal and informal discussion with well-chosen examples.

Jellyfish metaphor. Knuth and co-authors used a jellyfish metaphor for a data structure (stems, polyps, tentacles, heads, nematocysts) to manage a very complicated reduction in a complexity paper. Mary-Claire asked whether he would regret not calling it a "stinger" if the structure became widely used. He admitted to having regret about other names (he wishes he had called LR(k) grammars "L(k) grammars").

Key ideas

• A question-and-answer (Socratic) format can present complex formal material more accessibly than standard exposition.
• Biological metaphors for data structures can be powerful for complex reductions, but term choice has permanent consequences.
• Good style examples are worth distributing before teaching them abstractly.
• The dialogue form is almost unused in computer science but has a long honorable tradition in mathematics and philosophy.

Key takeaway

Good style is best learned by reading examples of good style; distributing them before discussing them is a more effective pedagogy than abstract rule-giving.

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Section 26 — Mary-Claire van Leunen on 'hopefully'

Central question

What is the history of "hopefully" as an adverb, and what does its evolution reveal about language change?

Main argument

This section is van Leunen's written essay on the word "hopefully," distributed as a style handout. Van Leunen traces how "hopefully" acquired its impersonal use ("Hopefully, rates will fall" rather than "He said hopefully") and argues carefully about whether this change is good or bad. The essay is included by Knuth less for its content than for its own style — he calls it "a particularly elegant essay" of the kind computer scientists and mathematicians rarely produce. The prose quality exemplifies the kind of writing he wants students to aspire to.

Key ideas

• Language change in adverbs like "hopefully" follows the general pattern of adverbs acquiring impersonal, speaker-attitude functions.
• Evaluating linguistic change requires distinguishing needed distinctions (which should be preserved) from superfluous ones (which can be released).
• The essay's style — clear, elegant, scholarly yet accessible — is the real lesson.

Key takeaway

Van Leunen's essay demonstrates that analytical precision and graceful prose are compatible; technical writers should aspire to both.

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Section 27 — Herb Wilf on Mathematical Writing

Central question

What practical principles for mathematical exposition does an experienced editor of two very different journals offer?

Main argument

Excerpts from class, October 28. Wilf edits the American Mathematical Monthly (expository) and the Journal of Algorithms (research). He is "older without feeling wiser" and uncomfortable setting down rules for "part brain and part hormone system," but offers several principles:

Get attention immediately. Snappy titles, arresting first sentences, lucid initial paragraphs. Gleason's "Trisecting the Angle, the Heptagon, and the Triskaidecagon" — attention-getting partly because Gleason is famous. Hugh Thurston's paper beginning "Can a graph be continuous and discontinuous?" — an immediately engaging paradox.

Get everything up front. "You can quintuple your readership if you will let them in on what it is that you are doing." Potential readers scan for theorem statements; if all text is discursive, they have nothing to latch onto. Use boldface (or "neon") to summarize results for page-flippers.

Motivation in moderation. Examples that do not yield the desired result are useful but lose charm quickly. "Whenever I see 'Consider the following …' I know the author really means 'Here comes something from left field bleachers.'"

Two extremes. Wilf shows two papers on the same combinatorial problem (counting necklaces under cyclic symmetry): a half-page version ("Enumeration of orbits of mappings under action of Cₙ") condensed to "evidently" at a key step, and a four-page version ("Counting Necklaces") that walks through every step. Both are legitimate; he is not saying one is superior.

Three exemplary books. Pólya and Szegő's Problems and Theorems in Analysis (self-contained pieces building to major results); Hardy and Wright's An Introduction to the Theory of Numbers ("short on motivation" but the subject is so attractive that only extravagant incompetence could make it dull); Rudin's Principles of Mathematical Analysis (rigorous; teaches what a proof is; leaves survivors feeling strong).

Key ideas

• Opening sentences and titles are the highest-leverage writing in any paper; invest heavily in them.
• Readers scan for theorem statements; make results visually prominent.
• There are two legitimate extremes of exposition density: the ultra-compressed and the fully elaborated. Know which you are writing.
• Good mathematical writing is "lean" not "dry"; the two are easily confused.
• Too much preliminary motivation annoys readers as much as too little.

Key takeaway

Wilf's contribution is the reader's-eye view from an editor who reads hundreds of papers: open with impact, front-load results, and accept that compression and elaboration are both valid — just be deliberate about which you choose.

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Section 28 — Wilf's First Extreme

Central question

What does maximally compressed mathematical exposition look like in practice?

Main argument

This section reproduces Wilf's short paper "From Acta Hypermathica" — a half-page exposition of the necklace-counting problem compressed to the point where the key symmetry argument is summarized in a single word ("evidently"). The paper represents one extreme of the compression-elaboration spectrum: it is formal, dense, and will only be fully understood by readers who can supply the missing steps themselves. It is intended as a concrete example for comparison with §29's expanded treatment.

Key ideas

• Compression to a single "evidently" can be intellectually satisfying to an expert reader; it is not an error.
• Mathematical writing permits extreme compression when the audience shares the background needed to fill gaps.

Key takeaway

Ultra-compressed exposition has a legitimate place in the literature, provided the audience is correctly targeted.

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Section 29 — Wilf's Other Extreme

Central question

What does fully elaborated mathematical exposition of the same problem look like?

Main argument

This section reproduces Wilf's four-page paper "Counting Necklaces" — an exposition of the same necklace-counting problem as §28, worked through at full detail. Every step that §28 compressed into "evidently" receives its own sentence or paragraph. The paper is intended for readers who cannot or do not wish to fill in the gaps themselves. The two papers together constitute a concrete demonstration of the compression-elaboration spectrum. Wilf notes that some readers are annoyed by the half-page treatment while others are annoyed by the four-page treatment — both responses are legitimate.

Key ideas

• Full elaboration serves readers who are new to the area or who prefer to read rather than re-derive.
• The choice of compression level is a targeting decision about the intended audience.
• Neither extreme is categorically superior; the choice depends on journal, audience, and purpose.

Key takeaway

The two Wilf papers together make the compression-elaboration spectrum visible and concrete; every author must decide where on that spectrum a given paper belongs.

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Section 30 — Jeff Ullman on Getting Rich

Central question

What are the economics of writing CS textbooks, and how can an author maximize both financial and intellectual return?

Main argument

Excerpts from class, November 18. Ullman's lecture is entitled "How to get rich by writing books" — illustrating the cover-design principle that titles attract readers with things not actually in the book. Back-of-envelope economics: a book is one megabyte of text at ~1 KB/hour of finished text = ~1000 hours. A good specialized textbook might sell 1500 domestic + 500 foreign copies at $40, netting $8000 over five years at 15% royalty. Possible to double royalty: CS authors have leverage because their books sell well.

Practical advice. Find a co-author (filters out idiosyncrasies; small improvements in a competitive market yield disproportionate market share). Use lots of examples ("Never saw a book with too many examples"). Write in the evening after reading about the topic by day — "you'll get the expository order right." Keep definitions and uses close together (don't define in Ch. 2 and use in Ch. 5). "The object of exposition is education, not showmanship."

Plagiarism. Ideas cannot be copyrighted, only expressions. Plagiarism is when a reasonable person could not believe you didn't have the other author's book open in front of you. Ullman borrowed a table of contents and expository scheme from a competitor's textbook without issue; but lifting notation and equations verbatim is damning.

State variable types. Always say "the set S" not "S" — name the type of your objects.

Key ideas

• CS textbook economics require ~1000 hours of writing and yield modest direct financial return; fame and field impact are the real rewards.
• Co-authors filter idiosyncrasies; small quality improvements have outsized market impact.
• Keep definitions close to their first use; readers cannot hold deferred definitions in mind.
• Examples: more is almost always better; even simple examples carry three-quarters of an idea.
• "Those who can't teach, show off" — exposition is education, not demonstration of the author's intelligence.

Key takeaway

Textbook writing is both a craft and a business; understanding the economics helps authors make better decisions about effort, audience, and positioning.

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Section 31 — Leslie Lamport on Writing Papers

Central question

When should one write a paper, and how does the craft of writing differ from the craft of mathematics?

Main argument

Excerpts from class, November 20. Lamport: "I have one thing to say about writing a paper: Don't. The market is flooded." He identifies two bad reasons to publish: building a publications list, and targeting a specific conference. One good reason: "You have done something that you are excited about." Learning to write is more like learning the piano than learning to type — you must spend more time reading than writing; "We should read great literature to learn to write good mathematical literature." "Bad writing comes from bad thinking."

Structure, not formatting. "Don't think about format. Do think about structure." Use macros for complex notation so that writing is not interrupted by typographic decisions. When writing a paper, front-load the conceit (Elizabethan sense: a fanciful central idea that organizes the paper), but be willing to abandon it if it intrudes on the ideas.

Examples before formalism. "It is better to have one solid example than a dry abstract paper." Examples keep you honest — Lamport rewrote an entire published theory upon discovering his original draft couldn't handle the example he wanted.

First sentences. "Avoid passive wimpiness; be simple and direct." Compare T. S. Eliot's first sentences to a random paper's. "When you come to sentence number 2079, you've got to keep socking it to them."

Structured proofs. Lamport shows two forms of the same proof: paragraph and tabular (like a high-school geometry proof). The tabular form is much easier to follow because it enforces structure that makes each step explicit. Writing a proof in tabular form first clarifies the logic before converting to prose.

Key ideas

• Publish only when you are excited about the work; the market is flooded with papers written for the wrong reasons.
• Bad writing comes from bad thinking; clear up the thinking first.
• Structure and organization are the author's work; formatting is a detail that can be automated.
• One solid example beats a dry abstract formulation.
• Tabular proofs clarify structure; write the tabular version first, then convert to prose.
• Sustained quality over a long paper requires maintaining excitement at every sentence.

Key takeaway

Lamport's contribution is the discipline side: the intellectual and ethical question of whether and what to write, before any sentence is drafted.

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Section 32 — Lamport's Handout on Unnecessary Prose

Central question

How does editing a co-author's draft improve prose by removing unnecessary words and restructuring sentences?

Main argument

This section is Lamport's annotated example of a co-author's draft with his own editing, showing specific cuts and rewrites. The lesson is that most first-draft prose carries unnecessary verbiage — hedges, throat-clearing phrases, redundant qualifications — that can be removed without loss. The before-and-after comparison shows the sentence length roughly halved while all substantive content is preserved. The handout complements §31's advice to "fix the idea, not just the sentence" by showing what sentence-level fixing looks like in practice.

Key ideas

• Much first-draft prose carries unnecessary hedges and qualifications that dilute impact.
• A 50% word-count reduction is often achievable without losing content.
• The edit should improve clarity, not merely reduce length.

Key takeaway

Editing for concision is a learned skill; seeing a before-and-after pair makes the operation concrete.

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Section 33 — Lamport's Handout on Styles of Proof

Central question

What does a structured (tabular) proof look like, and why does it sometimes outperform paragraph proofs?

Main argument

This section reproduces Lamport's handout showing the same proof in two forms: paragraph style (as published in most journals) and structured proof style (with numbered statements and explicit reasons, like a geometry proof). The structured form makes each logical dependency visible. Lamport argues that structured proofs reveal latent errors in reasoning that paragraph proofs conceal behind fluent prose. Don and Paul Halmos disagreed about the value of symbolic proof structure (§42), but Lamport's handout shows at minimum that the tabular form is a useful drafting tool even if the final publication uses paragraphs.

Key ideas

• Paragraph proofs can conceal logical gaps behind rhetorical fluency.
• Structured proof format makes each step explicit and checkable.
• Writing the structured form first and then converting to prose is a useful workflow.
• Tabular proofs are not a publication format for most mathematical journals — they are a drafting discipline.

Key takeaway

The structured proof handout demonstrates that proof presentation is a separate skill from proof discovery; choosing the right format affects both readability and error detection.

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Section 34 — Nils Nilsson on Art and Writing

Central question

In what sense is technical writing an art, and what artistic habits of mind does it require?

Main argument

Excerpts from class, November 23. Nilsson begins with two photographs — Edward Weston's snail-shell print and Ansel Adams's "Aspens in New Mexico" — and introduces his key formula: Composition = Organisation + Simplification. Writing shares this with photography: both require the creator to compose, organize, and simplify so that the viewer or reader participates actively. Another key word: Joy. Writing should be fun; the writer should feel a thrill as Knuth does when pinpointing a reference.

Nine principles:

1. Start early — impressionable minds are best.
2. Write, rewrite, rewrite, rewrite ("A work of art is never completed, only abandoned"; Mozart was the exception — Beethoven rewrote obsessively).
3. Read — reading sharpens style and critical faculties.
4. Model the reader — operationalize this with background "dæmons" that automatically catch misunderstandings. A split infinitive "should light up in red." These skills are motor skills, acquired through practice.
5. Master the medium — vocabulary, index design, typography, tables.
6. Master the material — writing reorganizes the internal model of what you know; you often discover what you think only when you try to write it.
7. Simplify — "lie if it helps"; provide correct details later. The 80-20 rule: 80% of truth from 20% of the difficulty. Mathematicians go the other way.
8. Avoid recycling — resist re-using old text; you are writing in a new context and are (hopefully) older and wiser.
9. Aim for excellence — "We are all apprentices in a craft where no one ever becomes a master" (Hemingway).

Key ideas

• Composition = Organisation + Simplification (from photography; applies equally to writing).
• Writing clarifies internal knowledge: "How do I know what I mean until I see what I say?" (E.M. Forster).
• Mastery of the "dæmons" — background error-detection habits — is a motor skill built through practice.
• Simplify first; lie if necessary; pay back the debt to truth gradually through elaboration.
• Recycled text is almost always wrong for the new context; rewrite it.

Key takeaway

Nilsson places mathematical writing inside the larger tradition of artistic craft, arguing that the same habits that make great painters and photographers great also make great expositors.

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Section 35 — Mary-Claire van Leunen on Calisthenics (1)

Central question

What structured exercises build writing skill at the level of vocabulary, syntax, and sentence construction?

Main argument

Excerpts from class, November 25. Van Leunen opens with the "magic pencil" parable: a pencil peddler on a Chicago bus who sells the ability to write the truth — "if you really believe, you can write the truth." Writing is the craft of explaining a new truth to someone you will never meet.

Vocabulary exercises. (1) Swap all content words (nouns, verbs, adjectives, adverbs) in a passage for new ones — reveals how vocabulary determines tone. This is a practice Greek boys did. (2) Write a thesaurus entry over a week; compare to actual thesaurus entries.

Syntax exercises. Syntax is more important than vocabulary; most writers use only 3–4 thousand words. Van Leunen's statistical analysis of student papers: average sentence length 15–24 words (under 20 is in the correct adult range); variation in sentence length is a measure of syntactic richness (aim for 9–10 different lengths per 10 sentences). Students lacked short punchy sentences. The "periodic sentence" (grammatical and physical end coincide; all adverbials out of final position) is a high literary form rarely appropriate in technical writing but useful as an exercise.

Manual labor exercises. Typing and copying other writers' text slows the brain down to the point where each word receives attention. W.H. Auden: "The proper way to show contempt for a poem is to copy it on a typewriter; the proper way to show admiration is to copy it in longhand." Memorize and recite.

Frozen sounds / "Marks on paper" / Stance, voice, tone. Read your prose aloud; form partnerships to read each other's work. Writing is "the most forgiving medium known to man — we can work on it until we get it right."

Key ideas

• Syntax is more important than vocabulary; variation in sentence structure is measurable and improvable.
• Under 20 words per sentence on average is the correct target for adult technical writing.
• The periodic sentence (end-heavy grammatical structure) is a useful exercise but wrong for most technical contexts.
• Manual copying slows reading speed to match careful attention; it is a form of close reading.
• Reading aloud reveals rhythm, register, and errors invisible in silent reading.

Key takeaway

Writing skill is built through systematic exercises targeting vocabulary, syntax, and the physical dimension of writing — treating it as a craft with drills, not just a talent expressed spontaneously.

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Section 36 — Mary-Claire's Handout on Composition Exercises

Central question

What is the full set of composition exercises that van Leunen has developed for building writing strength?

Main argument

This section is the full text of van Leunen's handout, organized into eight categories: Vocabulary, Syntax, Manual labor, Frozen sounds, Marks on paper, Stance/voice/tone, Observation, Same as and different from, Invention, Scansion, Précis, and "Nearly real." The final category ("Writing a joke is exposition at its purest: things aren't funny unless they are well written") exemplifies the principle that constraints force clarity. The "Ben Franklin exercise" — rewriting a master's passage from memory and limited written hints, then comparing — is highlighted. The scansion exercises (writing verse) are described as "hard but very important"; van Leunen argues one must write verse to learn to write mathematics.

Key ideas

• Writing verse builds awareness of rhythm, syllable weight, and the physical sound of sentences.
• The joke is a model of exposition: it must be correctly set up, correctly timed, and exactly worded.
• Ben Franklin's exercise (reconstruct from memory, compare to original) builds intuition for word choice.
• Précis exercises force reduction without meaning loss; the point at which further reduction destroys structure varies by text.

Key takeaway

The full composition-exercise program provides a structured curriculum for building writing strength outside of actual writing projects.

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Section 37 — Comments on Student Work

Central question

What recurring errors appear in student term papers after a full quarter of instruction, and why do they persist?

Main argument

Excerpts from class, November 30. Eleven of thirteen submitted papers contained "wicked whiches" — this after sustained instruction. Knuth administers a quiz (leaving the room per Honor Code). Most students recover. Van Leunen explains: people almost never use "which" incorrectly in speech; it is only under the pressure of writing that they resort to it, which makes it a marker of nervousness. Translations of the Bible are full of wicked whiches. The passage of the New English Bible — which its editors take pride in — commits this error because British writers consider the which/that distinction "unmitigated nonsense."

Other issues from student work. Subsection titles should not be "integral" to the running text — the prose should make sense if titles are skipped. Consistent definition of graph-theory terms (path, walk, simple path) is essential because different authors use the same term differently. Tense: an algorithm's worst-case complexity is a timeless fact (present tense); a problem's known complexity at a given time is historical (past tense may apply). Abbreviation-style references should be parenthetical so the sentence makes sense without them. "Less is more" — the authors of a paper could replace a three-part expression with a two-part one by observing that the middle term appears nowhere else.

Key ideas

• Persistent errors like wicked whiches often stem from writing-register pressure, not ignorance.
• Section titles should not be grammatically required for sentence coherence.
• Timeless mathematical facts use present tense; historical states of knowledge use past tense.
• Define graph-theory terms explicitly; different authors use the same term inconsistently.
• "Less is more" — eliminate intermediate symbols, objects, or steps that appear only once.

Key takeaway

Students who have mastered a rule in isolation will still violate it under the pressures of drafting; repeated exposure and real feedback on real work are necessary to internalize it.

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Section 38 — Mary-Claire van Leunen on Which vs. That

Central question

What is the historical origin of the which/that distinction, and why does it matter?

Main argument

Excerpts from class, December 2. Van Leunen traces the which/that distinction from the 17th century, when both were relative pronouns used interchangeably, through the 19th century (written English evolved toward spoken patterns), to the Fowler brothers' The King's English (1906 — conversational rhythm should be reflected in writing) and Fowler's Modern English Usage (1926 — the surviving brother makes the rule explicit). The modern rule: use "that" for restrictive relative clauses ("the book that I bought") and "which" for non-restrictive clauses set off by a comma ("the book, which I bought yesterday"). Speakers do not use "which" as a relative pronoun at all — spoken sentences are too short for non-restrictive clauses. The Fowlers were not inventing a rule but describing where spoken and written English had already converged.

Fowler as theoretician of the semicolon. Fowler saw the semicolon as a structuring device operating between the sentence level and the paragraph level — a purely graphical punctuation mark with no spoken equivalent.

Key ideas

• The which/that distinction reflects a convergence of written and spoken English that began in the 19th century; the Fowlers documented it, they did not invent it.
• Restrictive clauses ("the paper that Knuth wrote") use "that"; non-restrictive ("the paper, which Knuth wrote last year,") use "which" with commas.
• Spoken English uses neither "which" nor non-restrictive clauses — they are purely written constructs.
• Language belongs to millions of people; it cannot be controlled by decree.
• Fowler's semicolon theory: the semicolon operates between the sentence and the paragraph, imposing graphical structure.

Key takeaway

The which/that rule is not arbitrary pedantry but a convergence between written and spoken English; understanding its history makes it easier to apply consistently.

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Section 39 — Mary-Claire van Leunen on Calisthenics (2)

Central question

What did van Leunen's encore lecture cover about books, language, and the deeper craft of writing?

Main argument

Excerpts from class, December 2 (continued). Van Leunen brought a collection of books to the desk that there was not time to discuss in her first lecture. The session addresses remaining topics from the composition exercises handout, with particular attention to the use of interesting books on language (Sidney Landau's Dictionaries, for understanding how to use dictionaries properly) and the philosophical underpinning of "the magic pencil" story: writing is the craft of transmitting truth across time and space to people you will never meet.

Key ideas

• Dictionaries are complex artifacts with their own methodologies; understanding how they are compiled improves their use.
• The magical dimension of writing is its ability to communicate across time — to people not yet born.

Key takeaway

Writing has both a pragmatic dimension (skill, craft, rules) and a larger dimension (truth, transmission, posterity) that motivates the craft.

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Section 40 — Computer Aids to Writing

Central question

What software tools are available for analyzing and improving written prose, and how should writers interpret their output?

Main argument

Excerpts from class, December 4. Knuth demonstrates UNIX tools available in 1987. The style program analyzes a text file and reports: average sentence length, percentage of passive-voice sentences, readability indexes (Flesch, Kincaid, etc.). Knuth runs it on several texts: Church's lambda-calculus papers score high on sentence length but the prose is "musical, not ugly"; Grimms' Fairy Tales has very few adjectives (Mark Twain distrusted adjectives too). The two most similar drafts (written 20 years apart) had nearly identical style scores.

The diction program flags 450 "questionable" words and phrases: "gratuitous," "number of," "due to." Knuth notes that copy editors prefer "because of" to "due to" in ordinary writing, but "due to" is elegant in mathematical attribution ("This theorem, due to Cauchy"). The word "very" should be replaced by "damn" as a test — "damn good advice!" The explain program expands diction's objections: replace "if" for "assuming that," "really" for "actually." Users accept about 50% of diction's suggestions.

Key ideas

• Automated style analysis can identify sentence-length distribution and passive-voice frequency as numerical measures of stylistic problems.
• "Style" scores can mislead: musical long sentences are not the same as ugly long sentences.
• The "replace 'very' with 'damn'" test filters out casual intensifiers.
• Copy-editing tools are useful assistants, not arbiters; accept roughly half their suggestions.
• Mathematical usage ("due to Cauchy") differs from ordinary usage ("due to bad weather"); tools calibrated for general writing need adjustment.

Key takeaway

Computer tools can surface measurable style properties (sentence length distribution, passive rate) that human re-reading misses, but their judgments must be calibrated against the specific conventions of mathematical writing.

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Section 41 — Rosalie Stemer on Copy Editing

Central question

What does a professional newspaper copy editor see when reading mathematical and technical writing, and what systematic errors does she find?

Main argument

Excerpts from class, December 7. Rosalie Stemer (San Francisco Chronicle, teaches copy editing at Berkeley) works through a newspaper story on paper deterioration and a selection of student term papers. Key observations: (1) Subject and verb should be close together in a sentence; long separating phrases make parsing hard. (2) Answer more questions than you raise. (3) "Nine out of ten" not "nine out of ten"; "out" is superfluous. (4) Whole phrases can often be deleted without loss: "he has heard" is deletable in "Dr. Shaffer said he has heard one of the largest manufacturers …" → just "Dr. Shaffer said one of the largest manufacturers …". (5) "Reason for" constructions are almost always improvable: "The reason for X is Y" → "We do X because Y." (6) Out with semicolons, in with periods; a semicolon should be used only when clauses are very closely related, and even then a period is often better (Zinsser: semicolons convey "a certain 19th-century mustiness").

The "only" exercise. "I hit him in the eye yesterday" — inserting "only" in each of eight positions produces eight sentences with eight different meanings. Misplaced "only" is one of the most common errors in technical writing.

Conversational idioms. "Anyway, …"; "pretty straightforward"; "again iterate" — all discouraged in technical writing. Stick to subject-verb-object structure to avoid subject-verb separation and passive voice.

Key ideas

• Subject and verb should be adjacent; long embedded phrases between them cause parsing difficulty.
• Delete whole phrases that add no information: "the fact that," "he has heard," "it is noteworthy that."
• "The reason for X is that Y" can almost always be replaced by "We do X because Y."
• "Only" modifies the nearest following element by default; misplace it and the meaning changes.
• Semicolons slow pace and suggest 19th-century formality; when in doubt, use a period.

Key takeaway

A professional copy editor's eye finds systematic sentence-level problems — subject-verb separation, false "reason" constructions, misplaced modifiers — that authors, focused on content, routinely miss.

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Section 42 — Paul Halmos on Mathematical Writing

Central question

What two master principles organize all advice about mathematical writing, and how do semantics, syntax, symbols, and style interrelate?

Main argument

Excerpts from class, December 9. Halmos opens with two principles — Do organize material; Do not distract the reader — and an outline: Semantics, Syntax, Symbols, Style.

Semantics. Language change is real and often good. "Hopefully" in its impersonal use may be needed; Halmos reluctantly concedes. "Imply" vs. "infer" and "disinterested" vs. "uninterested" are valuable distinctions being lost — if communication requires those distinctions, avoid the now-ambiguous words entirely when writing for a large audience. Using "reference" as a verb is unacceptable (there is no such verb; compare "I quotationed the author").

Syntax. Grammar is logic — not entirely, but more than linguists admit. "Cases are good things" (nominative/accusative matter; "whom is the President?" is always wrong regardless of style). The rule against terminal prepositions is "reactionary grammarian balderdash" — Winston Churchill's "the sort of nonsense up with which I will not put."

Symbols. Place quotation marks logically, every time, despite typographers' conventions (Turabian is "The Antichrist" on this matter). Use '1' (numeral) not "one" (word) when you mean the number, to avoid ambiguity in sentences like "not one or a prime."

Style. Avoid both passive voice and "we" by using imperative and indicative moods more creatively: "We can now prove the following result" → "Consequence: A implies B." Organization is the hardest part: the plot of an exposition is rarely a straight line; branches and alternative threads must be woven together. Proof by contradiction should be avoided whenever a direct proof is available: "Suppose the variables are linearly dependent …" → just prove linear independence directly.

Key ideas

• Do organize; do not distract — these two principles subsume all other advice.
• Distinguish needed linguistic distinctions ("imply/infer") from harmless ones ("hopefully"); fight only for the former.
• The preposition-at-end rule is nonsense; terminal prepositions are often natural and cleaner.
• Logical placement of quotation marks (inside where logically required, outside where not) is worth fighting for in mathematics.
• Direct proofs are preferable to proofs by contradiction whenever a direct proof exists.
• Organization is non-linear; exposition must weave multiple threads together, not just proceed in sequence.

Key takeaway

Halmos distills the entire course into two principles — organize, don't distract — and demonstrates that every specific rule either serves these principles or should be discarded.

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Section 43 — Final Truths

Central question

What do course-end reflections, unsolicited suggestions, and final editorial debates reveal about the nature of writing style?

Main argument

Excerpts from class, December 11. Nelson Blachman (father of a class member) sent suggested improvements to the early handouts. Knuth liked some, disagreed with others. Key lesson: "The main thing I get from this is that the style has to be your own. You will write things that someone else will never write." Knuth and his wife Jill write alternate paragraphs in their annual Christmas letter because "there is no way that we can write a sentence acceptable to both of us."

Nelson's suggestions and their outcomes: "The proof above" vs. "the above proof" (Knuth agrees); spelling "hiccups" vs. "hiccoughs" (his dictionaries prefer the shorter form); "programming language notation" → "programming-language notation" (Knuth disagrees, for readers in the CS field the hyphen looks wrong, just as "random-number generator" and "floating-point arithmetic" look odd despite potential ambiguity).

Stacked vs. slashed fractions in typesetting. A novice typesetter tends to produce stacked fractions with tiny numerals; the slashed form n(n+1)(2n+1)/3 is almost always more readable in running text.

The "God is in the details" postscript. The first edition's search (mentioned in §22) reached a conclusion in the 1989 postscript: E. H. Gombrich's 1970 biography of Aby Warburg shows that Warburg used "Der liebe Gott steckt im Detail" as the motto of a 1925 Hamburg seminar series; Mies probably learned it from Warburg. But Flaubert and Nietzsche cannot be ruled out until their works are fully available in electronic form. The final line of the book (and possibly of the course): "Particulars, as every one knows, make for virtue and happiness; generalities are intellectually necessary evils."

A poetic computer program. An anonymous student contributed a limerick that is also a working bit-counting program for the GE635: "This algorithm to count bits / Rotates VALUE one left and sums its / two's-comp negation / in a zeroed location / Repeats WORDLENGTH times, then exits."

Key ideas

• Personal style is ultimately your own; no external rule overrides what genuinely works for you.
• Even expert stylists (Knuth and his wife) cannot agree on every sentence; collaborative writing requires negotiation.
• Stacked fractions degrade readability in running text; slash them.
• The trace of "God is in the details" leads to Aby Warburg (Hamburg, 1925), most likely via Mies van der Rohe.
• The course's final lesson: particulars matter; generalities are necessary but not sufficient.

Key takeaway

Final Truths closes the course with the reminder that style is ultimately personal, that rules are means not ends, and that attention to particulars — in mathematics, in exposition, in the quotation that has no confirmed origin — is the practitioner's highest virtue.

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The book's overall argument

1. Section 1 (Minicourse on Technical Writing) — establishes 27 specific rules covering notation, grammar, flow, and layout: the core ruleset that all subsequent sections apply, test, and extend.
2. Section 2 (An Exercise on Technical Writing) — moves immediately from rules to practice by presenting an egregiously bad proof and asking the reader to improve it, demonstrating that the rules matter in live mathematical prose.
3. Section 3 (An Answer to the Exercise) — resolves the exercise and adds the deeper lesson that proof strategy (direct vs. contradiction) is part of exposition, not just proof correctness.
4. Sections 4–5 (Comments on Student Answers) — surfaces the most common first-attempt errors and discusses the pronoun and tense questions that pervade every mathematical paper.
5. Sections 6–8 (Preparing Books for Publication, 1–3) — zooms out from single papers to the book production process: designers, copy editors, revision cycles, typographic conventions, and the economics of professional production.
6. Section 9 (Handy Reference Books) — identifies the reference shelf every serious writer needs, from the OED to Webster's Dictionary of English Usage.
7. Section 10 (Presenting Algorithms) — introduces the special challenges of computer science writing: algorithm presentation, the program-as-literature thesis, and the birth of WEB.
8. Sections 11–12 (Literate Programming, 1–2) — extends the program-as-literature idea into a full methodology: WEB, module-naming, module length, and the priority of the reader over the compiler.
9. Section 13 (User Manuals) — applies reader-first principles to the genre most often written for non-expert audiences, using the Digiflash student-manual case studies as evidence.
10. Section 14 (Galley Proofs) — walks through the production-stage mechanics of correcting a journal paper, including typographic symbol confusion and bibliography verification.
11. Sections 15–17 (Refereeing, 1–2; Hints for Referees) — argues that refereeing is cooperative teaching, not gatekeeping, and provides both the historical case (calibrating referees with the same paper) and the documental evidence (the Hints for Referees handout).
12. Sections 18–19 (Illustrations, 1–2) — establishes when diagrams earn their cost and demonstrates through TAOCP examples how numerical tables and fold-out comparisons serve purposes graphs cannot.
13. Sections 20–21 (Homework: Subscripts and Superscripts; Solutions) — proves by example that notation complexity causes mathematical error; four different solutions show that notation simplification admits multiple strategies.
14. Section 22 (Quotations) — treats the finding of quotations as a library skill with specific techniques: Bartlett, the OED, full-text electronic search.
15. Sections 23–24 (Scientific American Saga, 1–2) — narrates the most extended case study in the book: 14 months of editing disputes that led both to a better final paper and to the creation of TEX.
16. Section 25 (Examples of Good Style) — shifts from corrective to exemplary pedagogy, distributing actual good writing by the upcoming guest speakers as preparation.
17. Section 26 (Van Leunen on 'hopefully') — demonstrates, through its own style, the standard of prose writing that mathematicians should aspire to.
18. Section 27 (Wilf on Mathematical Writing) — offers the editor's perspective: front-load results, open with impact, and understand the two legitimate extremes of compression and elaboration.
19. Sections 28–29 (Wilf's Two Extremes) — makes the compression-elaboration spectrum concrete with two papers on the same problem.
20. Section 30 (Ullman on Getting Rich) — grounds the course in the pragmatics of the textbook market and offers practical advice about examples, co-authors, and the reader-first principle.
21. Section 31 (Lamport on Writing Papers) — raises the ethical and philosophical question of when and why to write, and argues that structure precedes formatting and that examples are the key disciplinary tool.
22. Sections 32–33 (Lamport's Handouts) — provides concrete before-and-after editing examples and the structured-proof format as a drafting tool.
23. Section 34 (Nilsson on Art and Writing) — situates all the specific rules inside a larger artistic framework: Composition = Organisation + Simplification; the craft must precede the art.
24. Sections 35–36 (Van Leunen on Calisthenics, 1–2; Composition Exercises) — provides the most systematic program for building writing strength: vocabulary, syntax, manual labor, scansion, précis, "nearly real."
25. Section 37 (Comments on Student Work) — shows how errors persist even after instruction, and identifies the conditions — subsection-title dependence, graph-theory vocabulary ambiguity, tense consistency — that survive a full quarter.
26. Section 38 (Van Leunen on Which vs. That) — gives the historical account that converts a rule into an understanding, making it easier to apply correctly.
27. Section 39 (Van Leunen on Calisthenics, 2) — extends the composition-exercises framework with van Leunen's encyclopedic collection of books about language and writing.
28. Section 40 (Computer Aids) — introduces quantitative tools and their limitations: useful for measuring what human re-reading misses, but requiring calibration against mathematical conventions.
29. Section 41 (Stemer on Copy Editing) — brings the newspaper copy-editing perspective: systematic attention to sentence structure, proximity of subject and verb, the "only" placement test.
30. Section 42 (Halmos on Mathematical Writing) — arrives at the synthesis: Do organize; do not distract. Every rule in the course either serves these principles or should be abandoned.
31. Section 43 (Final Truths) — closes with the personal dimension: style is ultimately your own; particulars are the substance of both mathematics and writing; "generalities are intellectually necessary evils."

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Common misunderstandings

Misunderstanding: The rules in §1 are a universal checklist to apply mechanically.

The rules are heuristics derived from Knuth's experience and taste. Halmos explicitly says that some of §1's rules are wrong (the terminal-preposition rule), and the Final Truths section emphasizes that personal style will lead you to write things that no rule requires and no rule prohibits. The rules are a starting point, not a sentence-level algorithm.

Misunderstanding: Literate programming is about adding comments to code.

Students who simply added a few comments to their Pascal programs and called it WEB "completely botched it." Literate programming requires reorganizing the entire program around the psychologically natural order of explanation — breaking it into named modules each introduced at the right narrative moment — not decorating an existing program with commentary.

Misunderstanding: The book teaches general writing skills.

The book explicitly distinguishes mathematical/CS writing from English-department writing: "the vast majority of the lectures were devoted to issues that are specific to mathematics and/or computer science." The minicourse acknowledges overlap with general English writing, but the book's purpose is to address the additional problems peculiar to the discipline.

Misunderstanding: Proof by contradiction is inherently suspicious.

Section 3 shows that Knuth's own solution used contradiction and then replaced it with a direct proof. The lesson is not that contradiction is wrong but that it is often chosen for discovery and then not replaced during revision. A direct proof is preferred for exposition when one exists; contradiction is not forbidden.

Misunderstanding: The SA saga shows that copy editors are adversaries.

Knuth explicitly acknowledges at the end of Section 24 that "this cooperative effort did much to remove the jargon and make the paper accessible to a general audience." He also acknowledges that the errors introduced by the copy editor showed him where his exposition was genuinely ambiguous. The saga is a model of productive friction, not of editorial villainy.

Misunderstanding: "Hopefully" in its impersonal sense is simply wrong.

Halmos, who dislikes the new usage, nonetheless says "arguably the change is a needed one" and "the English language won't collapse if the other side wins." Van Leunen's essay analyses the word's history in full and reaches a nuanced position. The book does not condemn the impersonal "hopefully"; it analyses it.

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Central paradox / key insight

The deepest insight in Mathematical Writing is that precision and accessibility are not opposites — they are the same requirement approached from two directions. The book's most memorable formulation belongs to Knuth (speaking of user manuals, but the principle generalizes): "If you try to write for the novice, you will communicate with the experts — otherwise you communicate with nobody."

This inverts the standard hierarchy. Most writers believe that technical precision is what makes writing hard for non-experts; the solution is therefore to simplify. But Knuth's argument is that what looks like technical precision is often actually unexplained jargon — conventions so familiar to the author that they are no longer visible as conventions. True precision — the kind that makes a proof checkable and a notation unambiguous — also makes the text readable, because it forces the author to make every step explicit, every variable named, every claim supported. The apparent conflict between precision and accessibility dissolves when the author actually achieves precision rather than merely affecting it.

Particulars, as every one knows, make for virtue and happiness; generalities are intellectually necessary evils.

The final quotation of the book (Aldous Huxley) names the same paradox from the opposite direction: the general rule is a convenience for the writer, never a substitute for the particular argument.

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Important concepts

Literate programming

A programming methodology in which a computer program is written as a document first — prose interleaved with code — organized in the psychologically natural order for human reading rather than the order required by the compiler. The WEB system produces either a typeset document (via WEAVE) or a Pascal source file (via TANGLE) from the same source. A module is named by an imperative phrase and introduced at the moment the reader is ready for it.

WEB

Knuth's specific literate programming system for Pascal, in which modules are introduced with imperative titles and surrounded by high-level prose. The TANGLE processor extracts Pascal; the WEAVE processor produces TeX. About 1400 modules constitute TEX: The Program.

Wicked which

A relative pronoun "which" used without a preceding comma or preposition in a restrictive relative clause, where "that" is correct (e.g., "a method which is faster" should be "a method that is faster"). Named and popularized in the course; van Leunen traces the rule to the Fowlers (1906/1926).

Periodic sentence

A sentence whose grammatical and physical end coincide — all adverbial modifiers appear before the final verb or object, giving the sentence a driven, forward-moving rhythm. Van Leunen introduces this as a writing exercise; it is a high literary form inappropriate to most technical writing but valuable for training.

Galley proofs

Typeset, pre-publication text returned by a publisher for author correction. Stage at which typesetting errors (symbols confused, line-breaks in formulas) must be caught; also the last chance to correct technical errors in the mathematics.

Facile generalization

Pólya's term for a generalization that is merely a mechanical manipulation of an argument without creating new insight. Referees who suggest facile generalizations are not actually improving a paper.

Reader-first principle

The organizing principle of the book: every decision about notation, sentence structure, proof presentation, or figure placement should be made by asking "What does the reader know so far? What does the reader expect next and why?" This is stated explicitly in §1 and repeated in every major section.

Composition = Organisation + Simplification

Nilsson's formula, derived from a photography teacher, applied to writing. All good exposition requires both organizing the material (structure, sequence, emphasis) and simplifying it (removing unnecessary complexity, lying if necessary and paying back the debt to truth gradually).

The 80-20 rule (in exposition)

80% of the truth can be conveyed with only 20% of the difficulty. Nilsson's argument for starting with a simplified model: get the reader to the essential insight first, then add correct details through elaboration. Mathematicians typically go the other way — generalize at every opportunity — which is often detrimental to exposition.

Structured proof

Lamport's format for presenting proofs: numbered statements with explicit reasons, in a tabular layout analogous to high-school geometry. Recommended as a drafting discipline to force explicit logical dependencies before converting to paragraph form.

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References and Web Links

Primary book and edition information

• Knuth, Donald E., Tracy Larrabee, and Paul M. Roberts. Mathematical Writing. Mathematical Association of America Notes, Vol. 14. MAA, 1989. ISBN 0-88385-063-X.
  • MAA Notes Volume 14 — Mathematical Writing (publisher page)
  • Stanford Technical Report STAN-CS-88-1193 (full PDF at JMLR/MIT)
  • Knuth's own page for the book at Stanford
  • Internet Archive scan (1988 Stanford report edition)
  • Google Books metadata page
  • Amazon listing with edition details

Background and overview

• Wikipedia article on Literate Programming
• Stanford CS209 course video recordings — finding aid at Online Archive of California
• Donald Knuth's Lectures on Mathematical Writing — YouTube playlist (video recordings of the 1987 lectures)
• MAA Review of Mathematical Writing (old MAA site)

Key source works cited in the book

• Halmos, Paul R. "How to Write Mathematics." In How to Write Mathematics, American Mathematical Society, 1973, pp. 19–48. — The essay Knuth calls "delightful"; the direct predecessor to this course.
• Strunk, William, and E. B. White. The Elements of Style. Macmillan, 1979. — Required background reading, "the nicest little reference for a quick tutorial."
• van Leunen, Mary-Claire. A Handbook for Scholars. Knopf, 1978. — Required text for CS209; covers footnotes, quotations, and scholarly reference conventions.
• Gillman, Leonard. Writing Mathematics Well. MAA, 1987. — The other required text; the homework assignment comes directly from its appendix.
• Tufte, Edward R. The Visual Display of Quantitative Information. Graphics Press, 1983. — Recommended for illustration principles.
• Fowler, H. W. A Dictionary of Modern English Usage. Oxford University Press, 1926. — Cited repeatedly; Fowler's the-oretical framework for the which/that distinction and semicolon usage.
• Knuth, Donald E. "Programming Pearls: Literate Programming." Communications of the ACM 29 (May 1986): 364–369. — The paper that introduced WEB to American readers (published by Jon Bentley with Knuth as "guest oyster").

Additional study resources

• Semantic Scholar record for Mathematical Writing
• ACM Digital Library book record
• Cambridge Core review of the book

These secondary sources are supplements to the primary text and should be consulted alongside it.