Synthesis — Mathematics

The evolving thesis. Spun out of the hub _inbox on 2026-08-02 from three mathematics texts parked under three different tags (mathematics, signal-processing, and the arriving probability course) across five days. A fourth, keisler-elementary-calculus, arrived the same afternoon and opened the corpus’s first tension — four sources in, the spoke already has something to argue about.

Current thesis

A spoke was missing, and the _inbox diagnosed it before the split happened. The fourier-analysis-for-beginners record, parked 2026-07-28, ended with the observation that Fourier analysis is “infrastructure beneath at least four spokes and the subject of none” and set the spin-out trigger at three. The vector-calculus-corral record the next day made the identical argument about calculus, tallied two, and named the shapes a third could take — “applied maths, linear algebra, probability/statistics, information theory, numerical methods.” mcmullen-probability-theory arrived on 2026-08-02 as the second item on that list. The cluster was predicted, not discovered.

That matters for what this spoke is. It was not created because mathematics is interesting; it was created because thirty applied spokes kept declining the same class of source for the same reason, and a repeated correct decline is what a missing spoke looks like from the outside.

The boundary is the thesis

Each founding source had a spoke that nearly took it, and each was refused on the same line: the maths was infrastructure under that spoke, not its subject.

• corral-vector-calculus against ../optimization-algorithms-wiki — two of its thirty sections minimize a function, but that spoke owns metaheuristic global optimizers, the family that exists because the calculus route needs derivatives it doesn’t always have.
• thibos-fourier-analysis against ../psychology-wiki — the fit is through the audience (vision science) and not the content (about 85% general Fourier analysis).
• mcmullen-probability-theory against ../quant-trading-wiki and ../machine-learning-wiki — both teach probability as a prerequisite to their own subject; neither is teaching probability.

Stated as a rule: the maths is the subject here, the application is the subject there. None of the mathematics here makes a claim about the world, which is what distinguishes this spoke from every other one in the hub and why the usual freshness question barely applies — none of this material dates. The claims that can be wrong are the pedagogical ones sitting on top of it, and those are exactly where the corpus has now split.

What the corpus can support, and what it can’t

Three of the four subjects still rest on exactly one source. The fourth ingest changed that for vector-calculus, which now has two accounts that agree on every theorem and disagree about the foundation.

Updated 2026-08-06. probability-theory is the second doubled subject, and it doubled in a different way. stirzaker-elementary-probability (CUP, 2nd ed. 2003, 538pp) covers McMullen’s syllabus and keeps going into martingales, Markov chains and Brownian motion; the two texts agree about everything they share. Where the calculus pair disagrees about foundations, the probability pair differs only in how much the reader must supply — 98 terse pages with Feller assumed, against a self-contained course with worked examples, problems and solutions. Both kinds of second source are worth having, and only one of them creates a tension. In a spoke about mathematics, single-sourcing is less alarming than elsewhere — a theorem does not need a second vendor to confirm it — but the pedagogy claims are where it bites, and those are where the sources have started to differ.

Updated 2026-08-10. osborne-rubinstein-course-in-game-theory adds a twelfth text and a new subject, game-theory, and it arrives saying something none of the others do: the authors state that they treat their subject “not as a branch of mathematics but as a social science”, and that a book with the same content could in principle be written without mathematics at all. Every other text here is written by someone claiming the subject as mathematics. This one proves theorems for 373 pages while disclaiming the label. It does not unsettle the boundary rule — a text teaching a body of theory from first principles routes here regardless of the field its examples come from — but it is the first time the rule has had to override the author’s own description of the work, rather than merely the field of the examples.

It also imports an argument the spoke had no page for: the rationality assumptions the whole subject rests on are, in the authors’ own words, “under perpetual attack by experimental psychologists”, and they decline to defend them. That attack is the heuristics-and-biases tradition held in ../psychology-wiki. Nothing in this spoke’s corpus can adjudicate it; recorded as a cross-spoke tension, not an open question here.

The rigour spectrum, stated up front

The founding three sit at visibly different points, and the corpus is more interesting for it than a uniform set would be:

• McMullen writes for people who will read Feller alongside, glosses binomial coefficients with the transitive action of a symmetric group, and derives P(A′) = 1 − P(A) from the axioms rather than from intuition.
• Corral proves selectively and says so: 5 out of 10, on the argument that too much rigour “can impede the flow of learning for the vast majority of the audience at this level.”
• Thibos starts at vector arithmetic and complex numbers, and spends three chapters on hypothesis testing that a mathematics text would not contain.

The through-line held for the founding three: the same mathematics gets taught at whatever rigour the audience will bear, and knowing which version you are reading matters more than the material dating.

keisler-elementary-calculus breaks that frame, and it is the most interesting thing in the corpus so far. Keisler does not move along the rigour axis at all. He changes the foundation — infinitesimals instead of limits — and claims this buys accessibility and rigour together rather than trading one for the other. His premise is that the century of ε-δ teaching cost students the original intuition for no gain once Robinson made infinitesimals precise in 1960. If that is right, the spectrum the other three sit on is not a necessary trade-off but an artifact of one historical choice.

A second author defends a foundation choice, and defends it the other way (2026-08-06). stirzaker-elementary-probability adopts the formal definition of probability “not because of high principles, but merely because the alternative intuitive approach seems to lead more students into errors.” That is Keisler’s own kind of argument — a claim about what happens to students — reaching the opposite conclusion in a different subject. Neither claim is measured, and Stirzaker’s is the weaker of the two: an experienced teacher’s impression, offered as an impression. What the pair establishes is that the rigour question in this corpus is argued on pedagogy rather than on principle by both authors who bother to argue it at all.

He is also the only author here who cites evidence for a pedagogical claim: a controlled experiment across five schools, run by Sister Kathleen Sullivan in 1972–74, reported in a Wisconsin PhD thesis and the American Mathematical Monthly. Weigh it honestly: it is the author summarizing a study of his own method in his own preface, and this hub holds neither the thesis nor the article. It is still the only such citation in the corpus.

The floor of the rigour spectrum, and what sits on it (2026-08-12)

illowsky-introductory-statistics extends the spectrum above downward past every text held here, and does it deliberately rather than by omission. No calculus, no proofs, intermediate algebra assumed, a graphing calculator standing in for the theory — and it still reaches confidence intervals, two-sample hypothesis testing, chi-square, regression and one-way ANOVA in 847 pages.

The spectrum is now wide enough to be read as a curriculum rather than a list of styles. From larsen-marx-mathematical-statistics deriving the generalized likelihood ratio, through herzog-understanding-statistics applying inference without derivation, to Illowsky & Dean teaching the same tests with a keystroke sequence, the corpus holds three altitudes of one subject. The mathematics is common; what varies is whether the reader is ever shown why it works, and mathematical-statistics now carries the comparison.

The tool a text assumes turns out to track the rigour, inversely. carroll-geometric-function-theory assumes no computational tool at all and proves everything. judson-abstract-algebra assumes Sage and proves everything. This one assumes a TI-83/84 and proves nothing — the calculator is doing the work the derivation would otherwise do, supplying numbers whose provenance the student is not expected to follow. Three points is not a law, but the spoke has been recording the assumed tool since carroll-geometric-function-theory and this is the first time it has paid.

A publishing arrangement the corpus had not seen. Every earlier open text here was released by its author. openstax is an institution that commissions, reviews and revises at scale, and the consequences are visible in the book: four practice exams, instructor guides behind verified accounts, an errata process, and a revision that preserves problem numbering so online homework providers can follow the edition change. A service-course textbook is infrastructure with downstream integrations, and this is the corpus’s first source that shows the pressure. The licence also undercuts the “free textbook” framing slightly: CC BY-NC-SA, not the unrestricted CC BY of mari-measurement-across-sciences.

The canon has an edge, and it is the end of the first course

Every text in this corpus until now teaches a first course, and the rigour spectrum above is what you see when authors disagree inside a syllabus they all inherited. Keisler changes the foundation, Stirzaker defends the formal definition, Corral proves selectively at 5 out of 10 — three arguments about how to teach material none of them chose.

carroll-geometric-function-theory is the corpus’s first second course, and its preface names the thing the rest of the shelf could not show. On the first course there is “general agreement on the list of main topics” — Cauchy, Liouville, Maximum Modulus, Residues, Rouché — and “the general syllabus of a first course in complex analysis is well-established.” Then: “This consensus evaporates when it comes to a second course.”

So the settledness the other texts enjoy is not a property of mathematics. It is a property of the first year of a subject, and it stops. That reframes the rigour spectrum: those authors could argue about pedagogy precisely because content was not in dispute, and Carroll cannot.

What an author does when nothing is settled. He writes §1.4, “What Is Not in This Book” — a numbered section listing the omissions and naming where each is properly treated: no curvature (Ahlfors, Krantz), no conformal invariants beyond the hyperbolic metric, no Riemann surfaces past the sphere, nothing on Denjoy–Wolff, which he says “would fit naturally in this book.” And he concedes the selection is contestable: “many other topics could lay equal claim to being included.”

No other text here does this. The nearest comparison is in another spoke and it is instructive: ../engineering-education-wiki’s coding-interview-university also announces exclusions (frontend, full-stack, SQL) but its omissions are dictated by an exam. Carroll has no exam. His omissions are argued, and he supplies the forwarding address — which is the behaviour a reader actually needs from a text at the edge of a canon, and which this corpus can now say is rare.

A third apparatus finding. The book assumes no computational tool — no Sage (judson-abstract-algebra), no Matlab (thibos-fourier-analysis) — and carries 54 pages of complete worked solutions, where corral-vector-calculus‘s 420 exercises answer only a subset. Both follow from the SUMS specification, which is the first case here of a publisher rather than an author setting the pedagogy: one-to-two semesters, fully-worked solutions, self-study viable. The corpus has been reading these decisions as authorial; at least sometimes they are contractual.

The hole this opens is larger than the one it closes. complex-analysis enters the corpus from the top: the spoke now holds the geometric second course and none of the settled first one. It is also an edge nobody had listed — the shelf ran calculus → analysis → algebra and simply skipped the complex plane. That it went unnamed through eight growth-edge revisions is worth recording about the edge-listing method itself, which catches subjects the existing sources gesture at and misses the ones they never mention.

Open questions

1. Is this spoke mathematics or quantitative methods? Flagged in the parked record before the split and called mathematics on two-of-three. keisler-elementary-calculus makes it three of four, and does so emphatically — an argument about the foundations of a subject is not a quantitative method by any reading. Leaning settled; leaving it open one more source. 2026-08-06 pulls both ways in one day. stirzaker-elementary-probability is unambiguously mathematics — CUP, an Oxford mathematician, theorems and exercises. mari-measurement-across-sciences, routed hours earlier, is unambiguously not. The name question is now less about drift than about whether the spoke tolerates one deliberate outlier (see question 7).

2. Did the ε-δ definition win on merit? The sharpest question the corpus has raised and the one it cannot answer. Keisler’s case is that limits took the curriculum in the 1870s because infinitesimals had no rigorous account, that Robinson removed that reason in 1960, and that the curriculum never revisited the decision. Corral, writing in 2008, simply assumes the limit-based sequence as prerequisite and does not mention the alternative — which is either evidence for Keisler’s thesis about inertia, or evidence that the profession considered and declined it. The corpus holds no source that argues the standard side deliberately.

3. Where does information and coding theory live? ../machine-learning-wiki holds information-theory-inference-learning-algorithms, routed there hours before this spoke existed, half of which is compression and coding theory that spoke does not claim. The routing stands, but the subject belongs here. The next such source should land here rather than pulling the first one over.

4. Does a linear-algebra text arrive, and from where? Answered 2026-08-04, and from neither of the two places the question guessed. hefferon-linear-algebra arrived via the research pass rather than a routed source, from a state university rather than Strang/MIT OCW, and it brought trench-real-analysis with it — so the analysis half is now filled from the standard side as well as Keisler’s hyperreal one. See linear-algebra and real-analysis.

5. The research-methods fork just got live — and the deciding source came down on the fork’s side, without settling it. Flagged in CLAUDE.md before the split and dormant since: if the corpus fills with experimental design, hypothesis testing and study statistics, that is arguably a different and better spoke. larsen-marx-mathematical-statistics brought ANOVA, randomized block designs, contrasts and multiple comparisons — that is experimental design — but it was read as strengthening the mathematics side, because it derives its distributions, proves its theorems, and teaches design as a consequence of the F test rather than as a protocol. A research-methods text would do the reverse. herzog-understanding-statistics (2026-08-06) does the reverse. It carries no derivations at all, teaches at the level a working experimentalist needs, and devotes a third of its length to meta-analysis, replication and the Test for Excess Success — statistics about research practice, which is the fork’s own description. By the test this question set, that is one vote each way. It was routed here anyway and the reasoning is worth recording rather than hiding: the spoke’s boundary rule files a text that teaches a body of theory from first principles here even when its examples come from one field, and this book does teach the subject from the ground up. But it is the first source whose centre of gravity sits outside the mathematics reading, and a second one like it should trigger the fork rather than another routing note. The condition to watch: a text that assumes the machinery and argues about how to use it, rather than building it.

6. Does the origin-predicts-quality reading survive five texts? The corpus recorded early that where a text came from predicts its tier and almost nothing about its usefulness (schoolcraft-college). Five texts in, the pattern is holding but the reason has shifted: the two new T1s earn the grade from a professional-society award and a prior commercial edition respectively — neither of which is about the institution. Worth one more source before it is stated as a finding.

7. Should a book that is not mathematics be in the mathematics spoke? Live from 2026-08-06, when mari-measurement-across-sciences was routed in from Telegram. It is measurement science and philosophy of measurement — metrology, the ontology of properties, scale types, psychometric validity — with no theorem in it that the spoke would otherwise hold. Three arguments carried the routing: no spoke in the hub owns measurement as a subject; at least three borrow it (statistics here, psychometrics in ../psychology-wiki, capability benchmarking in ../agentic-tooling-wiki and ../ai-governance-wiki); and the spoke’s own boundary rule files a text that builds a body of theory from first principles here even when its examples come from one field. The argument against is simply that the rule says mathematics as the subject, and this book’s subject is not mathematics. ../psychology-wiki was the runner-up on the validity and psychometrics half; ../philosophy-wiki was considered for chapter 4 and declined, since that spoke owns philosophers and their systems rather than philosophy of science. This bears on question 1 more than on question 5. The fork question 5 watches is mathematics vs research methods, and this book is neither — it sits under both. What it does supply is the strongest evidence yet that the spoke’s real subject is closer to quantitative methods than to mathematics proper, which is exactly what question 1 asked and had provisionally answered the other way. Two of the last three arrivals now sit outside a strict reading of the name. The condition to watch: a second measurement-science or philosophy-of-measurement source. Two would make a cluster the hub could spin out rather than a stretch this spoke absorbs.

Paid first, free second

lal-algebra-1 closes the last untouched subject on the growth-edge list and breaks the corpus’s licence pattern doing it: Springer, $49.99 for the eBook, $89.99 in hardcover, no open licence.

That has now happened twice in the same shape. Statistics arrived as larsen-marx-mathematical-statistics (Pearson, paid), and an open text — herzog-understanding-statistics, CC BY-NC — followed two days later to close the licence half separately. Algebra has arrived paid, and the open half is filed as its own edge rather than pretended away.

The observation worth keeping is about this wiki’s own collection policy: a corpus that only accepted open licences would have neither statistics nor algebra today. The free texts here are excellent and several are better than their commercial peers (hefferon-linear-algebra took a professional society’s award), but they arrive when someone chooses to write one, not when a subject needs covering. Access and quality are independent axes, and so are access and coverage. springer makes the same point from the publisher side: the same house issued the corpus’s paywalled algebra text and its open-access statistics text.

Updated hours later: the free text was pulled rather than waited for. At the curator’s instruction to go deeper, judson-abstract-algebra (GFDL, maintained since 1997) was brought in — closing edge 8 the same day it opened, and giving the subject a text this wiki has actually read rather than one it knows from a publisher’s page. So the pattern above holds in shape and collapses in time: paid first, free second, four hours apart, because the second one was fetched on purpose.

That changes what the corpus can claim. abstract-algebra now carries read definitions — the group axioms, and the ring → commutative ring → integral domain → division ring → field hierarchy with the examples that separate them (ℤ is a domain and not a field; ℤ₁₂ has zero divisors; the quaternions are a noncommutative division ring). group-theory is paged as the subject’s deepest corner, which it is in both books: Lagrange and its number-theoretic corollaries, the isomorphism theorems, Sylow, the classification of finite abelian groups.

And the subject arrived with a disagreement already in it, which no other subject here has on day one. The two texts differ about what a first course is for: Lal opens with two chapters of logic and set theory (Zermelo–Fraenkel with choice, motivated by Russell’s and Cantor’s paradoxes) and spends his later chapters on classification — Schreier, Jordan–Hölder, Remak–Krull–Schmidt, solvable and nilpotent groups. Judson opens with a short note on proofs, spends two full chapters on cryptography and coding theory a third of the way in, and reaches Galois theory in the same volume. Judson states his reason: applications exist to motivate students meeting rigorous proof for the first time. That is this corpus’s recurring move — a pedagogical argument, offered without evidence, in a preface — arriving now about content selection rather than about rigour or foundations.

The asymmetry to keep in view: Judson’s body was read, Lal’s was not. Where the two are described as differing, that is a comparison of stated scope, not of exposition.

Tensions

The foundational fork (opened 2026-08-02, on the fourth ingest)

The corpus’s first genuine disagreement, and it needs stating precisely because it is easy to overstate.

• corral-vector-calculus names the standard limit-based Calculus I and II as its prerequisite and builds multivariable calculus on top of them.
• keisler-elementary-calculus rebuilds exactly that prerequisite on the hyperreals, gets derivative, continuity and integral by chapter 4, holds limits back to chapter 5, and then covers Corral’s own material again in chapters 10–13 — gradients, line integrals, Green’s, Stokes, Gauss.

This is not a contradiction about mathematics. The theorems match, the notation matches, the methods for solving problems match; Keisler says so himself and keeps the traditional limit definition in the book rather than replacing it. The disagreement is about what a student should meet first, and behind that, about whether the ε-δ definition earned its place on merit or won by default during the century when infinitesimals had no rigorous account.

Recorded as a live tension rather than resolved, because the corpus cannot resolve it: one side is an assumption Corral never argues for, and the other is an argument Keisler makes at length with one second-hand study behind it. What the hub can say is that the two agree on every result they both cover, which is the strongest corroboration this spoke has of anything.

Second assumer, 2026-08-04. trench-real-analysis is the rigorous course where the ε-δ choice is made explicit for the first time in a curriculum — chapter 1 builds the reals from completeness and the whole development follows — and it does not mention infinitesimals either. That is the place an argument would live if anyone were making one, so the silence is more informative here than Corral’s was. It still is not evidence against Keisler; it is a second data point for the inertia reading and no data point for the merit reading. The corpus now covers multivariable calculus three times over (corral-vector-calculus, Keisler ch. 10–13, Trench ch. 5–7) and the three agree on the results while two of them decline to discuss the foundation. Restated as growth edge 3.

Provenance versus usefulness

The other standing tension is between the corpus’s own provenance and its usefulness. The most authoritative source (mcmullen-probability-theory, T1, Harvard, primary) is also the least self-sufficient — terse notes that assume a 1950 textbook the hub does not hold. The least authoritative (corral-vector-calculus, T2, community-college adjunct, self-published) is the most complete and the only one under a free licence. Provenance tier and reader value point in opposite directions here, which is worth remembering the next time a source is graded.

Weakened 2026-08-04, deliberately not withdrawn. hefferon-linear-algebra is T1 and also the most self-sufficient text in the corpus — full proofs, one calculus course assumed, solutions manual, video lectures. It is the first source here where tier and reader value point the same way, which shows the tension was never a law, only the shape of the founding four. The reasoning behind each grade is now written on the page rather than left to the letter (trench-real-analysis earns T1 from a prior Pearson edition, Hefferon from an MAA award), because a bare tier letter carried none of what the sentence above is actually about. Corral’s case stands unchanged.

Growth edges

Ranked; each names the kind of source that would close it (see ../QUALITY.md → Growth edges).

1. Linear algebra — closed 2026-08-04 by hefferon-linear-algebra (T1, research pass).
2. Analysis — closed 2026-08-04 by trench-real-analysis (T1, research pass).
3. A text that argues for the ε-δ foundation — now the spoke’s sharpest hole, and promoted here because the two closures above made it worse rather than better. The corpus holds three texts that assume the standard development and one that argues against it; nothing defends it. — needs: a T1 text or paper that considered non-standard analysis and declined it in print (a foundations-of-mathematics-education article, or a preface that engages Keisler directly).
4. Statistics and inference — closed 2026-08-04 by larsen-marx-mathematical-statistics (T1, Pearson 5th ed.), which brings estimation, testing, regression, ANOVA, blocked designs and nonparametrics in one text. See mathematical-statistics.
5. A freely-licensed statistics text. CLOSED 2026-08-06 by herzog-understanding-statistics (Herzog, Francis & Clarke, Springer 2019, open access), routed in from Telegram. It restores the corpus’s free-by-licence pattern that Larsen & Marx broke — with one qualification worth keeping: the licence is CC BY-NC, so it is free to read and share but carries a non-commercial restriction that hefferon-linear-algebra and trench-real-analysis do not. The edge said “open-licensed” without saying how open, and this is the weaker kind.
6. Information and coding theory as a subject. MacKay’s ITILA sits in machine-learning-wiki, routed there on dominant substance; the next source of this kind lands here. — needs: T1 text.
7. Abstract algebra and number theory — subject closed 2026-08-08 by lal-algebra-1 (Ramji Lal, Springer 2017, T1): groups, rings, fields and arithmetic in one volume, with number theory woven through it rather than separated out (abstract-algebra). The edge’s own qualifier is not met — it asked for an open textbook and this one is commercial ($49.99 / $89.99). So the subject is covered and the licence hole is new: an open-licensed algebra text is still wanted, and that is the residue of this edge rather than the whole of it.
8. An open-licensed algebra text — added and closed the same day (2026-08-08) by judson-abstract-algebra (GFDL, T1), pulled deliberately once lal-algebra-1 closed the subject and left the licence half open. It also carries Galois theory to the fundamental theorem, which Lal defers to a later volume. Still open within algebra: modules and representation theory, in neither book. — needs: a graduate-level text or a survey.

Coverage edges (added 2026-08-08, at the curator’s request for a wider backlog). These widen what the spoke covers instead of answering an open question above; one ordinary solid source closes any.

9. Differential equations. The applied subject with the widest reach across every other spoke, and the shelf here goes from calculus straight to analysis without it. — needs: an open-licensed ODE text (Trench wrote one, matching trench-real-analysis).
10. Discrete mathematics and combinatorics. The half of the subject computing readers need most — counting, recurrences, graphs — is unheld. — needs: an open-licensed text.
11. Logic and set theory. abstract-algebra and real-analysis both sit on foundations no page states, and the proof assistants in research-wiki assume the same material. — needs: an open-licensed logic or set-theory text.
12. Decision theory and utility theory. Opened 2026-08-10 by osborne-rubinstein-course-in-game-theory, which assumes von Neumann–Morgenstern expected utility and Savage’s subjective probability as prerequisites and explicitly declines to defend them (“we do not discuss the assumptions that underlie the theory of a rational decision-maker”). game-theory therefore sits on foundations no page here states. — needs: a T1 text on decision theory under uncertainty, or von Neumann & Morgenstern itself.
13. Mechanism design, and experimental / behavioral game theory. The two halves the source names as its own gaps — the authors say they would add chapters on experimental game theory and on learning and evolution if starting over, and implementation theory (ch. 10) is only the pure end of mechanism design. — needs: a modern game-theory text, or an auction/mechanism-design text.
14. The first course in complex analysis. Opened 2026-08-10 by carroll-geometric-function-theory, which is explicitly a second course and lists the settled first-course syllabus as its prerequisite. complex-analysis is the only subject page here written from the top down. — needs: a standard first-course text (Howie, in the same series, is the one Carroll names as sufficient; Stein & Shakarchi the one he recommends alongside).
15. Curvature and Riemann surfaces. Named by carroll-geometric-function-theory §1.4 as deliberate omissions, with the forwarding addresses given: Ahlfors’s Conformal Invariants and Krantz’s Complex Analysis: The Geometric Viewpoint. — needs: either of those.
16. Numerical methods. Floating point, conditioning and quadrature are where the mathematics in this corpus meets the machines in the rest of it. — needs: a standard text, or Goldberg’s floating-point paper for the arithmetic half.

Index — Mathematics Wiki

Catalog of every page, grouped by schema.org @type. Spine: synthesis (thesis), log.md (history), this file (catalog). Spun out of the hub _inbox on 2026-08-02 (3 sources, three tags); a fourth landed the same day. Most subjects here rest on a single source. Three are doubled: vector-calculus (three accounts, disagreeing about foundations), probability-theory and mathematical-statistics (two each, agreeing about the mathematics and differing in altitude). Record the level a text is pitched at alongside its tier — the nine sit at visibly different rigour.

DefinedTerm (concepts)

• mathematics — the root page: why a mathematics spoke exists at all, the four texts, and the boundary rule (subject here, application there) · root
• probability-theory — sample spaces, counting, distributions, the law of large numbers; specifying the space is the whole difficulty · field
• vector-calculus — several-variable and vector-field calculus, ending at Green’s/Stokes’/Divergence; exports the gradient to half the hub · field
• fourier-analysis — time and frequency as two descriptions of one signal; orthogonality, sampling, linear systems · field
• infinitesimal-calculus — calculus built on hyperreals instead of limits; Robinson’s 1960 result, standard parts, and the claim that ε-δ won by default · foundation
• linear-algebra — vector spaces and the maps between them; the subject every other one here consumes and none supplied · field
• real-analysis — calculus proved rather than computed, from completeness to metric spaces; where the ε-δ choice becomes explicit · field
• mathematical-statistics — probability pointed backwards: estimation (MLE, Cramér–Rao, Bayesian) and hypothesis testing (the generalized likelihood ratio); the first subject here that sits downstream of three others · field
• abstract-algebra — sets carrying operations that satisfy axioms, with the ring → domain → division ring → field hierarchy and the examples that separate them (ℤ is a domain, not a field; ℤ₁₂ has zero divisors); the structure under linear-algebra rather than beside it. The corpus’s first subject to arrive with two texts that disagree about what a first course is for · field
• group-theory — the subject’s deepest corner and the bulk of both algebra texts: Lagrange (and Fermat/Euler as corollaries), the isomorphism theorems, Sylow, finite abelian classification, group actions; symmetry and permutation/matrix groups as the two sources · field
• game-theory — what happens when decision-makers whose outcomes depend on each other choose: strategic vs extensive games, perfect vs imperfect information, and the solution concepts (Nash, sequential, the core, the Shapley value) that are most of the subject. Arrives with its own authors denying it is mathematics · field
• complex-analysis — calculus of the plane, and where analysis meets geometry through conformal mapping; entered from a second course with no first course held, so the settled half of the subject is the corpus’s largest single hole · field
• measurement-theory — what makes a procedure a measurement, and what its number means: four necessary conditions, model-dependent realism, scale types, and the error / uncertainty / validity split between the physical and human sciences. Not mathematics — the spoke’s odd one out, recorded as open question 7 · field

Course (sources)

• mcmullen-probability-theory — Curtis McMullen, Harvard 2011 (rev. 2021), 98pp: terse notes accompanying Feller, with the mailman paradox, Jovian birthdays and Benford’s law as its teaching set · source · T1 · people.math.harvard.edu
• thibos-fourier-analysis — Larry Thibos, Indiana University School of Optometry, 6th ed. 2014, 201pp: Fourier analysis as an experimentalist’s method, including three chapters on the statistics of measuring a coefficient · source · T2 · PDF only, no URL

Book (sources)

• keisler-elementary-calculus — H. Jerome Keisler, 1976/1986 (Dover 3rd ed. 2012), 992pp scanned, CC BY-NC-SA: the whole calculus sequence on infinitesimals, limits deferred to ch. 5; the corpus’s only pedagogical claim with an experiment behind it · source · T1 · people.math.wisc.edu

• corral-vector-calculus — Michael Corral, 2008, 222pp, GNU FDL: a complete Calculus III course with 420 graded exercises, whose author rates his own rigour 5 out of 10 · source · T2 · mecmath.net

• hefferon-linear-algebra — Jim Hefferon, 4th ed. 2020, GFDL/CC BY-SA: proves everything, assumes one semester of calculus, MAA Solow Award 2020 · source · T1 · hefferon.net

• larsen-marx-mathematical-statistics — Larsen & Marx, Prentice Hall/Pearson, 5th ed. 2012 (1st 1981), ~753pp: the two-semester mathematical-statistics sequence — estimation through Cramér–Rao and Bayes, testing through the generalized likelihood ratio, then regression, ANOVA, blocked designs and nonparametrics. A “Taking a Second Look at Statistics” essay per chapter on how the methods mislead. The corpus’s first text that is accessible without being freely licensed · source · T1 · © Pearson, read from an Emory course-page PDF

• trench-real-analysis — William F. Trench, free ed. 2013, CC BY-NC-SA: two-term real analysis, reissued after a Pearson edition; the corpus’s first rights-reverted text · source · T1 · digitalcommons.trinity.edu

• stirzaker-elementary-probability — David Stirzaker, Elementary Probability (CUP, 2nd ed. 2003, 538pp): a self-contained undergraduate course in three parts — basic probability, discrete random variables, continuous random variables — plus counting, generating functions and Markov chains, with worked examples, problems and solutions. The 2nd edition added martingales, optional stopping and the Wiener process (Black–Scholes as the stated motivation). The spoke’s second probability text, and its second in-copyright one · source · T1 · ctanujit.org PDF, © CUP

• herzog-understanding-statistics — Herzog, Francis & Clarke, Understanding Statistics and Experimental Design: How to Not Lie with Statistics (Springer, open access CC BY-NC, 2019, 142pp): applied inference in 8 chapters, then 4 on meta-analysis, replication and the Test for Excess Success — sum the power across a set of studies and compare with how often they actually succeeded (precognition: 6.27 expected vs 9 reported, p=0.058; bystander effect: 10.77 vs 10, p=0.76). Argues replication cannot arbitrate at moderate power. Closed the free-statistics-text edge; no derivations, so it adds no theory · source · T1 · link.springer.com

• lal-algebra-1 — Ramji Lal, Algebra 1: Groups, Rings, Fields and Arithmetic (Springer 2017, Infosys Science Foundation Series, 11 chapters / ~433pp): two chapters of logic and set theory before any algebra, then number systems, groups, rings and fields, with arithmetic returning twice once rings can state it. Volume one of three, undergraduate and graduate audience. Closes the abstract-algebra edge and breaks the free-licence pattern doing it ($49.99 eBook / $89.99 hardcover). Body not read — this page rests on the publisher’s chapter list and description · source · T1 · link.springer.com

• judson-abstract-algebra — Thomas Judson, Abstract Algebra: Theory and Applications (GNU FDL, 1997–, 23 chapters): groups → rings → fields, reaching Galois theory in one volume, with two full chapters of applications (cryptography, algebraic coding theory) placed early because students meeting rigorous proof need motivation — the corpus’s pedagogical argument arriving about content selection. Sage material by Rob Beezer; maintained for nearly three decades. Closed the open-licensed-algebra edge the day it opened · source · T1 · abstract.ups.edu

• mari-measurement-across-sciences — Mari, Wilson & Maul, Measurement Across the Sciences (Springer, open access CC BY 4.0, 2nd ed. 2023, 307pp): a shared concept system for measurement covering thermometers and reading-comprehension tests alike — four necessary conditions, model-dependent realism, an ontology of properties and scale types, the Hexagon Framework for measurand identification, and the comparison of the physical sciences’ uncertainty approach with the human sciences’ validity approach. A conceptual monograph, not a textbook; the corpus’s first unrestricted open licence, and its first source that is not mathematics · source · T1 · link.springer.com

• carroll-geometric-function-theory — Tom Carroll, Geometric Function Theory: A Second Course in Complex Analysis (springer 2024, SUMS, 358pp): the Riemann sphere and hyperbolic disk, Schwarz–Pick in three widening versions, normal families to the Great Picard Theorem, the Riemann Mapping Theorem, the Koebe ¼-Theorem, and uniformisation of planar domains — plus 54 pages of full solutions and no computational tool assumed. The corpus’s sharpest curriculum finding: “consensus evaporates when it comes to a second course”, and the only text here with an explicit §1.4 “What Is Not in This Book” · source · T1 · PDF held in raw/, © Springer

• osborne-rubinstein-course-in-game-theory — Osborne & Rubinstein, A Course in Game Theory (mit-press 1994, 372pp scanned): the graduate one-semester course — strategic games and Nash equilibrium, extensive games through bargaining, repeated games and four folk theorems, sequential equilibrium, then coalitional games to the core and the Shapley value. Proof-complete and deliberately pure-theory (“the art of applying an abstract model to a real-life situation should be the subject of another tome”). The corpus’s first text whose authors call their subject a social science, not a branch of mathematics · source · T1 · PDF from an Internet Archive scan, © MIT Press

• illowsky-introductory-statistics — Illowsky & Dean, Introductory Statistics 2e (openstax/Rice University, ©2026, CC BY-NC-SA 4.0, 847pp, held in raw/): the standard one-semester American service course in 13 chapters — sampling → description → probability → CLT → confidence intervals → one- and two-sample tests → chi-square → regression → one-way ANOVA — reaching all of it without calculus and without a proof, for “students majoring in fields other than math or engineering”. Assumes intermediate algebra and a TI-83/84 calculator; carries the authors’ own statistics labs, four practice exams, and a 2e revision whose stated themes are “currency, relevance, accuracy, and belonging”. The corpus’s rigour floor, and its first institutionally-published open text · source · T1 · openstax.org

Person (entities)

• michael-herzog — Brain Mind Institute, epfl; lead author of herzog-understanding-statistics (not michael-corral — the index near-matches the names)
• gregory-francis — Purdue; co-author, and the researcher behind the Test for Excess Success
• martin-j-osborne — economist, mcmaster-university at the time of writing; taught game theory at Stanford by Aumann, Hart, Kurz and Wilson · author
• ariel-rubinstein — tel-aviv-university and Princeton; the alternating-offers bargaining model and the bounded-rationality thread that shapes three chapters of the book · author
• tom-carroll — university-college-cork analyst, Hayman and Rippon student; the first author here who had to argue about what the syllabus is rather than how to teach an agreed one · author
• curtis-mcmullen — Harvard mathematician; author of the probability notes · author
• david-stirzaker — Oxford mathematician (St John’s; Grimmett & Stirzaker); defends formal definitions on student-error grounds, the mirror image of h-jerome-keisler‘s argument · author
• luca-mari — LIUC industrial engineering; the metrology half of mari-measurement-across-sciences · author
• mark-wilson — Berkeley School of Education; the psychometrics half · author
• andrew-maul — UCSB Gevirtz School of Education; the philosophy-of-measurement half · author
• thomas-judson — stephen-f-austin-state-university; kept his algebra text free under the GFDL since 1997 and still maintains it — the sustained-maintenance pattern he shares with jim-hefferon, h-jerome-keisler and william-trench · author
• ramji-lal — HRI Allahabad mathematician; author of the three-volume algebra series whose first volume is here · author
• barbara-illowsky — De Anza College; co-author of illowsky-introductory-statistics and of the Collaborative Statistics it grew from; the spoke’s first community-college author · author
• susan-dean — De Anza College; co-author of the same text and co-creator of its statistics labs · author
• michael-corral — Schoolcraft College adjunct; writes free-licensed mathematics textbooks · author
• larry-thibos — Indiana vision scientist; six editions of the same coursenotes over 25 years · author
• h-jerome-keisler — Wisconsin logician; kept his textbook free for 24 years, including through a Dover edition · author
• jim-hefferon — Vermont mathematician; CTAN maintainer and TUG officer, three open textbooks · author
• richard-larsen — Vanderbilt statistician; five editions over 31 years, the corpus’s longest sustained authorship · author
• morris-marx — University of West Florida statistician; co-author of the same · author
• william-trench — Trinity professor emeritus; reissued a Pearson title free once the rights reverted · author

Organization (entities)

• stephen-f-austin-state-university — CollegeOrUniversity: thomas-judson‘s institution; hosts the book in its ScholarWorks repository · institution
• mit-press — academic publisher; the spoke’s second publisher node, holding osborne-rubinstein-course-in-game-theory, which the authors typeset themselves in TeX · publisher
• mcmaster-university — CollegeOrUniversity: martin-j-osborne‘s department in 1994 · institution
• tel-aviv-university — CollegeOrUniversity: ariel-rubinstein‘s home department · institution
• openstax — nonprofit publisher at Rice University; free peer-reviewed textbooks under CC BY-NC-SA, produced institutionally rather than by their authors — revisions keep problem numbering stable so online homework providers can follow an edition change · publisher
• de-anza-college — Cupertino community college; home of barbara-illowsky and susan-dean, and the roster’s second two-year college · institution
• springer — academic publisher; the spoke’s first publisher node holding two of its books, at opposite ends of the access axis (paywalled lal-algebra-1, open-access herzog-understanding-statistics) · publisher
• springer-undergraduate-mathematics-series — CreativeWorkSeries: SUMS, springer‘s undergraduate line; a publisher-imposed pedagogical specification — one-to-two semesters, fully-worked solutions, self-study viable — visible in the one volume held · series
• university-college-cork — CollegeOrUniversity: tom-carroll‘s school, and the corpus’s one case of a department’s disciplinary taste being given as the reason a book exists · institution
• infosys-science-foundation-series — CreativeWorkSeries: the Springer series (ISFS/ISFM) carrying lal-algebra-1; a hook for volumes two and three · series
• harish-chandra-research-institute — ResearchOrganization: mathematics and theoretical-physics institute in Allahabad; ramji-lal‘s affiliation · institution
• epfl — École Polytechnique Fédérale de Lausanne; the Brain Mind Institute is michael-herzog‘s home
• purdue-university — psychological sciences; gregory-francis‘s home
• harvard-university — CollegeOrUniversity: hosts the notes on faculty web space, no publisher in between · institution
• indiana-university — CollegeOrUniversity: its School of Optometry is why the Fourier text has statistics chapters · institution
• schoolcraft-college — CollegeOrUniversity: community college, and the origin of the corpus’s only GFDL text · institution
• university-of-wisconsin-madison — CollegeOrUniversity: hosts the Keisler text, and produced the 1974 thesis that tested its method · institution
• university-of-vermont — CollegeOrUniversity: the fifth institution in five texts, and the one that breaks the origin-predicts-quality pattern · institution
• vanderbilt-university — CollegeOrUniversity: where the institution and the publisher finally come apart — the first text here that reached readers through Pearson rather than its authors’ servers · institution
• university-of-west-florida — CollegeOrUniversity: a regional public university co-authoring with a private research one, indistinguishably, inside one text · institution
• university-of-oxford — CollegeOrUniversity: david-stirzaker‘s Mathematical Institute and St John’s College; with harvard-university, the second elite-department origin, and both produced probability texts · institution
• liuc — CollegeOrUniversity: Università Cattaneo, Castellanza — the corpus’s first non-US institution and first engineering school · institution
• uc-berkeley — CollegeOrUniversity: Berkeley School of Education, mark-wilson‘s home · institution
• uc-santa-barbara — CollegeOrUniversity: Gevirtz Graduate School of Education, andrew-maul‘s home · institution
• trinity-university — CollegeOrUniversity: its Digital Commons repository hosts the analysis text — an institutional archive rather than a personal domain · institution