# References — Chapter ᚆ(H): The observation car *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-8, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch08.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 8.1 {#r-8.1} The Iverson bracket [P] is 1 when P is true and 0 when P is false; Iverson introduced it in 1962, restricted to a single relation in parentheses, and Knuth introduced the square brackets and the generality, campaigning for them in Two Notes on Notation, American Mathematical Monthly 99 (1992), 403–422. *Status: confirmed* - [Wikipedia, Iverson bracket](https://en.wikipedia.org/wiki/Iverson_bracket) — defines the bracket as the indicator of a proposition and attributes its 1962 introduction to Kenneth Iverson; the history section records Knuth’s later advocacy. - [Donald E. Knuth, Two notes on notation, arXiv:math/9205211](https://arxiv.org/abs/math/9205211) — Knuth’s article free on arXiv, advocating the notation [P] and extending Iverson’s idea; journal reference Amer. Math. Monthly 99 (1992), no. 5, 403–422. ### 8.2 {#r-8.2} George Boole published the algebra of logic in 1847 and 1854, with Boolean operations represented arithmetically by 0 and 1. *Status: confirmed* - [Wikipedia, Boolean algebra](https://en.wikipedia.org/wiki/Boolean_algebra) — records Boole’s The Mathematical Analysis of Logic (1847) and The Laws of Thought (1854), and the Boolean operations and two-element algebra. ### 8.3 {#r-8.3} Claude Shannon showed in 1937 that Boolean algebra models switching circuits. *Status: confirmed* - [Wikipedia, Boolean algebra](https://en.wikipedia.org/wiki/Boolean_algebra) — history section identifies Shannon’s 1937 master’s thesis as the application of Boolean algebra to electrical switching circuits. ### 8.4 {#r-8.4} Fermat’s Last Theorem says that for every integer n>2 there are no positive integers x,y,z with x^n+y^n=z^n; Fermat wrote the claim in a margin around 1637 and the 1670 edition was published by his son. Whether Fermat had a valid general proof is not known and appears unlikely; the proof by infinite descent he did leave behind covers the exponent 4. *Status: confirmed* - [Wikipedia, Fermat’s Last Theorem](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) — infobox and overview give the integer statement, c. 1637 margin, and posthumous 1670 publication in Samuel Fermat’s edition of Arithmetica; on Fermat’s own proof, “It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely,” and the descent proof he did leave is the one for fourth powers. ### 8.5 {#r-8.5} Euler proved the n=3 case, Sophie Germain proved the first case of the theorem for a broad class of prime exponents, and Kummer proved the theorem for regular primes in ~1847. The two standard histories date it differently and both are given below. *Status: confirmed* - [Wikipedia, Fermat’s Last Theorem](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) — historical-development section lists the proof of exponent 3, Germain’s theorem for a class of primes, and Kummer’s proof for regular primes, without a year. - [MacTutor, Fermat’s last theorem](https://mathshistory.st-andrews.ac.uk/HistTopics/Fermat's_last_theorem/) — supports 1847 — “By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular … if”. - [Wikipedia, Regular prime](https://en.wikipedia.org/wiki/Regular_prime) — supports 1850 — “In 1850, Kummer proved that Fermat’s Last Theorem is true for a prime exponent p if p is regular”, i.e. the published general proof, Crelle 40. ### 8.6 {#r-8.6} Andrew Wiles worked on the theorem alone for seven years, in the attic office of his house in Princeton, and announced a proof in three lectures on 21–23 June 1993. Peer review exposed a gap: the Euler system used to extend the Kolyvagin–Flach method was incomplete. He spent almost a year on the repair, by himself and then with his former student Richard Taylor, and found the way around the gap on 19 September 1994. The corrected proof fills the May 1995 issue of the Annals of Mathematics as two papers: Wiles, “Modular elliptic curves and Fermat’s Last Theorem,” 141(3), 443–551, and Taylor and Wiles, “Ring theoretic properties of certain Hecke algebras,” 141(3), 553–572. *Status: confirmed* - [Wikipedia, Fermat’s Last Theorem](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) — the three lectures “on 21–23 June 1993 at the Isaac Newton Institute for Mathematical Sciences”; the error in peer review; Nick Katz “alerted Wiles on 23 August 1993”; the two manuscripts submitted on 24 October 1994; “the entirety of the May 1995 issue” of the Annals of Mathematics. - [Wikipedia, Wiles’s proof of Fermat’s Last Theorem](https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem) — Katz, as referee, “asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap”; “the Euler system used to extend Kolyvagin and Flach’s method was incomplete”; Wiles “spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor,” with the breakthrough on 19 September 1994; the papers are cited as Annals of Mathematics 141(3), 443–551 and 553–572. - [Encyclopedia.com, Andrew J. Wiles](https://www.encyclopedia.com/history/encyclopedias-almanacs-transcripts-and-maps/andrew-j-wiles) — “He spent seven years working on it in the attic office of his home in Princeton.” ### 8.7 {#r-8.7} In Boolean algebra x∧¬x=0 and x∨¬x=1; these express non-contradiction and excluded middle, both of them Aristotle’s, and Aristotle calls non-contradiction the firmest of all principles in Metaphysics Book IV. L. E. J. Brouwer challenged the unrestricted validity of the classical laws in 1908, in a paper whose title translates as The untrustworthiness of the principles of logic. Tradition names three laws of thought: identity, non-contradiction and excluded middle. ⊤ (tee, verum) denotes the constant true and ⊥ (up tack, falsum) the constant false. *Status: confirmed* - [Wikipedia, Boolean algebra](https://en.wikipedia.org/wiki/Boolean_algebra) — gives complement identities and identifies Boolean algebra as an algebra of truth values. - [Wikipedia, Law of excluded middle](https://en.wikipedia.org/wiki/Law_of_excluded_middle) — identifies excluded middle as a classical logical law, discusses its Aristotelian formulation, and gives Brouwer’s intuitionist refusal of it, but no year for it. - [L. E. J. Brouwer, Wikipedia](https://en.wikipedia.org/wiki/L._E._J._Brouwer) — read as raw HTML, quoting Kleene: “in 1908: ‘… Brouwer, in a paper titled “The untrustworthiness of the principles of logic,” challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384–322 B.C.) have an absolute validity, independent of the subject matter to which they are applied’.” - [Stanford Encyclopedia of Philosophy, Contradiction](https://plato.stanford.edu/entries/contradiction/) — introduces Aristotle’s law of non-contradiction as “the firmest of all principles,” quoting Metaphysics 1005b19–23 in Kirwan’s translation: “For the same thing to hold good and not to hold good simultaneously of the same thing and in the same respect is impossible.” - [Wikipedia, Law of noncontradiction](https://en.wikipedia.org/wiki/Law_of_noncontradiction) — gives Aristotle’s formulations in Metaphysics Book IV, including “The most certain of all basic principles is that contradictory propositions are not true simultaneously” (1011b13–14). - [Law of thought, Wikipedia](https://en.wikipedia.org/wiki/Law_of_thought) — names the three: “the law of identity (LOI), the law of non-contradiction (LNC), and the law of excluded middle (LEM)”; and, quoting Russell, “for no very good reason, three of these principles have been singled out by tradition under the name of ‘Laws of Thought’.” - [Tee (symbol), Wikipedia](https://en.wikipedia.org/wiki/Tee_(symbol)) — “The tee (⊤, \top in LaTeX), also called down tack (as opposed to the up tack) or verum, is a symbol used to represent: The truth value of being true in logic, or a sentence (e.g., formula in propositional calculus) which is unconditionally true.” - [Up tack, Wikipedia](https://en.wikipedia.org/wiki/Up_tack) — “The truth value ‘false’, or a logical constant denoting a proposition in logic that is always false. (The names “falsum,” “absurdum” and “absurdity” come from this context.)” ### 8.8 {#r-8.8} The four denominator brackets evaluate to 1: 1+1=2, √2 is irrational, there are infinitely many primes, and ℵ₀<𝔠; therefore their ordinary sum is 4 and adding excluded middle gives 5. *Status: confirmed* - [Wikipedia, Irrational number](https://en.wikipedia.org/wiki/Irrational_number) — states that √2 is irrational. - [Wikipedia, Euclid’s theorem](https://en.wikipedia.org/wiki/Euclid%27s_theorem) — states that there are infinitely many primes. - [Wikipedia, Continuum hypothesis](https://en.wikipedia.org/wiki/Continuum_hypothesis) — states that the cardinality of the integers is strictly below that of the reals. - **Own computation** (the author's own work) — Each displayed proposition is true, so its Iverson bracket is 1; 1+1+1+1=4 and the Boolean excluded-middle term contributes 1, giving denominator 5. ### 8.9 {#r-8.9} The irrationality of √2 is credited by tradition to the Pythagoreans, in the fifth century BC; no original proof survives, and the reference works that report the tradition also record that neither the discoverer nor the date can be fixed. √2 ≈ 1.4142 is the diagonal of a unit square and probably the first number known to be irrational; historians do not agree on how the original proof went. *Status: judgement* - [Wikipedia, Irrational number](https://en.wikipedia.org/wiki/Irrational_number) — gives the traditional story, hedged: “The first proof of the existence of irrational numbers is attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram.” The historians’ disagreement over how the original proof went is on the arXiv line. - [MacTutor, Pythagoras](https://mathshistory.st-andrews.ac.uk/Biographies/Pythagoras/) — presents the Pythagorean historical tradition but does not establish a precise pre-400 BC document date. - [Square root of 2, Wikipedia](https://en.wikipedia.org/wiki/Square_root_of_2) — value, diagonal and history: “The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2.”; “Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational.”; “Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned.” - [Negrepontis et al., The anthyphairetic reconstruction of the original Pythagorean proof of incommensurability, arXiv:2501.08708](https://arxiv.org/abs/2501.08708) — the original proof is reconstructed, not extant: “Unquestionably the greatest discovery of the Pythagoreans is the existence of incommensurable magnitudes, most probably the incommensurability of the diameter to the side of a square, but there is no agreement among historians of Greek mathematics on their method of proof.”; “In this chapter we present novel arguments not only for an anthyphairetic reconstruction of the original Pythagorean proof of incommensurability” Editorial note: The tradition is universally reported and nowhere dated to a document. Wikipedia gives the Pythagorean discovery story together with the uncertainty about who proved it and how; MacTutor gives the tradition without a text behind it. So the page attributes the proof rather than dating it, and the entry is a judgement because “before 400 BC” rests on that tradition. The mathematics is not at issue: √2 is irrational, which is all the bracket in the denominator asserts. ### 8.10 {#r-8.10} Euclid’s proof that there are infinitely many primes appears in Elements IX.20 and is conventionally dated to about 300 BC. *Status: confirmed* - [Wikipedia, Euclid’s theorem](https://en.wikipedia.org/wiki/Euclid%27s_theorem) — “Euclid offered a proof in his work Elements (Book IX, Proposition 20)”; the c. 300 BC date is on the Euclid’s Elements line; the proof step, “Since q is either prime or not: If q is prime, then there is at least one more prime that is not in the list, namely, q itself. If q is not prime, then some prime factor p divides q”; and “Euclid is often erroneously reported to have proved this result by contradiction beginning with the assumption that the finite set initially considered contains all prime numbers, though it is actually a proof by cases, a direct proof method.” - [Euclid’s Elements, Wikipedia](https://en.wikipedia.org/wiki/Euclid%27s_Elements) — “The Elements … is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid.” ### 8.11 {#r-8.11} Cantor proved the real numbers uncountable in his article of 1874, and again by the diagonal argument of 1891, so ℵ₀<𝔠; he stated the continuum hypothesis in his paper of 1878. *Status: confirmed* - [Wikipedia, Continuum hypothesis](https://en.wikipedia.org/wiki/Continuum_hypothesis) — the hypothesis “was first introduced by Georg Cantor in his 1878 paper,” and “Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers.” - [Wikipedia, Cantor’s first set theory article](https://en.wikipedia.org/wiki/Cantor%27s_first_set_theory_article) — dates the article to 1874 and gives its proof that the set of real numbers is uncountable. - [Wikipedia, Cantor’s diagonal argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument) — dates the diagonal proof to 1891. ### 8.12 {#r-8.12} Gödel’s incompleteness theorems, published in 1931, say that sufficiently strong consistent effectively axiomatized theories have undecidable arithmetic truths and cannot prove their own consistency. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Gödel’s Incompleteness Theorems](https://plato.stanford.edu/entries/goedel-incompleteness/) — explains the 1931 first and second incompleteness theorems and their hypotheses: “The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F”; the standard statement is given “Accommodating an improvement due to J. Barkley Rosser in 1936.” ### 8.13 {#r-8.13} The continuum hypothesis is the first of the twenty-three problems Hilbert set in 1900. Gödel announced in 1938 and showed in full in 1940 that it is consistent with ZFC (relative to consistency), and Cohen showed in 1963 that it is independent of ZFC. Gödel’s model of ZFC satisfies CH and Cohen’s forcing model does not. *Status: confirmed* - [Wikipedia, Continuum hypothesis](https://en.wikipedia.org/wiki/Continuum_hypothesis) — attributes relative consistency to Gödel’s constructible universe and independence/non-provability to Cohen’s forcing work; its references give Gödel’s announcement as “Gödel, Kurt (1938). “The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis.” Proceedings of the National Academy of Sciences. 24 (12): 556–557.” - [Hilbert’s problems, Wikipedia](https://en.wikipedia.org/wiki/Hilbert%27s_problems) — read as raw HTML: the 1902 Bulletin of the AMS translation heads the list with “Cantor’s problem of the cardinal number of the continuum,” and the table of problems gives “1st: The continuum hypothesis: there is no set whose cardinality is strictly between that of the integers and that of the real numbers.” - [List of statements independent of ZFC, Wikipedia](https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC) — “Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC.” ### 8.14 {#r-8.14} Peano published his arithmetic axioms in 1889; Zermelo axiomatized set theory in 1908, and Fraenkel and Skolem supplied the later 1922 refinements represented in ZFC. Whitehead and Russell reach 1 + 1 = 2 by way of proposition *54.43, on page 379 of the first volume of Principia Mathematica in 1910, which remarks only that it will follow once addition has been defined; the proof is completed in the second volume, of 1912. From Peano’s axioms, with addition defined by recursion, 1 + 1 = 2 follows in three steps. *Status: confirmed* - [Wikipedia, Peano axioms](https://en.wikipedia.org/wiki/Peano_axioms) — gives Peano’s 1889 publication and the basic successor/induction framework; and “The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF..” - [Wikipedia, Zermelo–Fraenkel set theory](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) — gives the history of Zermelo’s 1908 axiomatization, Fraenkel’s and Skolem’s 1922 work, and ZFC. - [Wikipedia, Principia Mathematica](https://en.wikipedia.org/wiki/Principia_Mathematica) — proposition *54.43, “From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2,” stands on page 379 of the first volume of the first edition and page 362 of the second; the proof is completed in the second volume; the three volumes appeared in 1910, 1912 and 1913; and “The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell.” - [11.2 Arithmetic, MATH1001 Introduction to Number Theory, University of Southampton](https://personal.soton.ac.uk/jhr/math1001/arithmetic.html) — Theorem 11.2 derives it from the successor function and the recursive definition of addition: “1+1 = 1+s(0) = s(1+0) = s(1) = 2.” ### 8.15 {#r-8.15} The single and double turnstiles express syntactic and semantic consequence; Gödel’s completeness theorem of 1929 makes them coincide for first-order logic, while incompleteness still separates arithmetic truth from formal proof. The quantifier and connective signs are datable too: ∃ is Peano’s, of 1897, and ∀ is Gentzen’s, of 1935, made by analogy with it; ∨ is Russell’s of 1908, ¬ and ∧ are Heyting’s of 1930, and ⇒ and ⇔ are Bourbaki’s of 1954; and Frege’s Begriffsschrift of 1879, where quantified variables first appear, has left nothing behind but the turnstile. *Status: confirmed* - [Wikipedia, Turnstile (symbol)](https://en.wikipedia.org/wiki/Turnstile_(symbol)) — describes ⊢ as syntactic derivability and ⊨ as semantic consequence. - [Wikipedia, Gödel’s completeness theorem](https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem) — dates and states the completeness theorem for first-order logic. - [Stanford Encyclopedia of Philosophy, Gödel’s Incompleteness Theorems](https://plato.stanford.edu/entries/goedel-incompleteness/) — distinguishes completeness from incompleteness and explains the latter’s arithmetic consequences. - [Wikipedia, Quantifier (logic)](https://en.wikipedia.org/wiki/Quantifier_(logic)) — “In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano’s ∃ symbol,” which Peano introduced “in 1897”; ∀ “did not become canonical until the 1960s.” - [Wikipedia, Logical connective](https://en.wikipedia.org/wiki/Logical_connective) — the history of notations gives ∨ in Russell 1908, ¬ and ∧ in Heyting 1930, → in Hilbert 1918, and ⇒ and ⇔ in Bourbaki 1954. - [Wikipedia, Begriffsschrift](https://en.wikipedia.org/wiki/Begriffsschrift) — the 1879 calculus “contains the first appearance of quantified variables,” and “Some vestige of Frege’s notation survives in the ‘turnstile’ symbol ⊢.” ### 8.16 {#r-8.16} De Morgan’s laws were published in modern form by Augustus De Morgan in 1847 and interchange negated conjunction and disjunction. The observation is older: Aristotle made a similar one, William of Ockham wrote the laws out in words in the fourteenth century, and Jean Buridan gives rules of conversion along the same lines in his Summulae de Dialectica. In circuit design, De Morgan’s laws underlie building every Boolean function from NAND gates alone. *Status: confirmed* - [Wikipedia, De Morgan’s laws](https://en.wikipedia.org/wiki/De_Morgan%27s_laws) — states both identities and gives De Morgan’s 1847 Formal Logic attribution. Re-read raw on 17 September 2026 for the precursors: “a similar observation was made by Aristotle, and was known to Greek and Medieval logicians. For example, in the 14th century, William of Ockham wrote down the words that would result by reading the laws out. Jean Buridan, in his Summulae de Dialectica, also describes rules of conversion that follow the lines of De Morgan’s laws. Still, De Morgan is given credit for stating the laws in the terms of modern formal logic.” - [NAND logic, Wikipedia](https://en.wikipedia.org/wiki/NAND_logic) — “The NAND Boolean function has the property of functional completeness. This means that any Boolean expression can be re-expressed by an equivalent expression utilizing only NAND operations.”; and “noting from de Morgan’s law that a NAND gate is an inverted-input OR gate.” ### 8.17 {#r-8.17} The therefore sign ∴ carries its modern meaning in Johann Rahn’s Teutsche Algebra of 1659; the exclusive pairing of ∴ for therefore with the inverted ∵ for because dates from the nineteenth century; Q.E.D. translates the ὅπερ ἔδει δεῖξαι of Euclid and Archimedes; and the solid tombstone that now closes a proof is Paul Halmos’s, taken from the end marks of magazine typography. *Status: confirmed* - [Wikipedia, Therefore sign](https://en.wikipedia.org/wiki/Therefore_sign) — gives the 1659 Rahn attribution for ∴ and the inverse because sign’s history. - [Wikipedia, Q.E.D.](https://en.wikipedia.org/wiki/Q.E.D.) — explains Q.E.D. and the modern tombstone/halmos symbol, including Halmos’s adoption. Editorial note: Two of these are attributions rather than datings. Rahn used the triangle in both orientations, and so did writers after him; what the nineteenth century added was the exclusive convention. The tombstone is credited to Halmos on his own account — he “claims to have pioneered” it — which is how the reference works put it, and how the page takes it. ### 8.18 {#r-8.18} The chapter’s displayed factor evaluates to (0+0)/5=0, with Boolean truth values returning as ordinary integers. *Status: confirmed* - **Own computation** (the author's own work) — Fermat’s bracket is 0 by 8.4–8.6; x∧¬x=0 by 8.7; the denominator is 4+1=5 by 8.8; hence (0+0)/5=0. An Iverson bracket has values 0 and 1, which are ordinary integers by 8.1. ### 8.19 {#r-8.19} The chapter epigraph is Fermat’s marginal note, written about 1637 (the note is undated, and every source gives the year as approximate) and printed by his son Clement-Samuel in the 1670 edition of Diophantus’s Arithmetica; the margin itself does not survive. *Status: confirmed* - [Fermat’s Last Theorem, Wikipedia](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) — read as raw HTML: the Latin is given as “Cubum autem in duos cubos … cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet” and translated “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain”; and “After Fermat’s death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father’s comments.” Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker, and to make the provenance explicit: the margin itself does not survive, and what exists is the 1670 edition his son Clément-Samuel printed. The chapter’s footer was reworded to say so, and to give the son’s full name, in the pass of 18 September 2026. ### 8.20 {#r-8.20} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Peano built on Dedekind’s 1888 analysis; Boole’s algebra (1847, 1854); Goedel’s incompleteness theorems of 1931; Fermat’s marginal note in his copy of Diophantus’s Arithmetica, 1637. *Status: confirmed* - [Peano axioms, Wikipedia](https://en.wikipedia.org/wiki/Peano_axioms) — read raw: “In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book The principles of arithmetic presented by a new method.” - [Boolean algebra, Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra) — read raw: “Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854)..” - [Gödel’s incompleteness theorems, Wikipedia](https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems) — read raw: “These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in philosophy of mathematics..” - [Fermat’s Last Theorem, Wikipedia](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) — read raw: “The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica..” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.