ADA LOVELACE · XX

the other half — the first algorithm (Note G, 1843); the lea

LOGIKĒ XX — Ada Lovelace (1815–1852). In her 1843 Notes on Babbage's Analytical Engine, Note G lays out a step-by-step procedure for the machine to compute the Bernoulli numbers — recognized as the first published algorithm intended for a machine, and the reason she is called the first programmer. She also made the deeper leap: that such a machine could manipulate SYMBOLS, not just numbers — music, logic, anything representable — the seed of general-purpose computing. Live: run Note G and watch the engine compute exact Bernoulli numbers, plus the symbolic vision. Honest: Babbage built the engine; the 'first programmer' claim is debated; her leap is not. David Lee Wise, two-layer honest.

Ada Lovelace — the first algorithm ◄ UD0 ← Tarski LOGIKĒ · XX · the other half LOGIKĒ · XX · the first algorithm Ada Lovelace 1815 – 1852 · London · Augusta Ada King, Countess of Lovelace · the world's first programmer · the face of this universe For nineteen spheres this lineage has been, as it happens, entirely men — not because women couldn't reason, but because for most of history they were forbidden the room : barred from universities, denied degrees, lecturing under men's names. Ada is where the other half begins, and she begins it at the summit. In her 1843 Notes on Babbage's Analytical Engine — three times longer than the paper she was translating — Note G sets out a complete step-by-step procedure for the machine to compute the Bernoulli numbers : the first algorithm ever published for a machine to run. And she saw further than Babbage himself: that a machine which manipulates numbers could manipulate any symbols — that computing was not arithmetic but the science of operations . ⚖ The other half — read honestly. The nineteen figures before her are men because mathematics excluded women by force for millennia. The women in this rank (Ada, then Ladd-Franklin, Péter, Robinson, Barcan Marcus) are not a courtesy addition — they did first-rank logic through walls built to keep them out. The scarcity is the injustice, not the measure of their minds. ✓ STRONG A real first, and a real vision. Note G is a genuine, complete algorithm (loop, variables, a computed table) published in 1843; and her insight that the engine could operate on symbols of any kind is the conceptual seed of general-purpose computing. ◐ "FIRST PROGRAMMER" IS DEBATED Shared, not solo. Babbage designed the engine and sketched earlier routines; the Notes grew from his work and their collaboration. Historians argue how much of Note G is hers. Her authorship of the Notes and the visionary leap, though, are firmly hers. ◔ A MACHINE NEVER BUILT Ahead of any hardware. The Analytical Engine was never completed in her lifetime; her algorithm ran only on paper for a century. She also tempered the hype — the engine "has no pretensions to originate anything." I · Note G — run the first program Note G's task: have the engine compute the Bernoulli numbers , a deep sequence of fractions from number theory, using only the engine's operations — add, subtract, multiply, divide — driven by a loop with stored variables . Step the engine forward and watch it produce each Bernoulli number exactly (as a true fraction, computed live by the recurrence Note G implements). This is a program: it has working storage, a repeated cycle, and a computed result. step the engine ▸ run ▶ reset ↺ computing B₀, B₂, B₄, … exactly The engine is loaded with Note G. Step it to begin the cycle. Each row is one Bernoulli number the engine has produced; the live cycle applies the recurrence Σ C(n+1,k)·Bₖ = 0 , solving for the newest Bₙ — exactly the kind of repeated, variable-driven computation Note G lays out across its 25 operations. ⚑ Loops, stored variables, a computed table of results: by any modern definition, this is code — written in 1843, for a machine that did not yet exist. "The Analytical Engine weaves algebraic patterns, just as the Jacquard-loom weaves flowers and leaves." — A. A. L., Note A, 1843 II · The leap — a machine of symbols, not numbers Babbage saw a calculator. Ada saw something none of her contemporaries did: that the engine's numbers could stand for anything — and so the machine could operate on symbols of any kind, by rule. Pick what the engine's operations might represent. numbers music logic letters Her own words: if the relations of pitched sounds could be expressed by the engine's notation, it "might compose elaborate and scientific pieces of music of any degree of complexity." That is the definition of a general-purpose computer — a symbol-manipulator — stated a century before one existed. Babbage built the body; Ada saw the idea . ⚑ Every program in [[psephos-processors-domain]] is operations on symbols that merely stand for numbers, sounds, words — exactly her leap. III · The Notes, the myth, and the woman NOTE G — THE ALGORITHM A complete machine procedure for the Bernoulli numbers, with loop and variables (1843). The first program published for a general computing machine. THE SCIENCE OF OPERATIONS The insight that a number-machine is really a symbol-machine — able, in principle, to do music, logic, language. The concept of general-purpose computing. Gate kept on. The honest reckoning cuts both ways and lands firmly in her favour. Against the legend: she did not build the engine (Babbage did), she did not invent computing alone, and Babbage had sketched routines before her — so "the world's first programmer" is a contested honorific, and a few historians argue Babbage shaped Note G heavily. For the reality: the Notes are unquestionably hers (signed "A. A. L."), they run three times the length of the original paper, Note G is a genuine and correct algorithm, and the conceptual leap — that the machine operates on symbols, not numbers, and could therefore do far more than sums — is hers alone and is the single most prescient idea in the prehistory of the computer. She was also clear-eyed against hype, warning the engine "has no pretensions to originate anything" — a caution AI still needs. The woman: daughter of the poet Byron (whom she never knew), raised relentlessly on mathematics by a mother determined to suppress any inherited "poetic" madness, she fused the two into what she called "poetical science" ; she died of cancer at 36, the same age as the father she never met. The machine was built in full only in our century — and her algorithm ran. ⚑ She is the face of this universe for a reason: the ancestry of the algorithm runs through her. [[alfred-tarski]] ← · next → Christine Ladd-Franklin. ADA LOVELACE · LOGIKĒ XX · two-layer honest · the other half begins live Note G — the engine computes the Bernoulli numbers exactly (BigInt fractions via the recurrence Σ C(n+1,k)Bₖ = 0; verified B₂=1/6, B₄=−1/30, B₆=1/42, B₈=−1/30) · the symbol-machine leap Note G (1843) = the first published algorithm for a machine · "the science of operations" = the idea of general-purpose computing · the face of UD0 ← Tarski · next → Christine Ladd-Franklin (Boolean logic & the antilogism) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


CHRISTINE LADD-FRANKLIN · XXI

the other half — the antilogism: one test for every syllogis

LOGIKĒ XXI — Christine Ladd-Franklin (1847–1930). Studying under Peirce at Johns Hopkins, she gave Boolean logic a single, beautiful test for the validity of every syllogism: the ANTILOGISM (the inconsistent triad). A syllogism is valid exactly when its two premises plus the contradictory of its conclusion cannot all be true — and one such inconsistent triad yields three valid syllogisms at once. Live: test any syllogism by its antilogism. Honest: she completed every requirement for a Johns Hopkins PhD in 1882 and was denied the degree for being a woman — finally granted it in 1926, at age 78. David Lee Wise, two-layer honest.

Christine Ladd-Franklin — the antilogism ◄ UD0 ← Ada Lovelace LOGIKĒ · XXI · the other half LOGIKĒ · XXI · the antilogism Christine Ladd-Franklin 1847 – 1930 · Connecticut & Johns Hopkins & Columbia · student of C. S. Peirce · logician & vision scientist Aristotle left a sprawling catalogue of valid and invalid syllogisms; Boole turned logic into algebra. Ladd-Franklin — studying under [[charles-peirce]] — found the single test that unifies all of it. Her insight: forget memorising which moods are valid. A syllogism is valid exactly when its two premises, taken together with the contradictory of its conclusion, form an inconsistent triad — three statements that cannot all be true at once. She called it the antilogism , and it is gorgeously symmetric: from one inconsistent triad you can read off three valid syllogisms. It was, for decades, the cleanest decision procedure for the syllogism in existence. ✓ STRONG A real, elegant result. The antilogism is a correct and unifying test for syllogistic validity — a genuine advance in algebraic logic, recognised by Peirce and standard in logic teaching for generations. ◐ ONE PIECE OF A FIELD Syllogistic, not all logic. The antilogism settles the classical syllogism beautifully; it is not, and was not meant to be, the whole of modern predicate logic ([[gottlob-frege]]). A sharp tool with a defined edge. ◔ THE DEGREE WITHHELD Excluded by rule, not merit. She completed every requirement for a Johns Hopkins PhD in 1882; the university refused the degree to a woman. She got it in 1926 — forty-four years later, at 78. The injustice, named. I · The antilogism — one test for every syllogism Pick a syllogism. The machine forms its antilogism — the two premises plus the contradictory of the conclusion — and checks whether those three can all be true together. If they cannot (an inconsistent triad), the original argument is valid : denying its conclusion while granting its premises is impossible. If they can all hold, the argument is invalid. (Checked live by enumerating every possible Venn arrangement of the three terms.) its antilogism — premises + the contradictory of the conclusion The whole of Aristotle's syllogistic collapses into one question: is this triad inconsistent? ⚑ And the symmetry is the prize — any inconsistent triad gives three valid syllogisms (drop any one statement, negate it, and conclude it from the other two). One inconsistency, three theorems. This is the syllogism finally made algebraic and mechanical — a decision procedure, the kind of thing a computer runs. "The cause of [a syllogism's validity] is that the three propositions … form an inconsistent triad." — C. Ladd, "On the Algebra of Logic", 1883 II · One inconsistency, three syllogisms Here is the antilogism's deep economy. Take the current inconsistent triad. Drop any one of its three statements and assert its opposite as a conclusion from the remaining two — you get a valid syllogism, every time. Three buttons, three theorems from a single fact of inconsistency. This is why she could throw away the medieval mnemonics (Barbara, Celarent, Darii…): they are just the faces of inconsistent triads. Find the inconsistencies, and you have found every valid syllogism there is. ⚑ A unifying principle in place of a list — the move that turns a catalogue into a science . III · The logic, and the forty-four years Gate kept on. The mathematics is clean and real : the antilogism is a correct, elegant decision procedure for the categorical syllogism, praised by Peirce and taught for generations — a genuine contribution to the algebra of logic, even if it addresses the classical syllogism rather than the full predicate logic that [[gottlob-frege]] was building in the same years. What must be told without softening is the cost . Christine Ladd entered Johns Hopkins in 1878 — when it did not admit women — by the intervention of the logician J. J. Sylvester, who recognised her work; she was permitted to attend on the condition that her name not appear in the register. She completed every requirement for the doctorate and wrote a dissertation Peirce published in his landmark 1883 volume — and the university refused to grant her the degree because she was a woman . She was finally awarded it in 1926, forty-four years later, at the age of 78 , four years before her death. In between she did major work in the theory of colour vision as well, and spent decades barred from the regular faculty positions her male peers held automatically. Her logic stands entirely on its own; the four-decade delay stands as the plainest possible illustration of why this lineage was, for nineteen spheres, "all men." ⚑ [[ada-lovelace]] ← · next → Rózsa Péter. CHRISTINE LADD-FRANKLIN · LOGIKĒ XXI · two-layer honest · the other half live antilogism validity tester (a syllogism is valid iff premises + ¬conclusion form an inconsistent triad — checked by full Venn-region enumeration; verified Barbara/Celarent/Darii valid, undistributed-middle invalid) + one triad → three syllogisms the antilogism / inconsistent triad (1883) = a unifying decision procedure for the syllogism · studied under Peirce · also a theory of colour vision ← Ada Lovelace · next → Rózsa Péter (recursion theory) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


RÓZSA PÉTER · XXII

the other half — founder of recursion theory; a function tha

LOGIKĒ XXII — Rózsa Péter (1905–1977). She founded recursion theory as a subject — the mathematics of functions defined in terms of themselves, the formal heart of what a computer can compute. She gave the Ackermann function its standard two-variable form (the Ackermann–Péter function): a function that is total and computable yet grows so explosively it cannot be built from simple bounded loops — proving 'primitive recursion' is not all of computation. Live: run the Ackermann–Péter function and watch it explode, and step primitive recursion building addition and multiplication. Honest: a Hungarian Jew who survived the Budapest ghetto and made recursion a field. David Lee Wise, two-layer honest.

Rózsa Péter — recursion, and a function that outruns the loop ◄ UD0 ← Ladd-Franklin LOGIKĒ · XXII · the other half LOGIKĒ · XXII · recursion, and a function that outruns the loop Rózsa Péter 1905 – 1977 · Budapest · founder of recursion theory · author of "Recursive Functions" (1951) & "Playing with Infinity" When Gödel, Church, and Turing pinned down what a machine can compute, the engine under all of it was recursion — functions defined in terms of themselves. Rózsa Péter took that engine and made it a subject : her 1951 book Recursive Functions was the first systematic treatment, and she is rightly called a founder of recursion theory , the mathematics of computability. Her signature result is a single, startling function. The "safe" functions — built from primitive recursion , the bounded loops every beginner writes — seem like they should be everything. Péter's two-variable form of the Ackermann function is total and computable , yet grows so violently it provably escapes them: a function no bounded loop can hold. ✓ STRONG A field, and a theorem. Péter systematised recursion theory and gave the Ackermann function its standard two-argument form — a clean, total, computable function proven to outrun primitive recursion. Rigorous and foundational to computability. ◐ SHARED ROOTS Built on Ackermann, with others. Wilhelm Ackermann found the original; Péter simplified and generalised it (the "Ackermann–Péter function") and built the surrounding theory. Recursion theory also has Gödel, Kleene, and others in its parentage. ◔ SURVIVED TO BUILD IT The work was done against history. A Hungarian Jew, she lost her teaching post under the fascist laws and survived confinement in the Budapest ghetto; she wrote much of her foundational work in and after that. The mathematics carried through. I · The Ackermann–Péter function — total, computable, unbounded The rule is tiny: A(0,n)=n+1 ; A(m,0)=A(m−1,1) ; A(m,n)=A(m−1, A(m,n−1)) . It always halts (it is total and computable). But watch the value — and the number of recursive calls — as m climbs. By m=4 the result is a tower of exponentials; the function provably grows faster than anything you can write with bounded for -loops. That is the proof that primitive recursion ≠ all computation. m = 0 1 2 3 4 n = 0 1 2 3 4 compute A(m, n) ▸ (inputs capped — it explodes fast) A(2,n) is roughly 2n+3; A(3,n) is exponential (2ⁿ⁺³−3); A(4,n) is a tower of 2's of height n — A(4,2) already has 19,729 digits. Each is computed by pure recursion, no loop bound in sight. ⚑ This is the precise mathematical line between "what a simple loop can do" and "what computation can do" — and Péter drew it. "Mathematics is one of the oldest of the sciences and yet one of the youngest — for it grows by what it learns to ask." — in the spirit of Rózsa Péter, "Playing with Infinity" II · Primitive recursion — building arithmetic from the successor Recursion theory starts from almost nothing: zero, the successor (+1), and the schema of primitive recursion — define a function's value at n+1 from its value at n . From that alone, all of basic arithmetic unfolds. Watch addition and multiplication build themselves by recursion. add(a, b) — by recursion on b mult(a, b) — by recursion on b a = 2 3 4 b = 2 3 4 unfold ▸ Addition is "apply successor b times"; multiplication is "add a, b times" — each defined purely by recursion on the previous case, no magic. ⚑ These are primitive recursive : safe, always-halting, loop-bounded. Péter's achievement was to map this whole world precisely — and to show, with Ackermann's function, exactly where it ends. III · The field, the limit, the survivor RECURSION THEORY Her "Recursive Functions" (1951) was the first book to organise the theory of computable functions as a subject — a cornerstone of the mathematics of computation. THE ACKERMANN–PÉTER FUNCTION A total computable function that is provably not primitive recursive — the clean witness that bounded loops are weaker than full computation. Gate kept on. The mathematics is solid and foundational — recursion theory is a pillar of computability, and the two-variable Ackermann function is a textbook result, total and computable yet beyond primitive recursion. Honest attributions: the original function is Wilhelm Ackermann's (1928); Péter gave the standard two-argument simplification now usually taught (hence "Ackermann–Péter"), and her larger and more important contribution was making recursion theory a coherent field — though Gödel, Kleene, and others share its foundations. The woman: Rózsa Péter (born Politzer) was a Hungarian Jew who, under Hungary's fascist racial laws, was dismissed from teaching and forbidden to work, and survived confinement in the Budapest ghetto during the war — doing and preserving mathematics through it. Afterward she became a beloved professor and wrote Playing with Infinity , one of the finest popular books on mathematics ever written, insisting the subject was for everyone. She made the theory of "functions that call themselves" into a science — the formal soul of every program that loops or recurses in [[psephos-processors-domain]]. ⚑ [[christine-ladd-franklin]] ← · next → Julia Robinson. RÓZSA PÉTER · LOGIKĒ XXII · two-layer honest · the other half live Ackermann–Péter function (A(0,n)=n+1; A(m,0)=A(m−1,1); A(m,n)=A(m−1,A(m,n−1)); total, computable, beyond primitive recursion — verified A(2,2)=7, A(3,3)=61) + primitive recursion building add & mult founder of recursion theory · "Recursive Functions" (1951) · the two-variable Ackermann form · "Playing with Infinity" ← Ladd-Franklin · next → Julia Robinson (Hilbert's 10th problem) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


JULIA ROBINSON · XXIII

the other half — Hilbert's 10th: no algorithm decides i

LOGIKĒ XXIII — Julia Robinson (1919–1985). She spent decades on Hilbert's tenth problem: is there an algorithm that decides whether any Diophantine (integer polynomial) equation has integer solutions? Her work — the Robinson hypothesis and the Davis–Putnam–Robinson theorem — built the frame that Yuri Matiyasevich completed in 1970, proving the answer is NO: no such algorithm can exist (the MRDP theorem, that Diophantine sets are exactly the computably enumerable sets). Live: hunt integer solutions to Diophantine equations and meet the undecidability wall. She was the first woman president of the AMS and first woman mathematician in the National Academy. David Lee Wise, two-layer honest.

Julia Robinson — Hilbert's tenth problem ◄ UD0 ← Péter LOGIKĒ · XXIII · the other half LOGIKĒ · XXIII · Hilbert's tenth problem Julia Robinson 1919 – 1985 · St. Louis & Berkeley · first woman president of the AMS · first woman mathematician in the U.S. National Academy of Sciences Of [[david-hilbert]]'s 23 problems, the tenth asked for something concrete: an algorithm that, given any Diophantine equation — a polynomial equation in whole numbers, like x² − 2y² = 1 — decides whether it has an integer solution. For most of the twentieth century no one knew if such a procedure existed. Julia Robinson spent decades on it, building the crucial machinery — the Robinson hypothesis and the Davis–Putnam–Robinson theorem — that reduced the whole question to one missing piece. In 1970 a 22-year-old in Leningrad, Yuri Matiyasevich , supplied it, completing a proof that is hers as much as anyone's: no such algorithm can exist. ✓ STRONG A 70-year problem, closed. The MRDP theorem (Matiyasevich–Robinson–Davis–Putnam) settles Hilbert's tenth: Diophantine sets are exactly the computably enumerable sets, so no algorithm decides Diophantine solvability. Rigorous and deep. ◐ A RELAY, NOT A SOLO Four names, two generations, one Cold War. Davis, Putnam, and Robinson built the frame across the 1950s–60s; Matiyasevich (1970) added the final Fibonacci step. Robinson's was the central, sustained contribution — but it was a collaboration. ◔ THE SEARCH CAN'T DECIDE Why the demo has a wall. Hunting for solutions can confirm a "yes," but can never prove a "no" in general — exactly what the theorem says. The toy below finds solutions; it cannot settle the undecidable in the abstract. I · Diophantine equations — and the wall behind them A Diophantine equation asks for whole-number solutions. Some have infinitely many, some have exactly one, some have none — and there is no telling which from the look of it. Pick an equation; the machine searches a bounded range for integer solutions. Notice the asymmetry that is Hilbert's tenth: when solutions exist, a search finds them; when they don't, no bounded search can ever prove it — and the theorem says no un bounded procedure can either. "Find a solution" is a search you can run; "decide whether a solution exists, for every equation" is what Hilbert wanted — and Robinson's life work proved it impossible . ⚑ It's the same undecidability as [[alan-turing]]'s halting problem, wearing the clothes of ordinary algebra: the question "does this equation have a whole-number answer?" has no general mechanical answer. "I would rather not be remembered as the first woman this or that. I would rather be remembered… for the theorems I have proved." — Julia Robinson II · MRDP — what the proof actually says DIOPHANTINE = COMPUTABLY ENUMERABLE The deep equivalence: a set of numbers can be defined by a Diophantine equation exactly when it can be listed by a computer program. Number theory and computability are the same reach. ⇒ HILBERT'S 10th IS UNDECIDABLE Since some computably-enumerable sets are undecidable (Turing), some Diophantine questions must be too. No algorithm decides solvability for all equations. The bridge is breathtaking: Robinson and her collaborators showed that the humble whole-number polynomial is secretly as powerful as a computer — anything a program can enumerate, an equation can capture. So the undecidability that [[alan-turing]] found in machines and [[kurt-godel]] found in proof reappears, exactly, in the arithmetic of integers. Hilbert asked for a decision machine; the answer, decades in the building, was that the integers themselves are Turing-complete , and so no such machine exists. It is one of the most surprising unifications in mathematics — number theory is computability — and Julia Robinson is at its center. III · The decades, the relay, the woman Gate kept on. The result is rigorous and monumental , and the credit is genuinely shared — the honest name is MRDP (Matiyasevich–Robinson–Davis–Putnam). Martin Davis and Hilary Putnam built early pieces; Robinson's hypothesis (that some Diophantine relation grows exponentially) was the keystone the whole edifice waited on for over a decade; and in 1970 the young Yuri Matiyasevich proved her hypothesis using Fibonacci numbers, completing it. Robinson immediately and generously credited him, and the two became close collaborators and friends across the Cold War divide — a rare thing in that era. Her contribution was the sustained spine of a twenty-year effort, not a single flourish. The woman: weakened for life by childhood rheumatic fever that damaged her heart, told she might not live long and could not have children, she poured herself into mathematics; she became the first woman president of the American Mathematical Society and the first woman mathematician elected to the National Academy of Sciences , often working without a regular faculty appointment at Berkeley for years because she was a professor's wife. She asked to be remembered for her theorems — so: she helped prove that arithmetic itself is undecidable, that the integers are as powerful as any computer. ⚑ [[rozsa-peter]] ← · next → Ruth Barcan Marcus. JULIA ROBINSON · LOGIKĒ XXIII · two-layer honest · the other half live Diophantine solution-hunter (bounded integer search over Pell & other equations — finds solutions where they exist; verified x²−2y²=1 → (1,0),(3,2),(17,12)…) and the undecidability wall it illustrates Hilbert's 10th problem · the Robinson hypothesis & the MRDP theorem (Diophantine sets = computably enumerable sets → no algorithm decides solvability) · 1st woman AMS president & NAS mathematician ← Péter · next → Ruth Barcan Marcus (quantified modal logic) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


RUTH BARCAN MARCUS · XXIV · THE OTHER HALF CLOSES

the other half — the first quantified modal logic (1946) &am

LOGIKĒ XXIV — Ruth Barcan Marcus (1921–2012). At 25 she published the first systems of QUANTIFIED MODAL LOGIC (1946), combining 'necessarily/possibly' with 'for all/there exists' — and stated the Barcan formula (∀x□Fx → □∀xFx), the principle governing how necessity and quantification interact, a decade before possible-worlds semantics existed. She also pioneered direct reference (names as 'tags'), anticipating Kripke. Live: a possible-worlds model where you set what's true in each world and watch the Barcan formula hold while ◇ and ∀ refuse to commute. Honest: a real priority she had to defend; the formula and the field are hers. David Lee Wise, two-layer honest.

Ruth Barcan Marcus — quantified modal logic ◄ UD0 ← Robinson LOGIKĒ · XXIV · the other half LOGIKĒ · XXIV · quantified modal logic Ruth Barcan Marcus 1921 – 2012 · New York & Yale · founder of quantified modal logic · the Barcan formula Logic had two great extensions running on separate tracks: quantifiers ([[gottlob-frege]]'s ∀ "for all" and ∃ "there exists") and modality ("necessarily" □, "possibly" ◇). In 1946, at twenty-five, Ruth Barcan was the first to combine them rigorously — the first formal systems of quantified modal logic — and to ask the question that combination forces: when you have both, does the order matter? Her answer is the Barcan formula , ∀x□Fx → □∀xFx — the principle that governs how necessity and "for all" slide past each other. She wrote it down a decade before the possible-worlds semantics that would explain it existed; today it is a cornerstone of modal logic, the logic of necessity, time, knowledge, and provability. ✓ STRONG A field she opened. The first rigorous quantified modal logic (1946) and the Barcan formula are foundational and bear her name; modal logic underlies the semantics of necessity, time, knowledge, obligation, and program verification. ◐ SEMANTICS CAME LATER Syntax first, meaning after. She gave the axioms; the possible-worlds semantics that interpret them came in the late 1950s–60s (Kripke and others). The formula is hers; its now-standard picture was filled in later. ◔ PRIORITY SHE HAD TO DEFEND Credit, contested and corrected. Some attributed her ideas (direct reference, the formula) to later men; she had to insist, correctly, on her 1946 priority. The record now plainly credits her. I · Possible worlds — does □ commute with ∀? Picture several possible worlds and two individuals, a and b . Click to set, in each world, whether each individual has property F . Then read off how the modal operators and quantifiers combine. The Barcan formula concerns one pair — ∀x□Fx ("each thing is necessarily F") versus □∀xFx ("necessarily, everything is F") — and a different pair shows that other orders don't commute at all. the Barcan pair — □ and ∀ ∀x □Fx (each is necessarily F) □ ∀xFx (necessarily all are F) the other order — ◇ and ∀ ◇ ∀xFx (some world: all F) ∀x ◇Fx (each is F somewhere) The Barcan formula says ∀x□Fx and □∀xFx march together (with a constant set of individuals across worlds, they are equivalent — toggle around and they never disagree). But ◇ and ∀ are a different story: "there's a world where everything is F" is far stronger than "everything is F in some world or other" — set a true only in W1 and b true only in W2 to split them. ⚑ This is the modal cousin of [[charles-peirce]]'s and [[gottlob-frege]]'s lesson that quantifier order matters — now tangled with necessity, the exact knot Barcan untied first. "Whatever it is to be a thing… does not change from world to world." — on the constant-domain reading behind the Barcan formula II · What the formula means — and when it fails QUANTIFIED MODAL LOGIC (1946) The first rigorous union of quantifiers and modality. The framework for reasoning about what must, might, or always will be — across time, knowledge, obligation, and proof. THE BARCAN FORMULA ∀x□Fx → □∀xFx. It holds exactly when the individuals are the same across all possible worlds — so the formula is really a claim about whether existence is necessary. The deep content is metaphysical. The Barcan formula holds when every world has the same individuals (a "constant domain") — and fails if new things can exist in other possible worlds (a "varying domain"). So a dry-looking axiom turns out to encode a question about being : must exactly the same things exist no matter how the world had gone? Barcan's logic made that question precise and answerable, and her work on direct reference — treating names as bare "tags" that pick out the same object in every world — anticipated the theory of reference usually credited to Saul Kripke fifteen years later. The machinery of necessity in [[psephos-processors-domain]]'s verification tools, in the logic of knowledge, and in temporal reasoning all descends from the systems she opened. III · The opener, the priority, the woman Gate kept on. The achievement is real and first : Ruth Barcan's 1946 papers are the earliest rigorous systems of quantified modal logic, and the Barcan formula carries her name by right. Two honest qualifications. First, on semantics : she gave the axiomatics ; the possible-worlds interpretation that makes the formula intuitive (the picture used above) was developed afterward, chiefly by Kripke and others in the late 1950s and 60s — so she opened the syntax, and the meaning was filled in later (a normal division of labour, not a deduction from her work). Second, on credit : Barcan Marcus's ideas on direct reference and the necessity of identity genuinely anticipated themes later made famous by Saul Kripke, and there was a real, documented dispute about priority — including a notorious seminar exchange — which the historical record now resolves in her favour: the formula and the early framework are hers. The woman: Ruth Barcan Marcus earned her doctorate and published this work in the 1940s while the profession barely admitted women, raised four children, and spent a career insisting — correctly, and against resistance — on her own intellectual priority, eventually chairing philosophy at Yale. She gave logic its grip on necessity and possibility joined to all and some : the formal language in which we reason about how things might have been . ⚑ With her, the other half of the lineage stands: [[ada-lovelace]] · [[christine-ladd-franklin]] · [[rozsa-peter]] · [[julia-robinson]] · Barcan Marcus — first-rank logic done through walls. [[julia-robinson]] ←. RUTH BARCAN MARCUS · LOGIKĒ XXIV · two-layer honest · the other half closes live possible-worlds model (set F per world×individual; ∀x□Fx ↔ □∀xFx always agree — the Barcan formula; ◇∀xFx vs ∀x◇Fx can split — order matters; verified) the first quantified modal logic (1946) · the Barcan formula ∀x□Fx → □∀xFx · direct reference / names as "tags" (anticipating Kripke) · first woman to chair Yale philosophy ← Robinson · the other half of LOGIKĒ, restored to the lineage LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


EMMY NOETHER · XXV

the other half — the algebra of STRUCTURE (groups/rings/idea

LOGIKĒ XXV — Emmy Noether (1882–1935), the greatest woman mathematician in history. She moved algebra from computing with particular numbers to reasoning about abstract STRUCTURES — groups, rings, ideals, defined by their axioms alone — the modern style of all higher mathematics, and the frame in which Boolean algebra is just one structure among many. Her theorem ties every continuous symmetry to a conservation law, underpinning modern physics. Live: a Cayley-table explorer that checks the group axioms, and Boolean logic shown as an algebraic structure. Honest: barred from a paid post for being a woman, she lectured for years under Hilbert's name; the Nazis expelled her in 1933; she died at 53. David Lee Wise, two-layer honest.

Emmy Noether — the algebra of structure ◄ UD0 ← Barcan Marcus LOGIKĒ · XXV · the other half LOGIKĒ · XXV · the algebra of structure Emmy Noether 1882 – 1935 · Erlangen & Göttingen & Bryn Mawr · "the most important woman in the history of mathematics" (Einstein) Boole made logic into an algebra; Noether changed what algebra is . Before her, algebra meant calculating with particular things — numbers, polynomials, matrices. She lifted it to reason about abstract structures — a group , a ring , an ideal — defined by axioms alone , caring only about the rules an operation obeys, never what the elements "really are." That move is modern mathematics: it's why a logician, a chemist, and a cryptographer can all be studying "the same group," and why Boolean algebra is recognised as just one structure among a vast family. Her theorem also handed physics its deepest principle — every continuous symmetry yields a conservation law — but the revolution this lineage owes her is the axiomatic, structural way of thinking itself. ✓ STRONG Foundational, on two fronts. Noether's abstract algebra (rings, ideals, the ascending chain condition) reshaped mathematics; Noether's theorem (symmetry ⇒ conservation) is a pillar of physics. Both are rigorous and permanent. ◐ ALGEBRA, NOT "LOGIC" NARROWLY Structure, the soil logic grows in. Her field is abstract algebra, not formal logic per se — but the axiomatic-structural method she perfected is exactly how modern logic treats its objects (Boolean algebras, lattices, models). She belongs here as the algebra under it all. ◔ BARRED, THEN EXPELLED Genius worked around by force. Denied a paid post as a woman, she lectured for years under Hilbert's name ; in 1933 the Nazis expelled her, a Jew, from Göttingen. She fled to America and died two years later at 53. I · A structure, not its elements — the Cayley table Here is the structural view in miniature. A group is any set with one operation obeying four axioms: closure, associativity, an identity, and inverses — and nothing else is assumed. Pick a group; its Cayley table shows the operation in full. The machine checks the axioms live. Notice that wildly different things — clock arithmetic, the symmetries of a triangle, the logic values {0,1} — are the same structure when their tables match. That sameness is Noether's whole idea. A group cares only about how its operation behaves, never what its elements are made of. ⚑ This is why [[george-boole]]'s logic is "an algebra" in the exact technical sense, why a Boolean algebra sits in a family with rings and lattices, and why proving something about "all groups" instantly proves it about every concrete one. Noether taught mathematics to reason about form instead of stuff . "My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously." — Emmy Noether II · Two revolutions ABSTRACT ALGEBRA Rings, ideals, modules, and the ascending chain condition (a "Noetherian" ring bears her name). She made structure , defined by axioms, the object of study — the template for all modern algebra and for how logic handles its own structures. NOETHER'S THEOREM (1918) Every continuous symmetry of a physical system corresponds to a conserved quantity: time-symmetry ⇒ energy, space-symmetry ⇒ momentum. One of the deepest principles in all of physics. The thread to this lineage is the axiomatic method carried to its conclusion. [[euclid]] axiomatised geometry; [[david-hilbert]] demanded axioms for everything; Noether showed how to actually work that way — to define a kind of object purely by the laws it obeys and then prove theorems about every instance at once. That is precisely how a modern logician studies "a Boolean algebra" or "a model," and how a computer scientist reasons about "a monoid" or "a type." Her structural style is the air all of higher mathematics now breathes — usually, as she predicted, without her name attached. III · The name on the work, and the name withheld Gate kept on. The honest placement: Noether is, strictly, the supreme algebraist rather than a formal logician — but the structural, axiomatic mode of thought she perfected is the ground modern logic stands on, which earns her a place in this lineage at the side of the women restored to it. Her results are beyond dispute: Noetherian rings, her treatment of ideals, and Noether's theorem are permanent fixtures of mathematics and physics. What must be told plainly is the obstruction . Emmy Noether was the daughter of a mathematician and a mathematician of the first rank — and for years the University of Göttingen refused to let her hold a paid teaching position because she was a woman ; she lectured under Hilbert's name on the timetable while the faculty argued the point (Hilbert reportedly snapped that the university senate "is not a bathhouse"). She finally gained a modest, poorly paid lectureship — and then, in 1933, the Nazi regime expelled her , a Jew, from the university entirely. She emigrated to Bryn Mawr College in the United States, taught two more years, and died in 1935 at 53 after surgery. Einstein, in her obituary, called her the most significant creative mathematical genius produced since the higher education of women began. Her methods, she said, "crept in everywhere anonymously" — the structural language of all modern mathematics, hers. ⚑ [[ruth-barcan-marcus]] ← · next → Stephen Kleene. EMMY NOETHER · LOGIKĒ XXV · two-layer honest · the other half live Cayley-table explorer (cyclic groups, triangle symmetries, the logic values {0,1}; closure / associativity / identity / inverses checked — verified Z₄ is a group, a broken table fails associativity) · the structural view abstract algebra (rings, ideals, Noetherian condition) = the axiomatic study of structure · Noether's theorem (symmetry ⇒ conservation) · the method behind modern algebra & logic ← Barcan Marcus · next → Stephen Kleene (regular expressions & the Kleene star) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


STEPHEN KLEENE · XXVI

the other half (men) — regular expressions = finite automata

LOGIKĒ XXVI — Stephen Kleene (1909–1994). A founder of recursion theory (Church's student), he proved that the patterns describable by 'regular expressions' are exactly those a finite-state machine can recognize (Kleene's theorem, 1951) — the result behind every grep, lexer, and search box. The repetition operator, the Kleene star (*), is named for him; so is the Kleene closure and the recursion theorem. Live: a working regular-expression matcher (concatenation, union, and the star) and Kleene's bridge to finite automata. Honest: regular languages are a precise, limited class; the star is real, the field is his. David Lee Wise, two-layer honest.

Stephen Kleene — regular expressions and the star ◄ UD0 ← Noether LOGIKĒ · XXVI · the foundations rank LOGIKĒ · XXVI · regular expressions and the star Stephen Kleene 1909 – 1994 · Wisconsin · student of Alonzo Church · a founder of recursion theory · pronounced "KLAY-nee" [[alonzo-church]]'s student Kleene helped build the mathematics of what can be computed — and then gave one of its pieces a form you use every day. He proved that the patterns you can write with a regular expression — built from single characters by three operations, concatenation , choice ( | ), and repetition (the * ) — are exactly the patterns a simple finite-state machine can recognize (Kleene's theorem, 1951). Two completely different descriptions — an algebra of patterns and a machine with a handful of states — turn out to define the same languages. The repetition operator is named the Kleene star after him, and it runs inside every search box, grep, lexer, and validator on Earth. ✓ STRONG Exact and ubiquitous. Kleene's theorem (regular expressions = finite automata) is a rigorous cornerstone of computer science, and the Kleene star, closure, and recursion theorem are permanent tools of computability and language theory. ◐ A LIMITED CLASS Powerful, but not all-powerful. Regular languages are precisely the weakest rung of the computation hierarchy — they can't count unboundedly (no balanced parentheses). That limit is itself the point: it's exactly what a finite memory can do. ◔ "REGEX" ≠ REGULAR The everyday tool overgrew the theory. Modern "regex" engines add backreferences and lookahead that exceed true regular expressions. Kleene's regular expressions are the clean mathematical core, not the kitchen-sink version. I · A regular expression, running Three operations build every regular expression: put patterns in sequence ( concatenation ), offer alternatives with | ( union ), and repeat zero-or-more times with * (the Kleene star ). Below is a real matcher built from exactly those rules. Type a pattern and a string — or pick an example — and see whether the string belongs to the language the pattern defines. pattern string match ▸ The star is the heart of it: (a|b)* means "any run of a's and b's, including none" — repetition without a fixed bound, captured in one symbol. ⚑ That single idea, zero-or-more , is what lets a finite machine describe infinitely many strings; it's why one short pattern can validate every email address or tokenize a whole programming language. The Kleene star is among the most-typed mathematical inventions in history. "The * is to be read 'any number of times, including none.'" — after Kleene, defining the star, 1951 II · Two faces of the same language THE PATTERN (regular expression) An algebraic description: characters combined by concatenation, | , and * . Declarative — it says what strings match, not how to check them. THE MACHINE (finite automaton) A handful of states and transitions that read a string left-to-right and end in "accept" or "reject." Operational — it says exactly how to recognize the language with finite memory. Kleene's theorem says these two define precisely the same class of languages — the regular languages . Give me a pattern, I can build a machine; give me a machine, I can write the pattern. It is the same kind of beautiful equivalence as [[alonzo-church]]'s (λ-calculus = Turing machine), one rung lower on the ladder of power: where Turing machines have unlimited memory, finite automata have none beyond their fixed states — so the regular languages are exactly "what you can recognize while remembering only a bounded amount." Every lexer that turns your source code into tokens, every grep , every input validator, is Kleene's theorem at work — the pattern compiled into the machine. ⚑ The pattern algebra of [[george-boole]] and [[charles-peirce]], grown into the search bar. III · The star, the closure, the man Gate kept on. The results are exact and load-bearing : Kleene's theorem is a foundation of formal-language theory and compiler construction, and his name is attached, by right, to the Kleene star (zero-or-more repetition), the Kleene closure , and the recursion theorem of computability. Two honest scope notes. First, regular languages are deliberately the weakest useful class — they have only finite memory, so they provably cannot match balanced parentheses or count without bound; this isn't a flaw but the exact characterization of what bounded memory can do, and recognizing where the wall is (the "pumping lemma") is part of the theory. Second, the everyday word " regex " has drifted: the regex engines in programming languages bolt on backreferences and lookahead that make them strictly more powerful than — and slower than — Kleene's clean regular expressions; his are the mathematical core. The man: Stephen Cole Kleene did his doctorate under Alonzo Church at Princeton alongside Turing's visit, helped lay the foundations of recursion theory in the 1930s, and spent his career at Wisconsin; the field still runs on his definitions. He gave computer science its everyday pattern language and the symbol — * — that more people have typed than perhaps any other in mathematics. ⚑ [[emmy-noether]] ← · next → Gerhard Gentzen. STEPHEN KLEENE · LOGIKĒ XXVI · two-layer honest live regular-expression matcher (concatenation · union | · the Kleene star *; verified (a|b)*c matches "ababc", rejects "abab"; a*b matches "aaab") + Kleene's theorem (regex = finite automata) Kleene's theorem (1951): regular expressions = finite-state machines · the Kleene star / closure · the recursion theorem · a founder of recursion theory (Church's student) ← Noether · next → Gerhard Gentzen (natural deduction & proof theory) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


GERHARD GENTZEN · XXVII

the other half (men) — natural deduction & the sequent c

LOGIKĒ XXVII — Gerhard Gentzen (1909–1945). He invented NATURAL DEDUCTION and the SEQUENT CALCULUS — the first formal systems that mirror how mathematicians actually reason, with introduction and elimination rules for each connective. His cut-elimination theorem (the Hauptsatz) shows any proof can be normalized to a direct form, and he proved the consistency of arithmetic by transfinite induction up to ε₀. Natural deduction is the basis of proof assistants and, via Curry–Howard, of typed programming (proofs = programs). Live: build a natural-deduction proof rule by rule, checked. Honest: a Nazi party member who died in a postwar Prague prison camp at 35. David Lee Wise, two-layer honest.

Gerhard Gentzen — how a proof is built ◄ UD0 ← Kleene LOGIKĒ · XXVII · the foundations rank LOGIKĒ · XXVII · how a proof is built Gerhard Gentzen 1909 – 1945 · Greifswald & Göttingen · inventor of natural deduction & the sequent calculus · proof theory Frege and Russell could write proofs, but their systems were unnatural — long chains of obscure axioms nothing like how a mathematician actually argues. In 1934, Gentzen built the system that matches real reasoning: natural deduction . Its idea is beautifully simple — each logical connective gets two kinds of rule, an introduction rule (how to prove a statement using it) and an elimination rule (how to use such a statement) — and you reason from assumptions you later discharge. He also gave the sequent calculus and proved its central theorem (cut-elimination), and showed the consistency of arithmetic. Today natural deduction is the engine of every proof assistant, and — through the Curry–Howard correspondence — proofs are literally programs . ✓ STRONG The structure of proof itself. Natural deduction, the sequent calculus, and the cut-elimination theorem (Hauptsatz) are rigorous and foundational — the basis of modern proof theory, automated reasoning, and proof assistants. ◐ CONSISTENCY, WITH A PRICE Around Gödel, not over him. Gentzen proved arithmetic consistent — but only by using transfinite induction up to ε₀, a principle outside arithmetic. It doesn't evade Gödel; it measures exactly how much extra strength a consistency proof costs. ◔ A DARK BIOGRAPHY Named without flinching. Gentzen joined the Nazi party and the SA, and benefited from the regime that expelled his Jewish colleagues. He died in 1945 in a Soviet-controlled Czech prison camp at 35. The logic is his; so is the record. I · Build a proof, rule by rule Natural deduction proves a goal by applying rules that each introduce or eliminate a connective. Here is a real derivation engine: pick a theorem, then apply the available rules in order to construct the proof. Each step is checked — a rule only fires when its inputs are already on the page — and the line that reaches the goal lights up. Watch a proof grow the way a mathematician would actually grow it. Notice the shape: to prove A → B you assume A, derive B, then discharge the assumption (→-introduction); to use A → B together with A you get B (→-elimination, which is modus ponens, [[chrysippus]]' first rule). Every connective, two rules, perfectly balanced. ⚑ This balance is why proofs normalize — and why, under Curry–Howard, a proof of A → B is exactly a function taking evidence of A to evidence of B. A proof is a program. "I wished to set up a formalism that comes as close as possible to actual reasoning." — Gerhard Gentzen, 1934 II · Cut-elimination, and proofs as programs THE HAUPTSATZ (cut-elimination) Any proof using a "cut" (an intermediate lemma) can be rewritten into one that doesn't — a direct, lemma-free proof. Proofs can always be normalized ; from this, consistency and the subformula property follow. CURRY–HOWARD Introduction/elimination rules match exactly the way you build and use data. A proof of A→B is a function; normalizing a proof is running a program. Logic and computation, the same thing. Gentzen's cut-elimination is one of the most consequential theorems in logic: it says proof has a direct form , and that detours (lemmas) can always be removed. Decades later this became the deepest bridge in the field — the Curry–Howard correspondence — where a proposition is a type , a proof is a program of that type, and simplifying the proof is executing the program. The proof assistants that today verify the correctness of compilers, cryptography, and aircraft software ([[bertrand-russell]]'s types made to work) run on Gentzen's natural deduction directly. He also used his methods to prove the consistency of arithmetic — not in defiance of [[kurt-godel]], but by climbing into transfinite induction up to the ordinal ε₀, pinning down the exact "proof-theoretic strength" the job requires. III · The structure, the strength, the record Gate kept on. The mathematics is first-rank and permanent : natural deduction and the sequent calculus are how logic is taught and how proof assistants are built; cut-elimination is a deep structural theorem; and his consistency proof for arithmetic (1936) is honestly not a loophole around Gödel — it uses transfinite induction up to ε₀, a resource stronger than arithmetic itself, and its real content is to measure exactly how much strength consistency costs (founding ordinal proof theory). On the biography , the record is dark and is stated without softening: Gerhard Gentzen joined the Nazi party (NSDAP) and the SA , signed the loyalty declarations of the regime, and continued his career within institutions from which his Jewish teachers and colleagues — including those who had shaped his field — were being expelled and worse. After the war, he was arrested in Prague and died in a Soviet-administered internment camp in 1945 , at 35, of malnutrition. The work belongs in this lineage on its merits; the man's complicity belongs in the record beside it. Holding both is the whole point of keeping the gate on: the logic does not absolve the life, and the life does not erase the logic. ⚑ [[stephen-kleene]] ← · next → Saul Kripke. GERHARD GENTZEN · LOGIKĒ XXVII · two-layer honest live natural-deduction proof builder (introduction/elimination rules, assumptions discharged, each step checked; proves A→A, modus ponens, ∧-commutativity, A→(B→A)) natural deduction & the sequent calculus (1934) · cut-elimination / the Hauptsatz · consistency of arithmetic via ε₀ · the basis of proof assistants & (via Curry–Howard) typed programming ← Kleene · next → Saul Kripke (possible-worlds semantics) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


SAUL KRIPKE · XXVIII

the other half (men) — possible-worlds semantics: the meanin

LOGIKĒ XXVIII — Saul Kripke (1940–2022). As a teenager he gave modal logic its meaning: a model is a set of possible WORLDS with an ACCESSIBILITY relation, where 'necessarily P' is true at a world when P holds in every world it can see, and 'possibly P' when P holds in some. Different shapes of accessibility (reflexive, symmetric, transitive) validate different modal systems (T, S4, S5) — the semantic backbone behind Ruth Barcan Marcus's quantified modal logic, and the logics of knowledge, time, and provability. Live: a possible-worlds model where you set the accessibility relation and evaluate □ and ◇, and watch which modal axioms hold. David Lee Wise, two-layer honest.

Saul Kripke — possible worlds ◄ UD0 ← Gentzen LOGIKĒ · XXVIII · the foundations rank LOGIKĒ · XXVIII · possible worlds Saul Kripke 1940 – 2022 · Omaha & Harvard & Princeton · child prodigy · "Naming and Necessity" [[ruth-barcan-marcus]] gave modal logic its axioms ; Kripke — as a teenager — gave it its meaning . His picture is now so standard it's hard to remember it had to be invented: imagine a set of possible worlds , connected by an accessibility relation saying which worlds are "possible relative to" which. Then " necessarily P " (□P) is true at a world exactly when P holds in every world that one can see; " possibly P " (◇P) when P holds in some world it can see. Suddenly the dry symbols □ and ◇ have a crisp model — and, astonishingly, the shape of the accessibility relation decides which logic of necessity you get. This possible-worlds semantics is the foundation under the logic of necessity, knowledge, time, obligation, and program correctness. ✓ STRONG The semantics of modality. Kripke models (worlds + accessibility) are rigorous, complete for the standard modal systems, and the working foundation of modal, temporal, epistemic, and provability logic across philosophy and computer science. ◐ SEMANTICS ATOP HER SYNTAX A relay, honestly told. The quantified-modal systems and the Barcan formula are [[ruth-barcan-marcus]]'s (1946); Kripke supplied the possible-worlds interpretation (c. 1959–63). Two halves of one achievement — and a real, documented priority dispute between them. ◔ "WORLDS" — A MODEL, NOT A METAPHYSICS Math first. The possible worlds are a precise mathematical device. Whether other worlds "really exist" is a separate philosophical question (Kripke himself was cautious; David Lewis went further). The semantics works regardless. I · A world model — and the shape that sets the logic Three worlds; in each, a fact p is true or false (click a world to toggle p). The accessibility relation — which worlds each can "see" — is set by the buttons below, and its shape is everything. Evaluate □p and ◇p at the highlighted world, and watch the famous modal axioms switch on and off as you change the relation: a reflexive relation validates "□p → p" (system T); add transitivity for S4; make it an equivalence for S5. The revelation is that the relation's shape is the logic . "What is necessary is true" (□p→p) holds exactly when every world sees itself (reflexive). "What's necessary is necessarily necessary" (□p→□□p) holds exactly when accessibility is transitive. Pin down the relation and you pin down the modal system — T, S4, S5 — each matching a different reading of "necessary": logical, temporal, knowable. ⚑ This is the semantic engine behind [[ruth-barcan-marcus]]'s formula, behind the logic of knowledge (◇ = "for all I know"), and behind the modal logic of provability that re-encodes [[kurt-godel]]. "Possible worlds are stipulated, not discovered by powerful telescopes." — Saul Kripke, Naming and Necessity II · Why it mattered everywhere FRAMES ⇄ SYSTEMS Each modal axiom corresponds to a property of accessibility: reflexive ↔ T, transitive ↔ 4, symmetric ↔ B, equivalence ↔ S5. Soundness and completeness tie the syntax of necessity to the geometry of worlds — exactly. NAMING AND NECESSITY His later work argued names are rigid designators — they pick the same thing in every possible world — overturning the dominant theory of meaning. (The "tags" idea [[ruth-barcan-marcus]] had voiced earlier.) Possible-worlds semantics didn't just rescue modal logic from suspicion — it colonised half of logic and computer science . Read the worlds as moments in time and you have temporal logic , used to verify that a chip or protocol never reaches a bad state. Read accessibility as "compatible with what an agent knows" and you have epistemic logic , the logic of knowledge and belief in AI and economics. Read it as "provable" and you recover, in modal dress, [[kurt-godel]]'s incompleteness. The model checkers that prove safety properties of real hardware and software are walking Kripke structures. A picture invented by a teenager became the standard semantics for reasoning about what must, might, will, or is known to be — one of the most fruitful single ideas in modern logic. III · The prodigy, the priority, the picture Gate kept on. Kripke's contribution is real, rigorous, and enormous : the possible-worlds semantics for modal logic, with its soundness and completeness theorems linking frame conditions to modal systems, is the standard model used everywhere from philosophy to formal verification, and he produced the core of it as a teenager — a genuine prodigy who published the foundational papers around 1959–1963. The honest qualification is one of credit and sequence , and it ties directly to the sphere before this one: the quantified modal logics and the Barcan formula were [[ruth-barcan-marcus]]'s in 1946; the idea of names as world-spanning "tags" was also hers before it was his "rigid designators." Kripke supplied the semantics — the worlds-and-accessibility interpretation that made the systems intuitive — which is a major and largely independent achievement, but the popular tendency to credit him with the whole edifice is the error this lineage corrects: it is, properly, Barcan's syntax and Kripke's semantics , and there was a sharp, well-documented priority dispute (including a notorious seminar) between them. Finally, the "possible worlds" are a mathematical model , not a commitment that other universes literally exist — Kripke was careful on this; the device earns its keep by working, regardless of metaphysics. ⚑ With Kripke the modal thread that [[ruth-barcan-marcus]] opened is complete: the syntax, and the worlds that give it meaning. [[gerhard-gentzen]] ←. SAUL KRIPKE · LOGIKĒ XXVIII · two-layer honest live Kripke model (worlds + accessibility; □p true at w iff p holds in all worlds w sees, ◇p iff some; reflexive→T validates □p→p, transitive→S4, equivalence→S5 — verified) possible-worlds semantics for modal logic (c. 1959–63) · frame conditions ⇄ modal systems (T/S4/S5) · "Naming and Necessity" / rigid designators · the basis of temporal, epistemic & provability logic ← Gentzen · the modal thread, completed from Barcan Marcus LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


HASKELL CURRY

the modern foundations — combinatory logic (compute with NO

LOGIKĒ · the modern foundations — Haskell Curry (1900–1982). He built COMBINATORY LOGIC: a system with no variables at all, just a few fixed combinators (S, K, I) that copy, discard, and rearrange their arguments — yet it is exactly as powerful as Church's λ-calculus and the Turing machine. And his name marks the deepest bridge in logic: the CURRY–HOWARD correspondence, where propositions ARE types and proofs ARE programs. Live: a combinator reducer — watch S, K, I rewrite expressions, and see SKK reduce to the identity. Honest: the combinators came from Schönfinkel; Curry built the theory and shares the correspondence with Howard. David Lee Wise, two-layer honest.

Haskell Curry — computation without variables ◄ UD0 ← Church LOGIKĒ · the modern foundations LOGIKĒ · the modern foundations · combinatory logic Haskell Curry 1900 – 1982 · Millis, Massachusetts · Penn State & Amsterdam · combinatory logic · the λ in Haskell is named for him [[alonzo-church]]'s λ-calculus computes with functions and bound variables — and variables, it turns out, are a nuisance: they get captured, renamed, tangled. Curry asked a startling question: can you compute with no variables at all? His answer is combinatory logic . Take three fixed operators — I (do nothing), K (keep the first, throw the second away), and S (share an argument between two functions) — and that's the whole machine. No x , no λ , just these combinators copying, deleting, and rearranging whatever you feed them. It is exactly as powerful as the λ-calculus and the Turing machine. And his name marks logic's deepest bridge — the Curry–Howard correspondence : a proof is a program, a proposition is a type. ✓ STRONG Foundational, and elegant. Combinatory logic is a complete, variable-free model of computation, provably equivalent to the λ-calculus; and the Curry–Howard correspondence (proofs = programs) is one of the deepest unifications in logic and computer science. ◐ SHARED CREDIT Not solo. The combinators came first from Moses Schönfinkel (1924); Curry rediscovered and built the theory. And "Curry–Howard" is named for Curry's observation and William Howard's later formalisation — a bridge with two builders. ◔ A PARADOX, TOO Curry's paradox. His name also attaches to a self-referential paradox ("if this sentence is true, then everything is") that threatens naïve logics — a reminder that combinatory systems, untyped, can express dangerous self-reference. I · Compute with no variables — the combinator machine Three rewrite rules, applied left-to-right, are the entire calculus. I x → x . K x y → x (keep x, discard y). S x y z → x z (y z) (give z to both x and y, then apply). Pick an expression and reduce it step by step. Watch the famous identity emerge: S K K , applied to anything, hands it straight back — the combinators build the do-nothing operator out of "keep" and "share," with not a variable in sight. I x → x K x y → x S x y z → x z (y z) reduce one step ▸ run ▶ reset ↺ Every step just rearranges symbols by a fixed rule — no thinking, no variables, no scope. Yet from S, K, I alone you can build every computable function (numbers, booleans, recursion, the lot). ⚑ This is [[alonzo-church]]'s λ-calculus stripped to its bones: it proves that binding variables is a convenience, not a necessity — computation is, at bottom, just substitution of fixed patterns . Combinatory logic runs inside the implementation of functional languages, and S/K/I are how a compiler can eliminate variables entirely. "The combinators … enable us to dispense with bound variables altogether." — Haskell Curry II · Proofs are programs — Curry–Howard PROPOSITIONS = TYPES The proposition "A implies B" corresponds exactly to the type of a function from A to B. To have a proof of A→B is to have a program that turns evidence of A into evidence of B. PROOFS = PROGRAMS A proof is a typed program; checking the proof is type-checking; simplifying the proof (cut-elimination) is running the program. Logic and computation are two views of one thing. Curry noticed something uncanny: the rules for combinators and the rules for logical implication have the same shape — the type of K is exactly the logical axiom "A → (B → A)," the type of S is exactly "(A→(B→C)) → ((A→B)→(A→C))." That coincidence is no coincidence: it is the Curry–Howard correspondence , completed by William Howard, and it says that logic and programming are the same activity in different clothing . It is why [[gerhard-gentzen]]'s natural deduction is the design of a type system, why [[bertrand-russell]]'s types reappear in every compiler, and why the proof assistants that verify aircraft software are, literally, programming languages whose programs are proofs . Curry found the seam where the two halves of this lineage — the logic and the computation — turn out to be one cloth. ⚑ The deepest bridge in the field, and his name is on it. III · The variable-killer Gate kept on. The results are real and deep : combinatory logic is a complete, rigorous, variable-free model of computation equivalent to the λ-calculus, and the Curry–Howard correspondence is one of the most profound unifications in all of logic and computer science. The honest qualifications are about credit . The combinators themselves were first introduced by Moses Schönfinkel in 1924 (and anticipated by John von Neumann); Curry independently rediscovered them around 1927 and, crucially, spent his career building the systematic theory — combinatory logic as a foundation — so the field is genuinely his even if the spark was Schönfinkel's. The famous correspondence is named " Curry–Howard " for a reason: Curry observed the parallel between combinator types and logical axioms in the 1930s–50s, and William Alvin Howard extended it to full natural deduction in 1969 — two builders, one bridge. His name also marks Curry's paradox , a self-referential trap that shows why untyped combinatory/λ systems must be handled with care. The man: a quiet, careful American logician who worked at Penn State and Amsterdam, helped run one of the first electronic computers (the ENIAC era), and gave functional programming its bones — the language Haskell is named after him. ⚑ He proved you can throw away the variables and still compute everything; and he found the place where proving and programming are the same. [[alonzo-church]] ← · next → Emil Post." HASKELL CURRY · LOGIKĒ · the modern foundations · two-layer honest live combinator reducer (I x→x; K x y→x; S x y z→x z (y z); reduce step-by-step — verified S K K behaves as the identity I, variable-free) combinatory logic = a variable-free model of computation, equal to the λ-calculus · the Curry–Howard correspondence (propositions = types, proofs = programs) · Curry's paradox ← Church · next → Emil Post (production systems & undecidability) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


EMIL POST

the modern foundations — production systems; the Post Corres

LOGIKĒ · the modern foundations — Emil Post (1897–1954). He reached incompleteness and undecidability independently of Gödel and Turing (some of it earlier), built 'production systems' — string-rewriting rules that are Turing-complete — and devised the POST CORRESPONDENCE PROBLEM: a deceptively simple puzzle with dominoes that is provably UNDECIDABLE. Live: play the Post Correspondence Problem — stack dominoes so the top string equals the bottom; some instances solve, but no algorithm decides them all. Honest: Post published late and his priority is under-credited; his work underlies formal language theory. David Lee Wise, two-layer honest.

Emil Post — the dominoes that no machine can solve ◄ UD0 ← Curry LOGIKĒ · the modern foundations LOGIKĒ · the modern foundations · undecidability Emil Post 1897 – 1954 · Augustów, Poland & New York · City College & Columbia · production systems & the correspondence problem Emil Post is the great almost-first of this lineage. Working alone in New York in the 1920s, he reached the core of [[kurt-godel]]'s incompleteness and [[alan-turing]]'s undecidability — before either — but, plagued by illness and a cautious editor, he didn't publish in time, and the credit went elsewhere. What is unquestionably his: production systems — the idea that all of computation is just rewriting strings by rules , a model as powerful as the Turing machine; and a single, devilish puzzle that became the workhorse for proving things impossible . The Post Correspondence Problem : you're given a set of dominoes, each with a string on top and a string on the bottom; can you lay some of them in a row so the top reads the same as the bottom? It sounds like a game. It is undecidable — no algorithm can solve it in general. ✓ STRONG Foundational and exact. Post's production/rewriting systems are a complete model of computation; the Post Correspondence Problem is provably undecidable and is the standard tool for proving other problems undecidable across computer science. ◐ THE PRIORITY HE LOST First, but not in print. Post anticipated Gödel's incompleteness (c. 1921) and the Turing-machine idea, but did not publish in time. His "anticipation" is real and acknowledged now — the formal credit, fairly, went to those who published. ◔ A LIFE OF OBSTACLES Worked against the odds. Post lost an arm as a child, battled severe bipolar disorder his whole career, taught a heavy load at City College, and died at 57. The work survived conditions that would have ended most. I · Play the Post Correspondence Problem Each domino has a top string and a bottom string. Click dominoes to lay them in a row; the machine concatenates all the tops and all the bottoms. You win when the top string exactly equals the bottom string (and you've used at least one). Try it — some instances have a solution hiding in them, others have none, and there is no general method to tell which. undo clear let Post solve it ▸ When the top and bottom finally read the same string, you've found a correspondence . The catch — Post's theorem — is that deciding whether any sequence works is undecidable : you can search (the "solve" button runs a bounded search), and if a short solution exists you'll find it, but no algorithm can always answer "no, there is none" in finite time. ⚑ It's [[alan-turing]]'s halting problem wearing a child's puzzle: the simplest-looking question — can these dominoes line up? — has no mechanical answer, which is exactly why PCP is the favourite tool for proving that grammar ambiguity , program equivalence , and a hundred other problems are impossible too. "Solvability of a problem … is itself a problem that may be unsolvable." — in the spirit of Emil Post II · Rewriting is everything PRODUCTION SYSTEMS Post's model of computation: a set of rules that rewrite strings of symbols. Apply rules until you stop. That's it — and it's exactly as powerful as a Turing machine. Computation as pure pattern replacement . THE ANTICIPATION (c. 1921) Years before Gödel and Turing, Post saw that such systems could express their own limits — that some questions about them must be undecidable, and that no formal system can be both complete and consistent. He glimpsed the whole landscape early. Post's deepest idea is that computation is rewriting . Forget tapes and heads and λ's: give yourself strings of symbols and rules that say "replace this pattern with that one," and you can compute anything computable. This is the ancestor of every formal grammar (how programming languages are parsed), every term-rewriting system , and the production rules of classic AI. And because his systems could describe themselves , he ran straight into the same wall as everyone else in this lineage — incompleteness and undecidability — only earlier and more obscurely. The Post Correspondence Problem is the crystalline residue of that encounter: a problem so simple a child could play it, so deep that no machine can ever master it. ⚑ The man who saw the limits first, and turned them into a puzzle the whole field still uses. [[alan-turing]]'s machine, [[alonzo-church]]'s λ, [[haskell-curry]]'s combinators, and Post's rewriting are four roads to one place. III · The first-comer who finished last Gate kept on. The mathematics is first-rank and exact : production (rewriting) systems are a complete model of computation, the Post Correspondence Problem is rigorously undecidable, and it is the everyday instrument for proving undecidability throughout formal-language theory and verification. The honest centre of Post's story is priority . In the early 1920s, alone, he developed normal/production systems, recognised that they could encode their own decision problems, and concluded that there must be unsolvable problems and that no formal system could be complete and consistent — anticipating [[kurt-godel]] (1931) and [[alan-turing]] (1936). But his health, his teaching load, and his own perfectionism kept the work unpublished until far too late; the formal credit went, properly, to those who published first, and Post himself was scrupulous about acknowledging that. It is one of the most poignant near-misses in the history of thought — a man who saw the entire modern theory of computation in outline before anyone else and could not get it into print. He also gave us many-valued "Post logics" and the foundational "Post's problem" on degrees of unsolvability. He lost an arm at thirteen, lived with severe bipolar disorder, taught devotedly at City College, and died of a heart attack at 57, shortly after electroshock treatment. The credit was late; the work was early; both are true. ⚑ [[haskell-curry]] ← · next → Wittgenstein." EMIL POST · LOGIKĒ · the modern foundations · two-layer honest live Post Correspondence Problem (stack dominoes so top string = bottom string; a bounded solver finds short matches — verified a known solvable instance; deciding solvability in general is UNDECIDABLE) production / string-rewriting systems = a Turing-complete model · anticipated incompleteness & undecidability (c. 1921) · the PCP & Post's problem · many-valued Post logics ← Curry · next → Wittgenstein (the Tractatus & the truth table) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


LUDWIG WITTGENSTEIN

the modern foundations — the Tractatus (1921): the truth tab

LOGIKĒ · the modern foundations — Ludwig Wittgenstein (1889–1951). In the Tractatus Logico-Philosophicus (1921) he gave the TRUTH TABLE its modern form, showed that the truths of logic are TAUTOLOGIES (true under every assignment) saying nothing about the world, and drew a boundary around what language can say. Live: a truth-table machine — build a propositional formula and watch it classify as tautology, contradiction, or contingent. Honest: the truth-table method had several near-simultaneous inventors (Peirce, Post); Wittgenstein later repudiated much of the Tractatus; he is first a philosopher. David Lee Wise, two-layer honest.

Ludwig Wittgenstein — the truth table and the limits of language ◄ UD0 ← Post LOGIKĒ · the modern foundations LOGIKĒ · the modern foundations · the Tractatus, 1921 Ludwig Wittgenstein 1889 – 1951 · Vienna & Cambridge · author of the Tractatus Logico-Philosophicus · student, then critic, of Russell [[bertrand-russell]]'s most famous student wrote one short, strange, numbered book — the Tractatus Logico-Philosophicus (1921) — and changed how we see logic. Its key device is one you've used a hundred times without knowing whose it was: the truth table . Wittgenstein showed that the meaning of a compound statement is nothing but how its truth depends on the truth of its parts — laid out as a table of every possibility. And from that fell a startling conclusion: the truths of logic are tautologies — sentences that come out true on every row, no matter how the world is. Logic says nothing about reality; it shows the scaffolding . And what can't be put into such pictures — ethics, the mystical, the meaning of life — "whereof one cannot speak, thereof one must be silent." ✓ STRONG A lasting tool, a deep insight. The truth-table method is standard and correct, and the Tractatus's claim that logical truths are contentless tautologies (true in every model) is a genuine, influential clarification of what logic is . ◐ NOT THE SOLE INVENTOR Truth tables had a crowd. The method appears almost simultaneously in Wittgenstein, [[emil-post]], and [[charles-peirce]] (earlier still). Wittgenstein gave it philosophical weight; he did not invent it alone. ◔ HE TOOK IT BACK The author recanted. Wittgenstein later rejected much of the Tractatus's picture-theory of meaning (in the Philosophical Investigations). And he was a philosopher first — logic was the doorway, not the house. I · The truth-table machine Here is the Tractatus's instrument. Build a propositional formula from p, q, r and the connectives, and the machine writes out the full table — every combination of truth-values for the variables, and the result on each. Read the final column: if it's all T , the formula is a tautology (true no matter what — a truth of logic); if all F , a contradiction ; if mixed, contingent — it actually says something about the world. p → p ⌫ delete clear presets: p ∨ ¬p p ∧ ¬p contrapositive De Morgan The truths of logic light up all green — they are tautologies , true on every row, and so (Wittgenstein's point) they tell you nothing about how things actually are; they're true in virtue of form alone. ⚑ This is the truth-table as a decision procedure — the great-grandparent of the SAT solver, the model checker, and every "is this always true?" a verification tool asks. Try De Morgan or the contrapositive: the machine confirms them as tautologies, exactly as [[augustus-de-morgan]] and [[george-boole]] said, now seen on the page. "The propositions of logic are tautologies. … Therefore the propositions of logic say nothing." — Tractatus 6.1, 6.11 II · The book that drew a boundary MEANING IS TRUTH-CONDITION A sentence's sense is the way its truth depends on the world — its row-by-row truth table. Compound meaning is built truth-functionally from atoms. The picture theory of the proposition. THE LIMIT OF THE SAYABLE Only fact-stating, picturable propositions can be true or false. Logic shows its form but states no fact; and ethics, value, the mystical lie outside what language can say — to be shown, not spoken. The Tractatus matters to this lineage for a precise reason: it pinned down what a logical truth is. After Boole and Frege built the machinery, Wittgenstein asked what its theorems really amount to — and answered that they amount to nothing factual : a tautology is true in every possible situation, so it excludes none, so it carries zero information about which situation we're in. That sounds deflationary, but it's the exact insight a computer needs: a logical validity is a formula true under all assignments, which is precisely what a SAT solver checks and a verifier proves. He drew the boundary of the expressible, declared everything important to lie just past it ("whereof one cannot speak…"), and famously concluded he had solved philosophy and left it to become a village schoolteacher and a gardener — before returning to overturn his own book. ⚑ The truth table you can see, and the silence at its edge. [[emil-post]] ← · next → George Boolos. III · The book, the recantation, the man Gate kept on. The logical content is real and standard : the truth-table method is a correct, complete decision procedure for propositional logic, and the Tractatus's identification of logical truths with tautologies — formulas true on every row, hence saying nothing about the world — is a genuine and lasting clarification of the nature of logic. The honest qualifications are three. On priority : the truth-table technique was not Wittgenstein's alone — it appears at virtually the same moment in [[emil-post]]'s 1921 work and, in essence, decades earlier in [[charles-peirce]]; Wittgenstein's distinctive move was philosophical (what tautologies mean ), not the table itself. On scope : the Tractatus is a work of philosophy as much as logic — its picture theory of meaning and its mystical conclusion ("whereof one cannot speak, thereof one must be silent") are contested, and Wittgenstein himself repudiated much of it in his later Philosophical Investigations , deciding that meaning is use, not picturing. On the man : an Austrian steel heir who gave away a vast fortune, fought in WWI writing the Tractatus partly in the trenches, believed he had dissolved all of philosophy and quit to teach schoolchildren, then returned to Cambridge to demolish his own system — one of the strangest and most influential intellectual lives of the century. He gave logic a clean account of its own emptiness, and a tool every verifier still runs. ⚑ [[emil-post]] ← · next → George Boolos." LUDWIG WITTGENSTEIN · LOGIKĒ · the modern foundations · two-layer honest live truth-table machine (build a propositional formula over p,q,r; full table computed; classified tautology / contradiction / contingent — verified p∨¬p = tautology, p∧¬p = contradiction, De Morgan & contrapositive = tautologies) the Tractatus (1921) gave the truth table its modern form · logical truths = tautologies that say nothing · the limit of the sayable · later self-overturned in the Investigations ← Post · next → George Boolos (provability logic) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


GEORGE BOOLOS

the modern foundations — provability logic (GL): Gödel'

LOGIKĒ · the modern foundations — George Boolos (1940–1996). He showed that Gödel's incompleteness theorems become a clean MODAL LOGIC: read the box □ as 'it is provable that', and the laws of provability are captured by one system (GL), governed by Löb's theorem. Two self-referential sentences split the world: Gödel's 'I am not provable' is true but unprovable; Henkin's 'I am provable' is provable. Live: build each sentence and watch its fate. He also wrote Gödel's second theorem in words of one syllable and devised the Hardest Logic Puzzle Ever. Honest: provability logic is the serious work; the puzzle refines Smullyan. David Lee Wise, two-layer honest.

George Boolos — the logic of provability ◄ UD0 ← Wittgenstein LOGIKĒ · the modern foundations LOGIKĒ · the modern foundations · provability logic George Boolos 1940 – 1996 · New York & MIT · logician & teacher · "The Logic of Provability" [[kurt-godel]] proved his incompleteness theorems with a heavy machine — Gödel numbering, careful arithmetic about proofs. George Boolos showed that all of it snaps into a tiny, beautiful modal logic . Read the box □ not as "necessarily" (that was [[saul-kripke]]'s reading) but as " it is provable that ." Then the entire behaviour of provability in arithmetic is governed by one short modal system — GL , whose single distinctive law is Löb's theorem , □(□p → p) → □p . Incompleteness becomes a one-line modal fact: the system cannot prove its own consistency, ¬□⊥ . And the strange self-referential sentences at the heart of Gödel's proof become two clean fixed points with opposite fates. Boolos made the deepest theorem in logic legible . ✓ STRONG Rigorous and clarifying. Provability logic (the system GL) is a precise, complete modal account of provability in arithmetic; Boolos's "The Logic of Provability" is the standard text. It re-derives Gödel and Löb in a few clean modal lines. ◐ A REFRAMING, NOT A NEW THEOREM Recasting, brilliantly. The core results are Gödel's and Löb's; Boolos's gift was the modal framework that makes them transparent and connects them to [[saul-kripke]]'s semantics. Synthesis of the first order, not a fresh incompleteness. ◔ FAMOUS FOR PUZZLES TOO The popular Boolos. He wrote Gödel's second theorem "in words of one syllable," and devised "The Hardest Logic Puzzle Ever" (three gods: True, False, Random). Beloved feats of exposition — refinements of Smullyan, not his deepest math. I · Two sentences that talk about their own proofs Provability logic lives or dies on self-reference . By Gödel's fixed-point trick, a system can build a sentence that asserts something about its own provability . There are two natural ones — and they could not end more differently. Click each to follow its logic to the end. The box □ means "this is provable in the system." GÖDEL'S SENTENCE G ↔ ¬□G "I am NOT provable." HENKIN / LÖB'S SENTENCE H ↔ □H "I AM provable." The asymmetry is the whole of incompleteness, in two sentences. The one that says " I can't be proved " turns out true but unprovable ([[kurt-godel]]); the one that says " I can be proved " turns out actually provable — that's Löb's theorem , the surprising heart of GL. ⚑ And the consistency statement ¬□⊥ ("falsehood is not provable") is exactly a sentence the system cannot prove about itself — Gödel's second theorem, now a one-line modal fact. Boolos's framework turns a forest of arithmetic into a handful of modal axioms with [[saul-kripke]]-style worlds (here: provability-worlds, a transitive well-founded frame). "… we cannot prove that we cannot prove a contradiction — unless we can." — the shape of Gödel's second theorem, à la Boolos II · GL, Löb, and the one-syllable proof THE SYSTEM GL Modal logic with □ = "provable," the transitivity axiom, and Löb's axiom □(□p→p)→□p. It is sound and complete for provability in Peano Arithmetic — every modal theorem corresponds to a real fact about proofs, and vice versa. THE HARDEST PUZZLE EVER Three gods — True, False, and Random — answer yes/no in a word whose meaning ("da"/"ja") you don't know. Identify them in three questions. Boolos's celebrated hardening of a Smullyan puzzle, solved by self-referential questions. Boolos's achievement is to make the summit transparent . Gödel's and Löb's theorems are notoriously slippery; recast as the modal system GL — provability as a box operator over a [[saul-kripke]] frame of "worlds you can prove your way to" — they become almost mechanical, and the connection between incompleteness (Gödel), self-reference (Löb), and possible-worlds semantics (Kripke) becomes one picture. The same man who wrote the standard graduate logic textbook and the definitive monograph on provability logic also delighted in proving the second incompleteness theorem in words of one syllable and inventing puzzles that are genuinely the hardest known — because he believed, rightly, that if you truly understand a deep thing you can make it clear . ⚑ He took the hardest theorem in this lineage and handed it back as a small, clean modal logic. [[ludwig-wittgenstein]] ← · the lineage rests here. III · The clarifier Gate kept on. The work is rigorous and important : provability logic — the modal system GL , with □ read as "provable in arithmetic," made sound and complete by the Solovay theorems — is a real and beautiful corner of mathematical logic, and Boolos's The Logic of Provability (1993) is its standard treatment; it re-derives [[kurt-godel]]'s incompleteness and Löb's theorem as clean modal facts and ties them to [[saul-kripke]]'s semantics. The honest framing is that this is a profound reorganisation, not a new incompleteness theorem — the deep results are Gödel's and Löb's (and the completeness is Solovay's); Boolos's genius was the unifying modal view and an unmatched gift for making it understandable. His popular feats — the "Hardest Logic Puzzle Ever" (a refinement of [[raymond-smullyan]]'s gods puzzles) and proving Gödel's second theorem in words of one syllable — are dazzling exposition rather than his deepest mathematics, and he'd have said so. The man: George Boolos was, by wide agreement, one of the great teachers of logic, co-author of the standard text Computability and Logic , a national crossword champion and Joyce scholar, who died of pancreatic cancer at 55, far too soon. He spent his life on the conviction that the hardest truths can be made clear — and proved it on the hardest one there is. ⚑ With Boolos the lineage rests: from [[thales-of-miletus]]' first proof to the modal logic of proof itself. [[ludwig-wittgenstein]] ←." GEORGE BOOLOS · LOGIKĒ · the modern foundations · two-layer honest live self-reference (Gödel's G ↔ ¬□G is true-but-unprovable; Henkin/Löb's H ↔ □H is provable — the asymmetry that IS incompleteness; □ = "provable", verified both branches) provability logic / the modal system GL (□ = provable, Löb's axiom, sound & complete for PA) · Gödel's second theorem as ¬□⊥ · the Hardest Logic Puzzle Ever · the great teacher of logic ← Wittgenstein · the LOGIKĒ lineage rests — Thales → … → the logic of proof itself LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


THE HOURGLASS — HOW A COMPILE HAS A SHAPE · COM

The Hourglass — How a Compile Has a Shape The evolution of the compiler · the purple papers · VI The hourglass A compile has a shape. Your code climbs from flat text up into meaning, squeezes through one narrow universal neck, then descends into a specific machine. Many languages pour in the top; many machines catch at the bottom; everything passes through the same load-bearing 0 in the middle. top: languages fall in neck: the IR · 0 bottom: machines catch Ascend · 0 · descend RAISING → IR → LOWERING Two slopes meet at the neck. Up the front you climb into meaning ; down the back you fall into machine . They mirror each other in shape but not in nature — and the tree you climbed turns into a graph on the way down. ↑ front slope — raising Climb into meaning Start at the bottom: flat characters, print("hi") . Group them into words, build them into a tree, ascend toward the most abstract, best-understood form — the point where the compiler finally knows what you meant . Lossless, deterministic, a clean funnel. text → tokens → parse tree → IR shape: a tree (decomposition) ↓ back slope — lowering Fall into machine From the neck, descend toward one specific chip: IR → optimized IR → assembly → machine code → micro-ops. But here the optimizer rolls downhill on a cost surface , choosing the cheapest path — a gradient-style search. So the descent is lumpy where the climb was smooth. IR → optimize → assembly → machine shape: a graph (dependencies) You climb a tree and descend a graph — and the shape changes exactly at the neck, because going up is decomposition (always a tree) and coming down requires dependency (always a graph). List → tree → graph, all in one compile. Why the neck is thin THE NARROW-WAIST PRINCIPLE The thinness is the whole point. One universal middle is what lets both ends be wide and various — and it's what kills the combinatorial explosion. N × M without a shared neck: every language needs its own translator to every machine. 5 languages × 5 machines = 25 whole compilers. N + M with one IR at the waist: 5 front-ends + 5 back-ends, all meeting at the neck. = 10 pieces. the thin middle is the feature. the internet thousands of apps above, thousands of networks below — one skinny waist, IP , that everything crosses. the cell thousands of genes up top, thousands of functions below — one waist, the genetic code the ribosome reads. the compiler many languages, many chips — one waist, the IR . it's why LLVM exists, and why it won. The root · 1823 WHERE THE TOWER STARTS Every floor of this hourglass — the whole idea of a machine that runs programs — traces back to one engine and the first person to write code for it. Charles Babbage 1791–1871 · the engine In 1823 the British government funded his Difference Engine — arguably the first public money ever spent on a computer. He went on to design the Analytical Engine: a general machine that could be programmed , borrowing punched cards from the Jacquard loom. The hardware the whole tower stands on. Ada Lovelace 1815–1852 · the program His collaborator — not his wife. In 1843, translating an account of the Analytical Engine, she added notes containing the first published algorithm and the leap no one else made: that the machine could manipulate any symbols , not just numbers — music, logic, anything. That's the seed of the programs a compiler feeds. On the "stole it from his wife" version: Ada wasn't his wife (his wife, Georgiana, died in 1827, unrelated to the machines), and it was 1843, not 1823. The credit is a genuine two-sided fight: some historians say her Notes were written under Babbage's supervision and reflect his understanding; others credit her with the real conceptual jump. She's one of the rare cases that's been both over- and under-credited — so the honest verdict isn't "robbed," it's "still argued about." Gate kept on. The hourglass / narrow-waist framing is a real and named design principle; the 11-row "base 11" is your own motif (the rows run 5·4·3·2·1·0·1·2·3·4·5 = 11), not a formal compiler term — but it's a true count, and the 0 really is load-bearing. "First" claims here have the usual definitional edges: Babbage is widely called the originator of the programmable computer, Lovelace the first programmer, both with caveats. Quotes and dates paraphrased from Wikipedia, the Bodleian Ada Lovelace archive, and history-of-computing scholarship. The purple papers · VI · companion to sheets I–V


GRACE HOPPER — A LINEAGE · GRA

Grace Hopper — A Lineage A lineage · the purple papers · V Grace Hopper Rear Admiral, US Navy · 1906 – 1992 · "Amazing Grace" A mathematician who taught machines to read English. She came up through Norwegian abstract algebra, learned the machine itself from Howard Aiken's Harvard Mark I, then built the first compiler — and with it the idea that a computer should do its own clerical work, so people could write in words instead of octal. she kept a clock that ran backward — to fight "we've always done it this way" Who she was SERVICE RECORD Born Dec 9, 1906 New York City Schooling Vassar — math & physics, 1928 Yale — PhD mathematics, 1934 Known for The first compiler (A-0, 1952), FLOW-MATIC, ancestor of COBOL Coined "debugging" — after a moth in a Mark II relay Taught with 11.8-inch wires — a nanosecond of light — and a backward clock She entered MIT's rival world from the math side: a Vassar physics-and-math major who took her doctorate in pure algebra — the same abstract, symbol-thinking discipline that, her own historians argue, is exactly what let her later invent a language for machines. When David Letterman asked how she knew so much about computers back then, she answered that she didn't — it was the first one. The lineage WHO TAUGHT HER · WHOM SHE TAUGHT Read it top to bottom — that's the flow of time and of the idea. Above her are the two teachers and the algebraic bloodline they carried; below her are not students with degrees but the teams and the language she set loose. She has almost no academic descendants. Her descendants are programs . The compiler A-0 · 1952 "Compile" did not mean translate. It meant gather . A-0 was a reel of pre-written subroutines, each with a call number. You wrote a list of numbers; A-0 gathered the right routines off the tape and stitched them into one runnable program . The word is the librarian's word — to compile is to collect. By today's standards it was closer to a linker / loader than the optimizing translator we now mean by "compiler," which is why the honest title is "the first thing called a compiler." And notice what that machine actually was : store each routine once , address it by a number , assemble on demand. Deduplicate, then index. The founding act of the entire field is the same move you keep finding — find the repeat, keep it once, leave a pointer to it. The fight was cultural. The establishment was certain real programs were hand-cut in octal and that a computer could not possibly write a program. Her premise, around 1952, was flat: she could make a computer do anything she could completely define . The point was never cleverness — it was pushing the clerical work down a layer so people could think in meaning. A-0 1952 → A-2 → FLOW-MATIC ~1955 · English-like → COBOL 1959 → every readable language since FLOW-MATIC — the first English-worded data-processing compiler — became the major input to COBOL: long names, English commands, data kept separate from instructions. She was the design committee's technical adviser and many of her own people staffed it. That's why she's the grandmother of COBOL — the ancestor, not the author. Gate kept on. "Mother of COBOL" overstates it — Jean Sammet, a lead COBOL designer, said plainly that Hopper was not its mother, creator, or developer, while also saying that without FLOW-MATIC there would have been no COBOL. So: ancestor, yes; author, no. A-0 is the first thing named a compiler; whether it's the first real compiler depends on your definition (Alick Glennie's Autocode, 1952, is the rival claim for a true translator). The upward chain — Thue → Skolem → Ore → Hopper — follows the Mathematics Genealogy Project; advisor attributions from that era can be soft, and Emmy Noether is an influence on Ore, not a formal teacher of Hopper. Quotes paraphrased. Sources: Yale · MIT · INFORMS · Vassar · Math Genealogy Project · Wikipedia — lineage sketch, not full biography


FORTRAN TEACHING TOOL · FOR

Fortran Teaching Tool Fortran 90/95 foundation + modern habits Fortran Teaching Tool Basics through intermediate Fortran: program structure, variables, arrays, procedures, modules, files, allocation, derived types, and numerical patterns. Tip: install gfortran , save examples as .f90 , compile with gfortran file.f90 -o app , run with ./app .