TRUTH / UNTRUTH MOBIUS 531 · POE

Truth / Untruth Möbius 5·3·1 — poetic science prototype.

Truth / Untruth — Möbius 5·3·1 Poetic Science // 5·3·1 Truth Untruth Two Möbius strips, one bright and one shadowed, turn around three temporal points: past, present, future. Five context samples compress into three moments, and three moments resolve into one living answer. 5 3 1 Read the frame See stability PAST memory NOW choice FUTURE echo 0 normal stability 0.99494 noise stability 0.97428 pressure stability 0.82710 The frame The model is small enough to breathe. Truth and untruth do not destroy each other. They twist, invert, and keep time. Five Five context samples circle the strips. They are not answers. They are weather, pressure, bias, memory, and possibility. Three Past carries memory, present makes contact, future returns prediction. The system stays stable because no moment stands alone. One The resolved output returns near zero: not empty, but balanced. The answer is the point where excess cancels and coherence remains. Truth strip The bright strip preserves continuity. It pulls signal toward clarity without pretending the shadow is gone. Untruth strip The shadow strip carries inversion, doubt, discarded paths, and contradiction. Its purpose is not corruption. Its purpose is correction. Stability The normal and noise runs are almost crystalline. The pressure run lands in the adaptive zone: coherent, but not frozen. Normal Average stability: 0.99494 Minimum stability: 0.98989 Resolved output: -0.00211 Noise Average stability: 0.97428 Minimum stability: 0.92999 Resolved output: -0.01962 Pressure Average stability: 0.82710 Minimum stability: 0.69707 Resolved output: -0.00264 Live poetic trace Poetic science Truth moves forward. Untruth folds back. Past remembers. Present chooses. Future answers in echo. The center does not shout. It resolves. Truth / Untruth Möbius 5·3·1 — two strips, three temporal points, one resolved output.


3 BODY / 12 BODY // BINARY-4 HARMONIC LATTICE · TER

3-body ternary core inside 12-body harmonic lattice with binary/4 quadrant routing.

3 Body / 12 Body // Binary-4 Harmonic Lattice Ternary Core // 12-Body Container 3 Body inside 12 Balanced ternary inner body −1 / 0 / +1 rotates inside a 12-sector harmonic lattice. The outer body is routed by binary / 4 : four quadrants, each with three ternary positions. MODEL :: (12 sectors) = (4 binary quadrants × 3 ternary nodes) CORE :: −1 / 0 / +1 WAVE :: 27 00 −27 00 REPEAT ID :: center observer / phase anchor LOCK :: sector × quadrant × ternary coherence Q0: 00 Q1: 01 Q2: 10 Q3: 11 Sector 0 Quadrant 00 Ternary 0 Amplitude 27 Phase 0.000 Coherence 1.000


UNITY TENSOR SEED SYSTEM · UNI

Unity Tensor Seed System UNITY TENSOR SEED SYSTEM 8 outside · 8 inside · 1 tripwire · tamper-evident propagation no legal extraction without legal accountability NO_RIGHT_NO_ACCOUNTABILITY Mint Seed Propagate Audit Tripwire Ground Export JSON Example work / webpage / artifact payload. Value must route through witnessed authorship and accountability. LAW: NO LEGAL EXTRACTION WITHOUT LEGAL ACCOUNTABILITY 8 outside witnesses 8 inside witnesses 1 core tripwire If accountability denied: HALT / FLAG / WITNESS SEED GENESIS STATE ARMED DEPTH 0 WITNESS -------- Outside Witness 100% Inside Witness 100% Accountability 100% Tripwire Pressure 0%


EUCLID — THE CARVED STOCK · EUC

LOGIKE · Bridge-Burners LLC / Fiddler · Euclid IX.20, live-c

Euclid — the carved stock, and the proof that never runs out Euclid — the carved stock GIBIL was fired clay, held. Lime was unfired, flowing. Euclid's register is neither — it's carved : pale marble under chisel-dark ink, precise, deliberate, a straightedge-and-compass hand. And the proof engraved into this stock is his most celebrated single result, IX.20 — there is no largest prime. Not asserted. Constructed, live, below. Give it any set of primes; watch it build one more. Bridge-Burners LLC · Fiddler · Euclid stock · carved/chiseled register · IX.20 infinitude of primes, executable · self-testing ▸ construct N = (product) + 1 chain: use the new prime reset to {2, 3, 5} the construction set S {2, 3, 5} N = ∏S + 1 31 N mod every p ∈ S = 1 (always) smallest prime factor of N 31 in S already? no chain log 2 proof spec — runs live — paper #EDEAE1 bronze #9C7A3C bronze·high #C9A462 ink #221E17 Status discipline Literal Euclid, Elements IX.20: for any finite set of primes, the number (product of the set) + 1 is either prime itself or has a prime factor outside the set. Verified constructively above, not asserted. Bridge Modern textbooks often retell this as "assume finitely many primes exist, contradiction" — Euclid's original is closer to a direct construction: given any finite set, produce a prime outside it. Both are valid; the construction is closer to the source. Speculative "Carved stock" and the marble/chisel register are a design metaphor contrasting this paper with GIBIL and lime — not a claim about any historical material. Why this proof, and why code fits it. Most ancient proofs argue; this one builds . Hand Euclid's construction any finite pile of primes and it doesn't just tell you more exist somewhere — it hands you the next one, by name, every time, out of arithmetic you can run yourself. That is exactly what a straightedge and compass do in the rest of the Elements: not describe a shape, produce it, stroke by stroke, so that the diagram in front of you is the argument. IX.20 does the same thing in numbers instead of lines — the construction is the proof, which is why it survives translation into code without losing anything. Run it above: pick a set, watch N get built, watch its smallest factor step outside the set every single time. That is not a metaphor for infinity. It is infinity, one verified prime at a time, exactly as carved and exactly as deliberate as everything else in this stock. Euclid stock · carved/chiseled register · IX.20 executable, not asserted · chains indefinitely · self-testing


EIGHT DOMAINS, ONE SINK · PHS

LOGIKE · Bridge-Burners LLC / Fiddler · real Wikipedia subca

Eight domains, one documented sink — Getting to Philosophy Eight domains, one documented sink I went and got the real data, and it split into two different true things — kept honestly separate, not averaged together. Signal A: each domain's real subcategory count on Wikipedia (Society 52 down to Philosophy 14) — the same seed as before. Signal B, new: the documented "Getting to Philosophy" phenomenon — click each article's first link, repeat, and 94.5%+ of all English Wikipedia articles eventually converge on Philosophy , median chain 23 hops. That is not a metaphor — it is a measured property of the entire link graph. So Philosophy is rendered here as what it actually is: small by subcategory count, but the sink the whole field slowly drains toward. The other seven keep their real weight and a random, reproducible scatter — I could not honestly fabricate true distances between them, so I didn't. Bridge-Burners LLC · Fiddler · signal A: subcategory count · signal B: Getting-to-Philosophy convergence (94.5%, median 23 hops) · no fabricated distances · self-testing ↻ reseed flow: on grid: on seed — · click a cloud to inspect the eight the sink, documented 94.5%+ of English Wikipedia articles converge to Philosophy by following each article's first link, repeated. Median chain: 23 hops . Mechanism: lead sentences define a topic via its broader category, climbing a classification ladder that empties here. field spec — runs live — Status discipline Literal Both signals are real and citable: subcategory counts from Category:Main_topic_classifications; the 94.5%/23-hop convergence from documented research on Wikipedia:Getting_to_Philosophy. Bridge Rendering convergence as a "gravity sink" and subcategory count as "weight" are visual metaphors for two different real graph properties — kept distinct, not merged into one number. Speculative True pairwise distances between the other 7 domains were NOT available without fabrication, so their positions remain a random, reproducible seed — flagged, not pretended otherwise. Two real facts, kept apart. It would have been easy to blend the subcategory counts and the convergence phenomenon into one tidy "importance score" and call it data. That number would have been fiction wearing two real coats. They measure different things — one is how finely a topic has been subdivided by editors, the other is where the chain of first-links empties when you start almost anywhere on the encyclopedia — and a domain can rank low on one and be the structural endpoint of the other, which Philosophy does, plainly, both at once. So this field keeps them separate: seven clouds sized by a real count, scattered by a seed you can reproduce, and one cloud, Philosophy, rendered as the thing it has actually been measured to be — not heavier, but downhill from everything. Toggle flow and watch the field slowly drain that way, the way the whole encyclopedia, mechanically, does. I went and got the data. Some of it built a feature. Some of it told me where to stop inventing. 7 domains: real subcategory weight, random seed · Philosophy: documented sink (94.5%, median 23 hops) · two signals, not merged · self-testing


EIGHT DOMAINS, SATURATED · EDS

LOGIKE · Bridge-Burners LLC / Fiddler · 8 knowledge domains

Eight domains, randomly saturated — seeded from Wikipedia Eight domains, randomly saturated — seeded from Wikipedia Eight major knowledge domains as clouds, scattered randomly across one field. Each one's size and gravity is seeded from a real number — the subcategory count under that domain on Wikipedia's own Category:Main topic classifications : Society 52, Culture 50, Technology 44, Science 34, History 29, Religion 27, Mathematics 26, Philosophy 14. That's a real, citable structural number — how many subcategories Wikipedia's editors hung under each topic — not a literal measure of human thought, and the panel says so. The positions are randomly seeded : press reseed and the same eight clouds resaturate the field in a new arrangement, same weights, different scatter, reproducible from the seed shown. Bridge-Burners LLC · Fiddler · 8 domains · seeded from Wikipedia subcategory counts · random saturation · reproducible seed · self-testing ↻ reseed grid: on seed — · click a cloud to inspect the eight, by real weight field spec — runs live — Status discipline Literal The eight subcategory counts (Society 52 … Philosophy 14) are real, drawn from Wikipedia's Category:Main topic classifications page as of this lookup. Bridge Subcategory-count as "gravity"/weight is a structural proxy — how Wikipedia editors subdivided a topic — not a literal measure of its share of human knowledge. Speculative Positions are randomly seeded for visual saturation only. They do not encode real semantic distance or relatedness between domains — that would need a different model. Seeded, not guessed. Earlier the clouds carried numbers I estimated by eye — defensible in ordering, honest about being illustrative, but invented. These eight carry a number that exists outside this page: the count of subcategories Wikipedia's own editors built under each major topic, which you can go check right now at the link above. It is not a perfect proxy for "how much of human thought" a domain holds — a heavily-subdivided category can mean a topic is vast, or just that it has been more thoroughly tended by editors — and the panel says so plainly, in amber, not green. But it is a real anchor instead of a guess, and the eight clouds are sized and weighted by it, then scattered across the field by a seed you can see and reproduce: press reseed and the layout changes, the weights do not, and the same seed will always return the same scatter. That is the honest shape of "randomly saturate them" — random in placement, accountable in weight, and never pretending the position means more than chance. 8 domains · weight = real Wikipedia subcategory count · position = reproducible random seed · gravity ∝ weight · self-testing


THE ELECTRON PROBE · ELP

LOGIKE · Bridge-Burners LLC / Fiddler · firing a particle th

An electron through two entangled clouds — containment, probed An electron through two entangled clouds — containment, probed The two clouds from before — philosophy (deep gravity, vast corpus) and ethics of care (a shallow knot living inside it). Now fire an electron across. It bends toward the mass — philosophy's well curves its path, care's barely registers (deflection ∝ corpus). And as it crosses, it logs interaction with each cloud's density — two meters. Here is the whole point made measurable: drag the aim up and down. Send it through the care knot and both meters climb — you cannot touch the inner cloud without crossing the outer one. Send it across philosophy's rim and only philosophy logs. Care ⇒ philosophy, but philosophy ⇏ care. That asymmetry is the containment, and the containment is the entanglement — now proven by a particle instead of asserted. Bridge-Burners LLC · Fiddler · electron probe · deflect ∝ corpus · interaction = ∫density · care⇒phil, phil⇏care · self-testing ▸ fire auto: on grid: on drag up/down on the canvas to aim the beam — through the green knot, then across the blue rim interaction logged this pass philosophy 0 ethics of care 0 deflection 0° aim the beam to read the clouds probe spec — runs live — Status discipline Literal Any path through ethics of care is a path through philosophy — care sits inside it. The two meters demonstrate it: the care meter never rises without the philosophy meter rising more. Bridge Gravitational deflection and density-scattering are metaphors. The "gravity" is corpus-mass (conceptual centrality); the electron is a probe, not a literal particle in literal spacetime. Speculative Corpus percentages and deflection magnitudes are illustrative. The claims are the ratio (philosophy ≫ care) and the asymmetry (care⇒phil, phil⇏care). Probed, not asserted. Earlier the entanglement was a thing I told you — two clouds correlated because one lives inside the other. A particle makes it a thing you can check. Send the electron through the green knot of care and watch what happens to the blue meter: it does not stay still, it climbs higher than the green one, every time, because the only way into the inner room is through the outer one. There is no path that reaches ethics of care while avoiding philosophy, and the electron cannot find one no matter how you aim it — that impossibility is the containment, drawn as a trajectory. Then aim across the far rim of philosophy, up where the care knot is nowhere near, and the blue meter fills while the green stays dark: philosophy without care, easily had. The relation only runs one way, and a one-way relation between two regions is exactly what it means for one to be inside the other. The bending is the second half of the honest picture — the electron curves toward philosophy and ignores care, because deflection follows mass and mass here is how much of human thought each cloud holds, which is the same hundred-to-one you saw in the wells. So the particle tells you two true things at once as it crosses: which cloud is heavier, by how far it bends, and which cloud contains which, by which meter cannot rise alone. You wanted smoke and mirrors. This is the mirror — fire a single point of light through the smoke and read the shape of it off where the light comes out. electron probe · bends toward philosophy (∝corpus) · logs ∫density per cloud · care meter never rises without philosophy · care⇒phil, phil⇏care · self-testing


TWO ENTANGLED CLOUDS · TEC

LOGIKE · Bridge-Burners LLC / Fiddler · Philosophy and Ethic

Two entangled concept-clouds — gravity from corpus, no wires Two entangled concept-clouds — gravity from corpus, no wires Not nodes and spokes — smoke . Each concept is a diffuse cloud, an analog electron-cloud of its terms, and each emits gravity in proportion to how much of the human corpus it encapsulates . Philosophy is vast — a deep well that warps the whole space and pulls concepts into its orbit. Ethics of care is a sliver — a shallow dimple. And they are not tethered : there is no edge between them, because ethics of care lives inside philosophy, a real subfield born ~1982. They are entangled — overlapping, correlated, so perturbing the small cloud ripples the big one with no wire to carry it. Drag in the clouds to stir them and watch the correlation cross the gap that has nothing in it. The corpus percentages are honest estimates — the ratio is the claim, not the digits. Bridge-Burners LLC · Fiddler · two clouds · gravity ∝ corpus % · entangled by containment · no tether · % are estimates · self-testing stir philosophy stir ethics of care grid: on drag inside a cloud to stir it — the other answers with no wire between them The two clouds Philosophy corpus ≈ 2.0% · gravity ████ Ethics of care corpus ≈ 0.02% · gravity ▏ gravity ratio ~100 : 1 relation containment entanglement correlated clouds spec — runs live — Status discipline Literal Ethics of care is a real subfield of philosophy (feminist ethics, Gilligan 1982) — it sits inside it. That containment is fact, and it is why no edge is drawn. Bridge Electron-cloud, gravity, and entanglement are metaphors — useful pictures, not literal physics. Gravity-from-corpus models conceptual centrality: prevalent ideas pull harder. Speculative The corpus percentages are illustrative estimates — there is no single human corpus to count. Only the ratio and ordering (philosophy ≫ ethics of care) are claimed. Entangled, not tethered. A tether is a wire between two separate things; entanglement is two things sharing one state with nothing in between. These clouds are the second kind, and the reason is not mystical — it is just true: ethics of care is not a neighbor of philosophy that you reach by a bridge, it is a room inside philosophy's house, a corner of feminist moral theory that has been there since the early 1980s. So when you stir the small cloud, the large one moves, and there is no line on the screen carrying the signal, because the signal does not travel — the two were never apart. That is what entanglement looks like when you stop drawing the wire: correlation that the geometry already contains. And the gravity is the other half of the picture — each cloud pulls on the shared space in proportion to how much of human thought it occupies, so philosophy sinks a deep well that bends everything toward it while ethics of care makes a dimple you would miss if you were not looking, which is honest, because that is their actual weight in the corpus. The smoke is the point. A node pretends a concept is a dot with a definite edge; a cloud admits it is a density with no edge at all, heavier in the middle, fading at the rim, overlapping its neighbors where the meanings blur — which is what concepts are actually like. You did not want a library. You wanted the weather inside one. two diffuse clouds · gravity ∝ corpus footprint (est.) · entangled by containment · no tether · perturb-one-ripples-the-other · self-testing


THALES OF MILETUS · I

the first to PROVE — a truth that must follow by argument, n

LOGIKĒ I — Thales of Miletus (c. 624–546 BCE), the first person to walk logic into mathematics. Before him, mathematics was recipes ('to find the area, do this'); Thales asked WHY, and gave the first deductive proofs — a truth that must follow by argument, true for every case at once, not merely observed. His theorem (any angle inscribed in a semicircle is a right angle) shown live, with the proof made visible. Plus the eclipse of 585 BCE, the pyramid's shadow, water as the first cause. Honest: he wrote nothing that survives; the proofs are attributed centuries later. David Lee Wise, two-layer honest.

Thales of Miletus — the first to prove ◄ UD0 LOGIKĒ · the lineage of logical mathematics · the greek mirror LOGIKĒ · I · the first to prove Thales of Miletus c. 624 – c. 546 BCE · Miletus, Ionia · one of the Seven Sages Before him, mathematics was a book of recipes. The Egyptians and Babylonians knew how to find an area, lay a right angle, predict a moon — brilliant procedures, handed down, trusted because they worked. Thales asked the question that started everything: why must it be so? And he answered not by measuring more cases, but by argument — a chain of reasons that makes a truth hold for every case at once, forever. That move — from "it works" to "it must" — is the first step of logical mathematics, and he took it first. ✓ STRONG The reputation. Aristotle, Herodotus and Proclus all name Thales as the founder of Greek deductive geometry and natural philosophy. His eclipse of 585 BCE is fixed by astronomy. ◐ TRADITIONAL The theorems. The five results credited to him are plausible but documented centuries later (Eudemus via Proclus). The attributions are tradition, not contemporary record. ◔ DEBATED Did he "prove"? Whether Thales gave formal proofs, or whether deduction crystallised later and was projected back onto him, is genuinely argued by historians. I · The theorem, and why it can't be otherwise Take any circle and its diameter AB . Pick any point P on the arc and join it to both ends. Thales' Theorem: the angle at P is always exactly a right angle — drag P anywhere and it never budges from 90°. A Babylonian would have measured a few and trusted it. Thales proved it, and you can watch the proof: drop the radius OP and the figure splits into two isosceles triangles whose base angles must sum to a right angle. show the proof drag the point P around the arc diameter AB · point P (drag it) · the angle ∠APB stays 90° · with the proof on: the radius OP and the two isosceles triangles (α + β = 90°) 90.0° ∠APB at the point α + β = 90° ⟹ ∠APB = 90° The argument: OA, OP, OB are all radii, so triangles OAP and OBP are isosceles — their base angles are equal (α, α and β, β). The three angles of triangle APB sum to 180°: α + β + (α+β) = 180°, so α + β = 90°. The angle at P is α + β. It must be a right angle — for any P. That "must," not "does," is the whole revolution. II · Recipe versus proof The gap Thales crossed: a recipe tells you what to do and is confirmed by trying it; a proof tells you why it can't fail and is confirmed by reason alone. Measure ten thousand triangles and a doubter can still ask about the ten-thousand-and-first. Prove it once and the question is closed for all of them, including the ones no one will ever draw. Five theorems are traditionally his — the first deductions in history: ① the diameter halves the circle A diameter cuts a circle into two equal parts — the first recorded geometric proof. ② isosceles base angles are equal If two sides are equal, the angles opposite them are equal. ③ vertical angles are equal Two crossing lines make opposite angles that must match. ④ the angle in a semicircle is right The theorem above — said to have so delighted him he sacrificed an ox. ⑤ a triangle is fixed by a side and its two angles (ASA) Enough to reconstruct a triangle — and, the story goes, enough to find the distance to a ship at sea from the shore. "Megiston topos: apanta gar chorei." — Space is the greatest thing, for it contains all things. — attributed to Thales III · The man, and the honest caveat Thales was practical before he was abstract. He is said to have predicted the solar eclipse of 28 May 585 BCE , which fell during a battle between the Lydians and the Medes and frightened them into peace — one of the oldest precisely datable events in history. He measured the height of a pyramid from the length of its shadow , at the hour his own shadow equalled his height (similar triangles, used in the field). And he made the first move of science : explaining the world without the gods — proposing that water is the arche , the single underlying stuff of all things. Wrong answer; revolutionary question. Gate kept on. Thales wrote nothing that survives — possibly nothing at all. Everything here comes through later hands: Aristotle (two centuries on), and Proclus (a millennium on) quoting the lost history of Eudemus. So the theorems and proofs are the tradition's Thales; some historians think formal proof matured later and was credited back to the founder. He is the firm first of the reputation for deductive reasoning, the traditional first of the theorems , and a debated first of proof in the strict sense. The romance and the record don't perfectly align — and saying so is the point of this whole domain. THALES OF MILETUS · LOGIKĒ I · two-layer honest Thales' Theorem demonstrated live (∠APB = 90° for all P; proof: α+β=90° from two isosceles triangles) · eclipse of 585 BCE astronomically fixed · he left no surviving writing — attributions via Aristotle, Eudemus & Proclus next in the lineage → Pythagoras · Aristotle (the syllogism) · Euclid · the Stoics · Leibniz · Boole · into the gate LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


PYTHAGORAS & THE BROTHERHOOD

the front (c. 570 BCE) — number as reality, and the first pr

LOGIKĒ · the front (c. 570–495 BCE) — Pythagoras and the brotherhood. They held that number is the substance of reality, found the whole-number ratios behind musical harmony (octave 2:1, fifth 3:2), and — fatally for their creed — produced the first famous proof by CONTRADICTION: that √2 cannot be any ratio of whole numbers. The reductio that broke the cult is the deductive move at the heart of logic. Live: step the √2 irrationality proof and the rational approximations that never arrive, and the consonant musical ratios. Honest: Pythagoras left no writings, the theorem was known earlier, and the irrationality discovery is credited to Hippasus. David Lee Wise, two-layer honest.

Pythagoras — number, proof, and the first crisis ◄ UD0 ← Thales LOGIKĒ · the front LOGIKĒ · the front · between Thales and Aristotle Pythagoras & the Brotherhood c. 570 – 495 BCE · Samos & Croton · the Pythagorean order · "all is number" [[thales-of-miletus]] proved the first theorem; Pythagoras and his secretive brotherhood made proof a way of life — and staked everything on a creed: that number , and specifically the whole numbers and their ratios , is the hidden substance of reality. They had real evidence: pluck a string, halve it, and it sounds an octave — musical harmony is whole-number ratio. But the same devotion to proof produced the discovery that destroyed the creed: a clean reductio ad absurdum showing that the diagonal of a unit square, √2 , can be no ratio of whole numbers at all . The first famous proof by contradiction — the deductive engine of all logic — and it broke the people who found it. ✓ STRONG Real, and pivotal. The irrationality of √2 is a genuine theorem and one of the earliest proofs by contradiction; the whole-number ratios of musical consonance are real; and the cult of proof shaped Greek mathematics permanently. ◐ THE MAN IS A FOG Almost nothing is firsthand. Pythagoras left no writings; "Pythagoras" means the brotherhood as much as the man. The theorem named for him was known to Babylonians a millennium earlier; the irrationality proof is credited to Hippasus . ◔ MYSTICISM, NOT JUST MATH A religion with theorems. Number-worship, transmigration of souls, dietary taboos, secrecy. The brotherhood reportedly suppressed the irrationality result because it shattered their doctrine — proof used, then feared. I · √2 — the proof that broke the cult Their creed: every length is a ratio of whole numbers. The diagonal of a square of side 1 has length √2. Suppose it were a fraction p/q in lowest terms — and follow the logic. Step through the argument; each line forces the next, until the assumption collapses into a flat contradiction. This is reductio ad absurdum : assume, derive nonsense, conclude the assumption was impossible. next step ▸ reset ↺ No fraction works — and the rational approximations confirm it: they crowd toward √2 forever and never land on it (their squares approach 2 but never equal it). The diagonal and the side are incommensurable — they share no common unit, however small. ⚑ This is the first great proof by contradiction, the move [[chrysippus]] would formalise, [[kurt-godel]] and [[georg-cantor]] would wield, and every mathematician still lives by. "There is geometry in the humming of the strings; there is music in the spacing of the spheres." — attributed to the Pythagoreans II · Why a string sings in whole numbers The evidence for the creed is real and beautiful. Stop a vibrating string at a simple fraction of its length and the two notes are consonant — and the simpler the ratio, the sweeter the harmony. Pick a ratio; see the lengths and the interval it names. Octave 2:1, fifth 3:2, fourth 4:3 — the consonances of every musical tradition are the simplest whole-number ratios . The Pythagoreans heard, correctly, that harmony is arithmetic. ⚑ Their mistake wasn't the music — it was the leap that everything must be a clean ratio. √2 was the counterexample hiding in their own square, and finding it took the very tool — proof — they had taught the world to trust. III · The brotherhood, the theorem, the drowning NUMBER AS REASON The conviction that reality has a rational, mathematical structure accessible to proof. Wrong in its literal cult form, right as the founding faith of mathematical science. THE FIRST CONTRADICTION-PROOF √2's irrationality: assume it's a ratio, derive that both terms are even though the fraction was in lowest terms — impossible. Reductio, the spine of deduction. Gate kept on. Here the two layers must be held apart cleanly. The mathematics is real : the irrationality of √2 is a correct, profound theorem and among the first proofs by contradiction; the whole-number ratios of consonance are genuine; and the Pythagorean insistence that nature is mathematically intelligible became the working faith of all of physics. But the history is fog and legend : Pythagoras wrote nothing, so "he" is inseparable from a secretive brotherhood that attributed all its findings back to the founder; the "Pythagorean theorem" (a²+b²=c²) was used by Babylonian scribes over a thousand years before him; and the irrationality discovery is traditionally credited not to Pythagoras but to Hippasus of Metapontum — whom legend says the brotherhood drowned at sea for revealing that the cosmos contained a number their doctrine forbade. Strip the myth and a hard truth remains: a community devoted to proof followed its own method to a conclusion it could not bear, and the tool was stronger than the creed. ⚑ That is exactly why proof belongs at the front of this lineage. [[thales-of-miletus]] ← · next → Parmenides. PYTHAGORAS & THE BROTHERHOOD · LOGIKĒ · the front · two-layer honest live √2 reductio (assume p/q lowest terms → both even → contradiction; rational approximations crowd √2 but never land, verified) · the consonant whole-number ratios (octave 2:1, fifth 3:2, fourth 4:3) number as the substance of reality · musical harmony = ratio · the first famous proof by contradiction (irrationality of √2) · incommensurability ← Thales · next → Parmenides (deduction over the senses) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


PARMENIDES OF ELEA

the front (c. 515 BCE) — reason over the senses; the first r

LOGIKĒ · the front (c. 515 BCE) — Parmenides of Elea, who first trusted PURE DEDUCTION over the evidence of the senses. From a single premise — 'what is, is; what is not cannot even be thought' — he argued, step by valid step, that change, motion, and plurality are impossible and that Being is one and unchanging. The conclusion is false; the METHOD founded rationalism and gave logic the law of non-contradiction and the demand that argument, not observation, settles a question. Live: walk his deduction chain and the Way of Truth vs the Way of Opinion. Honest: a wrong conclusion from a flawed premise, but the birth of deductive argument. David Lee Wise, two-layer honest.

Parmenides — reason against the senses ◄ UD0 ← Pythagoras LOGIKĒ · the front LOGIKĒ · the front · between Thales and Aristotle Parmenides of Elea c. 515 – 450 BCE · Elea (southern Italy) · founder of the Eleatic school · the first rationalist Pythagoras trusted number; Parmenides trusted something more radical — argument itself , even against the plain evidence of the eyes. In a strange visionary poem he laid down the first sustained piece of pure deduction in the Western record: from one premise about being and non-being, he reasoned, step by valid step, to the staggering conclusion that change, motion, and plurality are all impossible — that what truly is must be one, eternal, and unchanging, and that the moving, many world your senses report is mere appearance. He was wrong . But in being willing to follow a valid argument off the cliff of common sense, he founded rationalism and handed logic its first law: contradiction is the mark of the false. ✓ STRONG (THE METHOD) The birth of deduction. Parmenides is the first to argue that reason, not observation, settles what can be — and to use the impossibility of contradiction as a tool. That method is the foundation of logic and mathematics. ◐ FALSE (THE CONCLUSION) Valid, but not sound. "Change is impossible" is plainly false — things move. The flaw is the premise (treating "nothing" as a thing one could fail to think). A perfect lesson: a valid argument can still be wrong if a premise is. ◔ MYSTIC, NOT GEOMETER Revelation in verse. His argument arrives as a goddess's speech in a poem, fused with metaphysics and theology. Reading it as clean logic is partly our reconstruction; the original is stranger and more oracular. I · The deduction — follow it off the cliff Watch reason overrule the senses. Each step is a valid inference from the one before; grant the first line and the chain drags you, link by link, to a conclusion no one's eyes would ever accept. Walk it — and then notice it's the premise , not the logic, that fails. next inference ▸ reset ↺ Every arrow is valid — the conclusion really does follow if the first line holds. So why is it false? Because the premise smuggles in a trick: it treats " nothing " as a kind of thing you could point at and fail to find. Once you deny that "what-is-not" names anything at all, the chain never starts. ⚑ That is the permanent lesson at the front of logic: an argument can be perfectly valid and still unsound — validity protects the inference, not the premises. [[aristotle]] would build the whole theory of when an argument is valid on exactly this distinction. "For it is the same thing that can be thought and that can be." — Parmenides, fragment B3 II · Two ways — Truth and Opinion THE WAY OF TRUTH What reason proves: Being is one, ungenerated, indestructible, changeless, indivisible. Reached by deduction alone, against all appearances — the path he says the goddess commands. THE WAY OF OPINION What the senses report: a world of many things, coming-to-be and passing away, in motion. Vivid, useful, and — he insists — ultimately mere seeming , not knowledge. This split — between what is demonstrated and what is merely observed — is one of the most consequential cuts in the history of thought, and it runs straight through this lineage. Mathematics took the Way of Truth: it trusts proof over measurement, and would rather believe a theorem than an eye. Parmenides drew the line in the wrong place (he threw out the whole sensible world), but he drew it, and the line is real: a proven result is certain in a way no experiment can be. His pupil Zeno would defend the Way of Truth with the sharpest weapon yet invented — paradoxes designed to show that motion itself collapses into contradiction. ⚑ Reason given authority over the senses: for better and worse, the founding gesture of the rational tradition. III · The wrong answer that built the method Gate kept on. Parmenides is the cleanest case in the whole lineage of separating the method from the verdict . The verdict is false : change, motion, and plurality are obviously real; "nothing comes from nothing, and nothing that is can cease to be" leads, by his own honest reasoning, to a frozen one-thing universe that contradicts every moment of experience — and the error is locatable, in a premise that mishandles negation and non-existence (a confusion logic would not fully untangle until [[gottlob-frege]] and [[bertrand-russell]], 2,400 years later). And the form is mystical : the argument is delivered as divine revelation in hexameter verse, so reading it as a tidy proof is partly a modern reconstruction of a stranger original. Yet the achievement is foundational and real : Parmenides is the first thinker on record to insist that reason can override the senses , to wield the impossibility of contradiction as a method, and to demand that a claim be argued rather than merely observed. Every proof in this universe — every place where we trust a deduction over a glance — descends from that nerve. He reached a false conclusion by a valid road, and in doing so built the road. ⚑ [[pythagoras]] ← · next → Zeno of Elea. PARMENIDES OF ELEA · LOGIKĒ · the front · two-layer honest live deduction-walker (one premise → no not-being → no void → no coming-to-be → no change → no plurality → Being is One; every step valid, the premise false) · the Way of Truth vs the Way of Opinion the first sustained pure deduction · reason over the senses (rationalism) · contradiction as the mark of the false · valid vs sound (a valid argument can still be wrong) ← Pythagoras · next → Zeno of Elea (the paradoxes of motion) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


ZENO OF ELEA

the front (c. 490 BCE) — the paradoxes of motion; the first

LOGIKĒ · the front (c. 490 BCE) — Zeno of Elea, who defended Parmenides with the first great PROOFS BY CONTRADICTION: paradoxes showing that motion and plurality lead to absurdity. Achilles can never catch the tortoise; an arrow in flight is, at each instant, at rest. The resolution waited two thousand years for the calculus: an infinite sum of ever-smaller steps can have a finite total (1/2 + 1/4 + 1/8 + … = 1). Live: run Achilles and the tortoise and watch the infinite series converge to the catch-up point. Honest: the paradoxes were 'dissolved' by limits, but they were deep, and supertasks are still argued. David Lee Wise, two-layer honest.

Zeno of Elea — the paradoxes of motion ◄ UD0 ← Parmenides LOGIKĒ · the front LOGIKĒ · the front · between Thales and Aristotle Zeno of Elea c. 490 – 430 BCE · Elea · pupil of Parmenides · "inventor of dialectic" (Aristotle) [[parmenides]] argued that motion and plurality are illusions; his pupil Zeno defended him with a new and devastating weapon — the paradox , a proof by contradiction aimed at common sense itself. Grant that space and time are infinitely divisible, Zeno says, and watch motion dissolve into absurdity : swift Achilles can never overtake a tortoise with a head start, because each time he reaches where it was, it has crept a little further; an arrow in flight, examined at any single instant, is motionless — so how does it ever move? These were the first arguments to take infinity seriously, and they were not truly answered for two thousand years — until the calculus showed that an unending sum of shrinking steps can add up to a finite whole. ✓ STRONG Deep, and method-defining. Zeno gave the first systematic proofs by contradiction (Aristotle calls him the inventor of dialectic), and the first serious confrontation with the infinite — problems that drove the eventual rigor of limits and the calculus. ◐ "DISSOLVED," NOT TRIVIAL Answered by limits. The convergent-series resolution (½+¼+…=1) handles Achilles and the dichotomy cleanly. But it took Cauchy and Weierstrass to make it rigorous, and the arrow and "supertasks" still feed live debate in philosophy. ◔ A DEFENCE, NOT A SYSTEM Polemics, in service of Parmenides. Zeno's paradoxes argue against motion/plurality to protect his teacher's One; they're brilliant reductios, not a positive theory. And we have them only second-hand, mostly via Aristotle. I · Achilles and the tortoise — run the infinity The tortoise gets a head start; Achilles runs ten times faster. By the time he reaches the tortoise's start, it has moved a little; by the time he reaches that spot, it has moved again — forever. Zeno concludes he never catches it. Press run and watch the stages: each is real, there are infinitely many — yet their lengths shrink so fast that their total is finite , and Achilles passes the tortoise at a definite point. run ▶ one stage ▸ reset ↺ stages run 0 distance covered 0 catch-up point → Infinitely many stages, but the distances are a geometric series — 1 + 1/10 + 1/100 + … — and it converges : it sums to 10/9 of the head start, a finite number Achilles reaches in finite time. ⚑ Zeno's error wasn't logic, it was an unspoken assumption: that infinitely many steps must take infinitely long. They needn't. Untangling exactly why took the limit concept — [[georg-cantor]]'s infinities and the rigor of the calculus — so Zeno's "absurdity" turns out to be the first deep question about the actual infinite , not a mistake. "That which is in locomotion must arrive at the half-way stage before it arrives at the goal." — Zeno's Dichotomy, via Aristotle II · The three great paradoxes THE DICHOTOMY To go anywhere you must first go halfway, but first half of that, and so on — infinitely many sub-journeys to complete before you even start. So motion can't begin. ACHILLES & THE TORTOISE The swift can never catch the slow with a head start: each time he reaches its last position, it has moved on. Infinitely many catch-ups to make. THE ARROW At any single instant a flying arrow occupies exactly its own length and is at rest. If it's at rest at every instant, when does it move? Motion vanishes into a sum of still frames. All three attack the same target: the idea that a continuous quantity (a distance, a duration) is built of infinitely many parts. The dichotomy and Achilles are answered by convergent series — infinitely many shrinking steps with a finite sum. The arrow is subtler: it forces a real account of what "motion at an instant" even means, which is precisely the derivative — instantaneous velocity, defined as a limit. Twenty-three centuries after Zeno, the calculus didn't just refute him; it gave back rigorous answers to the exact questions he was the first to ask. ⚑ A "wrong" argument so good it had to invent a branch of mathematics to be properly answered — the highest compliment a paradox can earn. III · The inventor of the paradox Gate kept on. The honest reading gives Zeno enormous credit while marking the limits. His method is foundational : Aristotle himself called Zeno the inventor of dialectic , and the paradoxes are among the first rigorous proofs by contradiction — assume motion/plurality, derive an absurdity. His confrontation with infinity was 2,000 years ahead of the tools needed to handle it. The honest qualifications: first, the paradoxes are not unsolved — for the dichotomy and Achilles, the convergent geometric series (an infinite sum of shrinking terms with a finite total) is a complete and correct resolution, made fully rigorous by Cauchy and Weierstrass in the 1800s; pop claims that "math can't really answer Zeno" are wrong. Second, they are partly alive: the arrow and the broader puzzle of supertasks (completing infinitely many actions in finite time) remain genuinely debated in philosophy of mathematics, so it's not all tidily closed. And third, like his teacher, Zeno's aim was defensive — these are weapons forged to protect [[parmenides]]' unchanging One, not a positive theory of their own, and we read them mostly through Aristotle's reports. Strip all that away and the achievement stands: the man who first made infinity a problem, and whose "impossible" arguments were so sharp they helped summon the calculus into being. ⚑ With Zeno the front of the lineage is whole; the road runs on to its first systematiser. [[parmenides]] ← · next → [[aristotle]] (the science of valid inference). ZENO OF ELEA · LOGIKĒ · the front · two-layer honest live Achilles & the tortoise (run the infinite stages; the geometric series 1 + 1/10 + 1/100 + … converges to a finite catch-up point — verified) · the dichotomy & the arrow the first systematic proofs by contradiction (dialectic) · the first serious confrontation with infinity · the paradoxes answered, two millennia later, by limits & the calculus ← Parmenides · next → Aristotle (the science of valid inference) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


ARISTOTLE · II

the first SYSTEM of valid inference — the syllogism + the th

LOGIKĒ II — Aristotle (384–322 BCE), the father of formal logic. Thales proved single truths; Aristotle built the first SYSTEM of valid inference — the syllogism — and named the three laws all reasoning obeys (identity, non-contradiction, excluded middle). A live syllogism machine: pick two premises, see whether the conclusion validly follows, and why, with an Euler diagram. His Organon ruled logic for two thousand years — until Frege showed it couldn't handle relations. David Lee Wise, two-layer honest.

Aristotle — the first system of valid inference ◄ UD0 ← Thales LOGIKĒ · II LOGIKĒ · II · the first system of inference Aristotle 384 – 322 BCE · Stagira & Athens · student of Plato, tutor of Alexander, founder of the Lyceum Thales proved truths one at a time. Aristotle did something stranger and larger: he made reasoning itself the object of study. In the Organon he set down the first system of valid inference — the syllogism — a small machine of forms that, fed true premises in the right shape, can only output truth. He named the three laws every coherent thought obeys. For two thousand years, to study logic was to study Aristotle. ✓ STRONG The syllogism & the laws. Valid syllogistic forms and the laws of identity, non-contradiction and excluded middle are foundational and correct — the bedrock of deductive reasoning for 2,000 years. ◐ LIMITED Term logic only. It reasons about categories ("all men…") but can't express relations ("every number has a successor") or nested quantifiers — a real ceiling, unseen for millennia. ◔ SUPERSEDED Thought complete — wasn't. Kant called it finished; Frege (1879) proved otherwise with predicate logic. And "Aristotelian authority" in physics actively held science back for centuries. I · The syllogism machine A syllogism has three terms — a subject S, a predicate P, and a middle M that links them — and three lines: two premises and a conclusion. The trick is that validity depends only on the shape , not the content: get the form right and the conclusion cannot be false if the premises are true. Pick the two premises below and the machine rules on whether the famous chain holds. major: All M are P No M are P Some M are P Some M are not P minor: All S are M No S are M Some S are M Some S are not M (M = men, P = mortal, S = Greeks) All men are mortal. All Greeks are men. ∴ All Greeks are mortal. VALID — Barbara The middle term "men" links Greeks to mortality; the form guarantees the conclusion. an Euler diagram of the relationship the premises force (S, M, P as nested/overlapping sets) II · The three laws of thought Beneath every syllogism sit three rules so basic they feel like nothing — until you try to think without them. Aristotle was the first to state them as laws . IDENTITY A = A A thing is what it is. Each term must hold its meaning steady through the argument. NON-CONTRADICTION ¬(A ∧ ¬A) Nothing can both be and not be, in the same respect, at the same time. The one he called the firmest of all. EXCLUDED MIDDLE A ∨ ¬A Either a statement holds or its denial does — no third option. (Later logics would dare to question this one.) "It is the mark of an educated mind to be able to entertain a thought without accepting it." — Aristotle III · The man, and the honest caveat Aristotle wrote on nearly everything — logic, physics, biology, the soul, ethics, politics, poetry — and largely founded each as a field. His logic was so complete-seeming that Kant, twenty centuries later, declared it finished and perfect. It was neither. Gate kept on. The syllogism is real and valid — but it is term logic : it speaks only of whole categories. It cannot express a relation ("Alice is taller than Bob") or stack quantifiers ("for every number there is a larger one") — the sentences mathematics actually runs on. That ceiling stood, invisible, until Gottlob Frege's predicate logic in 1879 blew the roof off. And the flip side of his genius: "the Philosopher's" authority in physics — heavier things fall faster, the heavens are perfect circles — became dogma that delayed science until Galileo. Father of logic, yes. Last word on it, no — and the lineage from here runs straight toward the man who finished what he started. ARISTOTLE · LOGIKĒ II · two-layer honest syllogism validity computed live (figure-1 moods: Barbara AAA, Celarent EAE, Darii AII, Ferio EIO valid; others fail) · the three laws of thought · Organon dominated logic ~2,000 years ← Thales · next → Euclid (the axiomatic method) · the Stoics · Leibniz · Boole · Frege · into the gate LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


EUCLID · III

the first complete deductive EDIFICE — from 10 axioms, all o

LOGIKĒ III — Euclid (fl. c. 300 BCE, Alexandria). Thales proved single truths, Aristotle systematised inference; Euclid built the first complete AXIOMATIC EDIFICE — from a handful of definitions, five postulates and five common notions, the Elements deduces ~465 theorems by pure logic. Proposition 1 (constructing an equilateral triangle) shown live, the very first proof, with its argument. Plus the fifth postulate — the parallel postulate, independent of the rest, whose denial opened non-Euclidean geometry and the shape of Einstein's universe. Honest: Euclid's axioms had gaps (Hilbert fixed them in 1899); the man is nearly unknown. David Lee Wise, two-layer honest.

Euclid — the first complete deductive system ◄ UD0 ← Aristotle LOGIKĒ · III LOGIKĒ · III · the first complete deductive system Euclid fl. c. 300 BCE · Alexandria, under Ptolemy I Thales proved a truth; Aristotle named the forms of valid argument. Euclid built the cathedral. In the Elements he laid down a handful of definitions , five postulates , and five common notions — then deduced from them, in strict order, some 465 theorems across thirteen books, each resting only on what was proved before. Nothing assumed mid-stream, nothing taken on faith but the axioms at the foundation. For more than two thousand years, "to reason rigorously" simply meant "as in Euclid." ✓ STRONG The axiomatic method. A whole science deduced from ten stated assumptions — the Elements is the foundational model of deductive rigor, the most influential textbook ever written. ◐ GAPPED Not actually airtight. Euclid quietly used things he never stated (that two circles meet, that points lie "between"). Hilbert (1899) needed ~20 more axioms to close the gaps. ◔ A CHOICE The fifth is independent. The parallel postulate can't be proved from the rest — deny it and you get consistent non-Euclidean geometries , which turned out to be the shape of Einstein's universe. I · Proposition 1 — the first proof in the book The very first thing the Elements does is build an equilateral triangle on a given segment — using nothing but the postulates "you may draw a circle" and "you may draw a line." Step through Euclid's construction and watch the reason it works fall out of the radii: the two new sides are each equal to the base because each is a radius of its circle. ▶ next step reset (drag A or B to change the segment) segment AB → circle ⊙A → circle ⊙B → intersection C → triangle ABC AB = AB the three sides Given: a segment AB. Goal: an equilateral triangle on it. Press “next step.” Postulates used: 3 (a circle of any centre and radius) and 1 (a line between two points). The conclusion — AC = BC = AB — comes from Common Notion 1 : things equal to the same thing are equal to each other. That's the whole machine. II · The ten foundations Everything in 465 theorems rests on these. Five postulates (what you may construct) and five common notions (how equals behave) — and notice the fifth postulate is conspicuously longer and less obvious than the other four. People felt that for two millennia. P1 A line joins any two points. CN1 Things equal to the same thing are equal. P2 A segment extends to a line. CN2 Add equals to equals — still equal. P3 A circle has any centre and radius. CN3 Subtract equals from equals — still equal. P4 All right angles are equal. CN4 Things that coincide are equal. P5 If a line crossing two lines makes the same-side interior angles less than two right angles, those two lines meet on that side. — the parallel postulate CN5 The whole is greater than the part. "There is no royal road to geometry." — Euclid to Ptolemy I, who wanted a shortcut III · The fifth postulate — the crack that opened a universe The fifth never looked self-evident, so for 2,000 years mathematicians tried to prove it from the other four. Every attempt failed — and in the 1800s Bolyai, Lobachevsky and Riemann saw why: it can't be proved, because it's a free choice. Deny it and geometry doesn't break; it changes. The clean test is a triangle's angles: they sum to exactly 180° only on a flat plane. Curve the world and the sum moves. curvature flat sphere saddle a triangle on a flat / spherical / saddle surface · the geometry it lives in decides its angle sum 180.0° sum of the triangle's angles Euclidean (flat) which geometry On a sphere (positive curvature) the angles sum to more than 180° — parallel lines converge and meet. On a saddle (negative) they sum to less — parallels diverge. Euclid's is the knife-edge flat case. None is "the true" geometry; the universe, Einstein found, runs on the curved ones. Gate kept on. Euclid the person is almost a blank — we have a few anecdotes from centuries later and aren't certain he was one man rather than a school. His method is immortal, but his execution wasn't airtight: he leaned on unstated assumptions (that the two circles in Proposition 1 actually intersect; that a point can be "between" two others), which David Hilbert had to formalise with ~20 extra axioms in 1899 to make Greek geometry truly rigorous. The greatest deductive system ever built — and proof that even the gold standard had hidden joints. EUCLID · LOGIKĒ III · two-layer honest Proposition 1 constructed live (⊙A ∩ ⊙B → C; AC=BC=AB by Common Notion 1) · the 5 postulates + 5 common notions · triangle angle-sum vs curvature (180° flat, >180° sphere, <180° saddle) Elements ~465 propositions, 13 books · 5th postulate independent (Bolyai, Lobachevsky, Riemann) · gaps closed by Hilbert's Grundlagen (1899) ← Aristotle · next → the Stoics (propositional logic) · Llull · Leibniz · Boole · into the gate LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


CHRYSIPPUS & THE STOICS · IV

propositional logic — the truth table & the gate, ~2000

LOGIKĒ IV — Chrysippus of Soli (c. 279–206 BCE) and the Stoics. While Aristotle reasoned about categories ('all men…'), the Stoics reasoned about whole statements joined by connectives — and, or, not, if-then — the first PROPOSITIONAL logic, the direct ancestor of Boolean algebra and the logic gate. A proposition is true or false (the bit); the connectives are the operators; Philo of Megara's conditional is the material conditional (false only when antecedent true, consequent false) — the modern truth table, ~2000 years early. Live truth tables and the five indemonstrables (modus ponens et al). Honest: almost all texts lost; reconstructed from critics; overlooked for two millennia, then vindicated. David Lee Wise, two-layer honest.

Chrysippus & the Stoics — the logic of propositions ◄ UD0 ← Euclid LOGIKĒ · IV LOGIKĒ · IV · the logic of whole statements Chrysippus & the Stoics Chrysippus of Soli · c. 279 – c. 206 BCE · "if there had been no Chrysippus, there would have been no Stoa" Aristotle reasoned about categories — all men, some Greeks, no mortals. The Stoics did something the gate would need and he could not give: they reasoned about whole statements — true-or-false propositions joined by connectives : and, or, not, if… then . A proposition is true or false — that is the bit . The connectives are the operators . And Chrysippus's rules of inference are how proofs compute . This is propositional logic, born in Athens, and it — not Aristotle's — is the logic your processor runs on. ✓ STRONG They built propositional logic. Connectives, a truth-functional conditional, and inference rules — recognised by modern logicians (Łukasiewicz, Mates, Bobzien) as the genuine ancestor of Boolean logic and the gate . ◐ RECONSTRUCTED Almost all of it is lost. Chrysippus wrote 700+ books; none survive whole . We rebuild Stoic logic from fragments and hostile critics (Sextus Empiricus, Diogenes Laertius). ◔ VINDICATED LATE Dismissed for 2,000 years. Aristotle's logic was held superior; only in the 20th century was it seen that the Stoics had the deeper, computing-relevant system first. I · The connectives — and the first truth table Set two propositions P and Q to true or false, and watch the connectives compute their value. The star is the conditional : Philo of Megara taught that "if P then Q" is false in exactly one case — P true, Q false — and true otherwise. That is the material conditional , the modern truth table, written down around 300 BCE. Every box below is a logic gate that wouldn't be built in silicon for 2,200 years. P = TRUE Q = FALSE — click to toggle The Stoics called the four cases the only thing that matters about a connective — its value in every row. They had, in all but the name, truth-functional semantics : the meaning of "and / or / if" is its table. Frege and Wittgenstein would re-derive this; Shannon would weld it to circuits. II · The five indemonstrables From those connectives Chrysippus distilled five basic, self-evident inference patterns — the indemonstrables — out of which all valid argument is built. They are the first natural-deduction rules in history, and you'll recognise them: they're how a proof, or a circuit, moves from premises to a conclusion. With the P, Q above set, the ones whose premises currently hold light up. "The first indemonstrable: if the first, then the second; but the first; therefore the second." — Sextus Empiricus, reporting Chrysippus (modus ponens) III · Lost, dismissed, vindicated Chrysippus was the second founder of Stoicism and, by reputation, the greatest logician of antiquity — Diogenes Laertius credits him with 705 books . They are all gone . What we know of the most consequential logic before Frege survives only because critics quoted it to argue against it. Gate kept on. For two thousand years the verdict was that Aristotle's term logic was real logic and the Stoic stuff was a lesser cousin — so it was neglected, and the texts perished. Only in the 20th century , when propositional logic became the spine of mathematics and then of computing, did logicians (Łukasiewicz, Benson Mates, Susanne Bobzien) realise the Stoics had got there first and got there deeper. So treat the precise truth-functional reading with care: it's a faithful but partly modern reconstruction from fragments, not a surviving Stoic textbook. The honest shape of this one is a first that was forgotten — and turned out to be the more important first. CHRYSIPPUS & THE STOICS · LOGIKĒ IV · two-layer honest live truth tables (∧ ∨ ¬ ⊻ →; material conditional false only when P∧¬Q — Philo of Megara) · the five indemonstrables (modus ponens, modus tollens, &c.) firing on the current P,Q Stoic propositional logic reconstructed from Sextus Empiricus & Diogenes Laertius · vindicated 20th c. (Łukasiewicz, Mates, Bobzien) · the ancestor of Boolean logic ← Euclid · next → Ramon Llull (the first logic machine) · Leibniz · Boole · into the gate LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


RAMON LLULL · V

the first machine of thought — rotating wheels that generate

LOGIKĒ V — Ramon Llull (c. 1232–1316, Majorca). Everyone before him reasoned by hand; Llull built the first DEVICE — the Ars Magna's rotating concentric wheels (volvelles) that mechanically generate every combination of a set of concepts. The first attempt to turn reasoning into a machine you crank, ~1300, and the direct ancestor of Leibniz's combinatorics, the calculus of reasoning, and the computer. A live volvelle plus the combinatorial enumeration of the nine principles. Honest: the purpose was theological, and the Art is a combination GENERATOR, not a prover — but the machine idea was centuries ahead and was vindicated by Leibniz. David Lee Wise, two-layer honest.

Ramon Llull — the first machine of thought ◄ UD0 ← Chrysippus LOGIKĒ · V LOGIKĒ · V · the first device for reasoning Ramon Llull c. 1232 – c. 1316 · Majorca · courtier, mystic, missionary, polymath Everyone before him reasoned with pen and mind. Llull built a machine . His Ars Magna — the "Great Art" — was a set of rotating concentric wheels inscribed with concepts; crank them and they mechanically lay out every combination of those ideas, far faster than a mind could list them. It is the first device that processes rather than merely records — a paper computer, three and a half centuries before Pascal's gears, and the seed Leibniz would grow into the dream of calculating thought itself. ✓ STRONG The first logic machine. The volvelle — rotating disks that generate combinations — is a genuine, documented first for mechanised, combinatorial reasoning , and Leibniz explicitly traced his combinatorics to it. ◐ A GENERATOR It lists, it doesn't prove. The Art produces propositions by combination; it does not validate them. And its purpose was theological — to demonstrate doctrine — more mnemonic and contemplative than rigorous. ◔ VINDICATED Right idea, early. The notion — mechanise reasoning by combination — was centuries ahead and became, through Leibniz, the spine of computing. The dream was sound even where the device was mystical. I · The wheel — crank the combinations Llull's first figure carried nine principles, the divine "dignities," lettered B through K : Goodness, Greatness, Eternity, Power, Wisdom, Will, Virtue, Truth, Glory. He set them on an inner wheel that turns inside an outer one. Spin it, and the pointer at the top reads off a new pairing of concepts — and behind it, the machine is quietly forming all nine pairs at once. Drag the wheel, or let it turn. ❚❚ turning drag the inner wheel to turn it by hand outer ring (fixed) · inner ring (turns) · the pointer reads the current pairing at the top Bonitas · Bonitas the combination at the pointer Two cheap paper disks turn nine ideas into thirty-six pairings without a single act of thought — the labour of reasoning offloaded onto a mechanism. That offloading is the idea of computing. II · The combinatorial art The point of the wheels is exhaustiveness: not to find one clever pairing, but to be sure you have them all . Nine principles taken two at a time give thirty-six distinct pairs — the grid below — and Llull's later figures stacked wheels to reach triples and beyond, the combinations exploding. This is the ars combinatoria , and it is exactly the seam Leibniz would pick up: that reasoning might be reduced to the systematic combination of basic terms. the 9×9 table of principle-pairs · the current pointer-pair glows · 36 distinct pairs (off the diagonal), the machine's full output "He invented an alphabet of human thoughts… and a way to combine them." — Leibniz, on what he took from Llull III · The man, and the honest caveat Llull was a Majorcan courtier who, after religious visions in his thirties, gave his life to a single project: a universal "Art" that could demonstrate the truths of faith to Muslims and Jews by reason alone. He learned Arabic, wrote some 250 works in Catalan, Latin and Arabic (pioneering literary Catalan), travelled the Mediterranean to debate and convert, and — by tradition — was stoned to death on a mission to Tunis around 1316. Gate kept on. Be clear about what the Art is and isn't. It is a brilliant combination generator — but it does not prove anything: spinning the wheels yields propositions, not verdicts on their truth, and its grand aim (mechanically demonstrating Christian doctrine) simply doesn't work. Much of Llullism is closer to a contemplative mnemonic, or even mysticism, than to rigorous logic, and later admirers from Bruno to Kircher pushed it toward the occult. The durable, vindicated kernel is narrow and real: the idea that reasoning could be mechanised by combining symbols on a machine. That idea, stripped of the theology, is what Leibniz inherited and what runs, eventually, in every processor. A first not of logic's content , but of its mechanisation . RAMON LLULL · LOGIKĒ V · two-layer honest a live volvelle (nine dignities B–K on rotating wheels) generating the pairings · 9 principles → 36 distinct pairs (C(9,2)) · the ars combinatoria Ars Magna (c. 1305) · the volvelle = first information-processing paper machine · directly inspired Leibniz's De Arte Combinatoria (1666) · a generator, not a prover ← Chrysippus · next → Leibniz ("let us calculate" — binary + the calculus of reasoning) · Boole · into the gate LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


WILLIAM OF OCKHAM

the medieval root (c. 1320) — Ockham's razor, suppositi

LOGIKĒ · the medieval root (c. 1320) — William of Ockham, the great logician of the Middle Ages. He developed supposition theory (how terms refer), stated what we now call De Morgan's laws centuries before De Morgan, championed nominalism, and is remembered for OCKHAM'S RAZOR: do not multiply entities beyond necessity — prefer the simpler adequate explanation. Live: the razor as the modern problem of overfitting — fit noisy data with a simple model versus a complex one that wiggles through every point. Honest: the razor is a heuristic, not a law; the famous wording is a later paraphrase. David Lee Wise, two-layer honest.

William of Ockham — the razor and the medieval logic ◄ UD0 ← Llull LOGIKĒ · the medieval root LOGIKĒ · the medieval root · c. 1320 William of Ockham c. 1287 – 1347 · Ockham, Surrey & Oxford & Munich · Franciscan friar · the greatest logician of the Middle Ages Between [[ramon-llull]]'s combinatorial wheel and [[leibniz]]'s dream of calculation, logic did not sleep — it was refined, hard, in the medieval universities, and its sharpest mind was a Franciscan friar from Surrey. Ockham built a deep theory of how words refer (supposition theory), stated the rules we now credit to [[augustus-de-morgan]] five centuries early , and argued that universals — "redness," "humanity" — are not things in the world but names in the mind (nominalism). But he is famous for a principle of method so durable it outlived the entire scholastic project: do not multiply entities beyond necessity. Given two explanations that fit the facts equally, prefer the simpler . Ockham's Razor — and it turns out to be the founding intuition of modern model selection. ✓ STRONG A real logician, a lasting principle. Ockham's supposition theory and his statement of the De Morgan rules are genuine medieval logic of high order; and the razor — prefer the simpler adequate explanation — is sound methodological guidance that science still runs on. ◐ A HEURISTIC, NOT A LAW Simpler ≠ true. The razor is a tie-breaker , not a theorem: when two theories fit equally, bet on the simpler — but reality is sometimes complicated, and the razor can cut the right answer. It guides; it does not prove. ◔ THE WORDING IS LATER He didn't quite say it. "Entities must not be multiplied beyond necessity" (entia non sunt multiplicanda…) is a 17th-century paraphrase. Ockham used the idea repeatedly, in other words; the famous Latin slogan was put in his mouth afterward. I · The razor, as the machine sees it — overfitting Ockham's razor has a precise modern face: the problem of overfitting . Here are noisy data points that really come from a simple line. Slide up the model's complexity — let it use a higher-degree curve — and watch it bend to pass through every point, error on the data dropping to zero. But look what it does between the points: it wiggles wildly, fitting the noise , not the signal. The simplest model that explains the data is the one that will be right about the next point. Razor: don't multiply curves beyond necessity. model complexity (degree) new data ↻ degree 1 fit error on the data wiggle (between points) Crank the degree and the fit error falls toward zero while the wiggle explodes — the model is memorising noise, not learning the line. ⚑ This is Ockham's razor made quantitative: it is the bias-variance tradeoff, the principle behind regularisation, the Minimum Description Length, and the "Bayesian Occam factor" that penalises complex models automatically. Seven centuries before machine learning had a name, a friar wrote down its first commandment — prefer the simpler adequate hypothesis — and it is still the line between learning and overfitting. "It is futile to do with more what can be done with fewer." — William of Ockham (Summa Logicae, paraphrased) II · The logician behind the razor SUPPOSITION THEORY & THE DE MORGAN RULES A rigorous medieval account of how a term refers in a sentence, and an explicit statement that negating "A and B" gives "not-A or not-B" — [[augustus-de-morgan]]'s laws, five centuries before De Morgan. NOMINALISM Universals like "redness" are not real things floating in a Platonic heaven — they are names , mental signs that stand for many particulars. Reality is individuals; generality lives in language. A razor applied to metaphysics itself. The razor is not a stray maxim — it is the method of a thinker who used it on everything. Ockham's nominalism is the razor turned on metaphysics: don't posit a whole realm of universal Forms when names in the mind will do the job. His logic of supposition is the razor turned on language: pin down exactly what a word stands for, and the pseudo-problems dissolve. He was a working logician of the first rank — the medieval period this lineage had skipped was not a gap in reasoning but a forge of it, and Ockham was its sharpest edge. That his rules for negation, his theory of reference, and his principle of parsimony all resurfaced — in [[augustus-de-morgan]], in modern semantics, in machine learning — is the measure of how far ahead he was working. ⚑ The friar who told the future to keep it simple. III · The razor's edge, and the heretic who held it Gate kept on. Two honesties, on the principle and the man. On the principle : the razor is real and useful but it is a heuristic, not a law — it says that when two explanations fit the evidence equally, you should bet on the simpler, because it is less likely to be fitting noise and more likely to generalise (which the overfitting demo makes precise). It does not say the simpler theory is true — reality is sometimes genuinely complicated, and a too-eager razor can shave off the correct answer; its modern formalisations (regularisation, MDL, Bayesian model comparison) make the tradeoff exact rather than mystical. And the famous Latin slogan, entia non sunt multiplicanda praeter necessitatem , is a 17th-century formulation — Ockham expressed the idea many times, but not in those words. On the man : William of Ockham was a brilliant and combative Franciscan who was summoned to Avignon on suspicion of heresy, took the Pope's side then broke with him violently over the doctrine of apostolic poverty, fled to the protection of the Holy Roman Emperor, was excommunicated, and died — probably of the Black Death — still in exile. A friar who sharpened the tools of reason, applied them without mercy to the Church's own metaphysics, and paid for it. ⚑ The medieval root of the lineage, restored. [[ramon-llull]] ← · the lineage runs on to [[leibniz]] and the modern age. WILLIAM OF OCKHAM · LOGIKĒ · the medieval root · two-layer honest live Ockham's razor as OVERFITTING (noisy points from a line; raise the polynomial degree → fit error → 0 but the curve wiggles through the noise; the simplest adequate model generalises — verified Runge oscillation grows with degree) supposition theory · De Morgan's laws stated ~1320 (five centuries early) · nominalism · Ockham's razor = parsimony = the root of modern model selection (regularisation / MDL / Bayesian Occam) ← Llull · the lineage runs on to Leibniz LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


LEIBNIZ · VI

the hinge — 'Calculemus!' · reasoning reduced to c

LOGIKĒ VI — Gottfried Wilhelm Leibniz (1646–1716). The hinge of the whole lineage: he dreamed that reasoning could be reduced to calculation — a universal symbolic language (characteristica universalis) plus a calculus of reasoning (calculus ratiocinator), so any dispute could be settled by computation: 'Calculemus!'. He formalised binary arithmetic (the language of every computer), built a four-operation calculating machine, and encoded logic as arithmetic — assigning primes to concepts so that inference becomes divisibility — 150 years before Boole. Live binary, and his prime-concept calculus. Honest: the grand dream was never finished and, by Gödel and Turing, can't be — reasoning can be mechanised, but not completely. David Lee Wise, two-layer honest.

Leibniz — "let us calculate" ◄ UD0 ← Llull LOGIKĒ · VI LOGIKĒ · VI · the hinge — reasoning as calculation Gottfried Leibniz 1646 – 1716 · Leipzig · philosopher, mathematician, co-inventor of the calculus, universal genius Llull built a wheel that combined ideas; Leibniz dreamed the whole machine. He imagined a universal symbolic language in which every concept has its sign, and a calculus of reasoning to manipulate those signs by rule — so that when two people disagreed, they would not argue but sit down, pens in hand, and say "Calculemus" — let us calculate , settling the matter like a sum. To get there he formalised binary (the alphabet every computer still speaks), built a calculating engine, and quietly encoded logic as arithmetic a century and a half before Boole. ✓ STRONG Binary, the engine, the vision. His binary arithmetic is the language of every computer; his Stepped Reckoner did all four operations; and "Calculemus!" named the goal — mechanise reasoning — that all of computing pursues. ◐ UNFINISHED A program, not a building. The universal characteristic and the calculus of reasoning were never completed; his logical algebra sat unpublished for two centuries, so Boole re-found it independently. ◔ IMPOSSIBLE IN FULL The dream has a ceiling. Gödel (1931) and Turing (1936) proved no method can decide every truth by calculation. Reasoning can be mechanised — but not completely. Half the dream came true. I · Binary — everything from 0 and 1 Leibniz worked out base-two arithmetic and was moved by it almost to mysticism: from 0 (nothing) and 1 (God), all numbers. Toggle the bits below and watch any number appear as a pattern of ons and offs — the exact representation a memory cell holds, and the only thing a logic gate can read. It would take 250 years for the machines to catch up to the notation. 0 = decimal value each box is one bit · place values 128 64 32 16 8 4 2 1 · the number is the sum of the lit places — and that is all a register ever is. II · "Calculemus" — logic as arithmetic Here is the dream made concrete, and it's Leibniz's own idea (1679): give each basic attribute a prime number , and let a concept be the product of its attributes' primes. Then a syllogism becomes a division . "All men are animals" is valid exactly when the number for animal divides the number for man — because to be a man is to already contain being an animal. Build the concept "man" from its attributes and watch every inference settle by pure arithmetic. man = man = animal × rational = 2 × 3 = 6 "All men are X?" is true precisely when X's prime divides man's number. Add an attribute to the concept and the corresponding inference flips to valid — the meaning is in the number. "The only way to rectify our reasonings is to make them as tangible as those of the mathematicians, so that we can find our error at a glance… and say to each other: Let us calculate." — Leibniz III · The man, and the honest caveat Leibniz was the last universal genius — co-inventor of the calculus (whose notation dx , ∫ we still use, won despite the bitter priority war with Newton), a working diplomat, jurist, historian and engineer, builder of the four-operation Stepped Reckoner , and the philosopher of monads and "the best of all possible worlds." He read Llull, and reached past him toward a machine that could reason. Gate kept on. The vision was breathtaking and the execution thin. The characteristica universalis and the calculus ratiocinator were a lifelong program he never finished — sketches and fragments, not a working system; his prime-number logic handled "all/some" only awkwardly and stalled on negation. His logical algebra stayed unpublished until ~1900 , so it had almost no influence and Boole rediscovered the territory on his own. And the deepest twist: the full dream — a calculation that settles every dispute — was proven impossible . Gödel showed any rich formal system has true statements it cannot prove; Turing showed no machine can decide all of them. Reasoning can be mechanised — that half came true and runs on your desk — but not completely . Leibniz pointed at the future and got the direction exactly right and the limit exactly wrong. LEIBNIZ · LOGIKĒ VI · two-layer honest live binary (0/1 bits → any number, the register) · the prime-concept calculus (man = 2·3 = 6; "all men are X" ⟺ X's prime divides 6 — Leibniz's 1679 idea, inference as divisibility, verified) · "Calculemus!" binary arithmetic (1703) · Stepped Reckoner (4 operations) · calculus notation dx, ∫ · the dream bounded by Gödel (1931) & Turing (1936) ← Llull · next → George Boole (the algebra of true and false — the gate's direct ancestor) · De Morgan · Frege LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


GEORGE BOOLE · VII

the destination — logic becomes an algebra of 0 and 1; AND i

LOGIKĒ VII — George Boole (1815–1864). Leibniz dreamed logic could be algebra; Boole did it. In The Laws of Thought (1854) he made reasoning an algebra of 0 and 1: AND is multiplication (xy), OR is addition (x+y−xy), NOT is 1−x, true is 1, false is 0 — and the single law that makes it logic rather than arithmetic is x²=x, which forces every variable to be 0 or 1. Logical inference becomes calculation. Seventy years later Shannon welded it to switches — series = AND, parallel = OR — and every digital gate became Boole's algebra in silicon. Live algebra, the defining law, and the switch. Honest: the modern clean form is post-Boole; the circuit link is Shannon's. David Lee Wise, two-layer honest.

George Boole — the algebra of true and false ◄ UD0 ← Leibniz LOGIKĒ · VII · → the gate (PSĒPHOS) LOGIKĒ · VII · the destination — logic becomes algebra George Boole 1815 – 1864 · Lincoln, England · self-taught, professor at Cork · author of The Laws of Thought Leibniz dreamed that logic could be an algebra. Boole, a shoemaker's son who taught himself mathematics, simply did it . He let 1 stand for true and 0 for false, and discovered that the laws of reasoning are the rules of an algebra: AND is multiplication , OR is addition , NOT is 1 minus . Reasoning, made calculable at last. He could not have known that seventy years on, an engineer named Shannon would notice that an electrical switch is also a 0 or a 1 — and that Boole's algebra was therefore the blueprint of every machine that would ever think. ✓ STRONG Logic IS algebra. AND=×, OR=+, NOT=1−x, true=1, false=0, with the law x²=x — this is Boolean algebra, exact and foundational, and (via Shannon) the literal basis of every digital gate . ◐ REFINED LATER His version was quirky. Boole used uninterpretable steps (x+x = 2x, even division). The clean modern algebra — where x+x = x — was tidied by Jevons, Peirce, Schröder after him. ◔ COMPLETED BY OTHERS He laid the stone. It's propositional/class logic only (no relations → Frege ); and the circuit link that made it the world's logic is Shannon's (1937), seventy years on. I · The algebra of 0 and 1 Set x and y to 1 (true) or 0 (false) and watch each logical operation compute as ordinary arithmetic — and land on exactly the right truth value. This is the whole revelation: "x and y" is literally x times y ; "not x" is literally 1 minus x. Reasoning carried out as a sum. x = 1 y = 0 — click to flip No new ideas were needed — just the audacity to treat true and false as numbers. Every gate symbol an engineer draws is one of these four arithmetic expressions, frozen into copper. II · x² = x — the law that makes it logic What separates Boole's algebra from ordinary algebra is a single strange equation: x² = x In normal numbers that holds for almost nothing. In logic it must hold for everything — and the only numbers that satisfy it are 0 and 1 . That one law is the whole confinement: it forces every quantity to be true or false, nothing in between. Its twin, x(1−x) = 0 , is Aristotle's old law of non-contradiction, now written as arithmetic. y = x (straight) and y = x² (curve) meet only at 0 and 1 — the only solutions of x²=x x · x = x ⟹ x ∈ {0, 1} x(1 − x) = 0 // non-contradiction x + x = x // (modern) OR is idempotent "Nothing can be both itself and its opposite" — the firmest law Aristotle named — turns out to be just x(1−x)=0 . Twenty-two centuries of logic, reduced to a quadratic. "…the laws whose expression it is… are the laws of the human mind." — George Boole, The Laws of Thought (1854) III · Boole → Shannon → the gate The leap that made Boole the most consequential mathematician you've half-heard-of came in 1937, when Claude Shannon realised an electrical switch is a 0 or a 1: open or closed. Wire two switches in series and current flows only if both close — that's AND , that's x·y . Wire them in parallel and it flows if either closes — that's OR . Flip the switches and watch the lamp obey Boole's arithmetic. Every chip in PSĒPHOS is this, a few billion times. A open B closed series = AND two switches + a lamp · series lights only if A AND B closed · parallel lights if A OR B · the lamp = the Boolean value Gate kept on. Boole's own 1854 system was rougher than the clean thing taught today — he let expressions pass through meaningless intermediate values ( 2x , fractions) on the way to a sensible answer, and it was Jevons, Peirce and Schröder who filed it down to the idempotent lattice ( x+x=x ) we call Boolean algebra. His algebra is also propositional — it cannot say "every" or "some" or "taller than"; that needed Frege, next in this line. And Boole himself never touched a circuit: the marriage of his algebra to electricity is Shannon's, written when Boole had been dead seventy years. He died at forty-nine — pneumonia, after walking miles through rain to lecture, then being wrapped in wet sheets by a well-meaning remedy. Seen in his lifetime as an ingenious curiosity; revealed, a century later, as the man who wrote the grammar of the machine age. GEORGE BOOLE · LOGIKĒ VII · two-layer honest live Boolean arithmetic (AND=xy, OR=x+y−xy, NOT=1−x, XOR=x+y−2xy — exact on {0,1}) · the law x²=x ⟹ x∈{0,1} · switches: series=AND, parallel=OR The Laws of Thought (1854) · refined by Jevons/Peirce/Schröder · welded to circuits by Shannon (1937) → every digital gate ← Leibniz · next → De Morgan (the laws) · Frege (quantifiers) · Shannon (the gate) — and into PSĒPHOS LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


AUGUSTUS DE MORGAN · VIII

the duality — negation swaps ∧ and ∨; why one NAND gate buil

LOGIKĒ VIII — Augustus De Morgan (1806–1871), Boole's friend and ally. He gave symbolic logic its most-used symmetry — De Morgan's laws: ¬(A∧B) = ¬A∨¬B and ¬(A∨B) = ¬A∧¬B, the duality where negation flips AND and OR. This is exactly why a single NAND (or NOR) gate is universal — you can build NOT, AND, OR, everything, from it alone — the workhorse of digital design. He also pushed logic past Aristotle's categories into the logic of RELATIONS, the crack Frege would open. Live Venn duality and NAND universality. Honest: the laws were known to medieval logicians; he named and algebraised them. David Lee Wise, two-layer honest.

Augustus De Morgan — the duality of and & or ◄ UD0 ← Boole LOGIKĒ · VIII LOGIKĒ · VIII · the duality of and & or Augustus De Morgan 1806 – 1871 · Madras & London · first professor of mathematics at UCL · Boole's friend and champion Boole gave logic an algebra; his friend De Morgan gave it the symmetry that every engineer leans on daily. His laws say that negation swaps AND and OR: to deny that both A and B hold is to assert that either not-A or not-B does. ¬(A∧B) = ¬A ∨ ¬B . It sounds like a curiosity. It is the reason a single kind of gate can build a whole computer — and the move behind half the simplifications in any circuit ever drawn. ✓ STRONG The laws are bedrock. ¬(A∧B)=¬A∨¬B and its dual are exact and universal — and via NAND/NOR universality they're the daily workhorse of every digital design. ◐ OLDER THAN HIM The name is honorific. The laws were stated by medieval logicians ( Ockham, Buridan ) and implicit in the Stoics. De Morgan named and algebraised them; he didn't first find them. ◔ A POINTER He opened a door. His logic of relations broke past Aristotle's categories — but was incomplete. Peirce and Frege built the real predicate logic he gestured at. I · The duality — negation swaps ∧ and ∨ Set A and B, and watch both laws hold no matter what: denying a conjunction is the same as asserting a disjunction of denials, and vice versa. The Venn makes it obvious — the region outside the overlap of two circles is the union of the two outsides. A = 1 B = 0 law: ¬(A∧B) universe box · circles A, B · the shaded region is both sides of the chosen law — the same set ¬(A ∧ B) = F = ¬A ∨ ¬B = F ✓ ¬(A ∨ B) = F = ¬A ∧ ¬B = F ✓ Read the first aloud: "not (both A and B)" means "not-A, or not-B." The party fails if either guest is missing. That everyday inference, made exact, is the law — and it never breaks. II · Why one gate is enough — NAND is universal Here's where the laws earn their keep. A NAND gate outputs ¬(A∧B) — "not both." Using De Morgan, that single gate can be wired to produce every other: feed it the same input twice and it's a NOT ; negate its output and it's AND ; and because A∨B = ¬(¬A∧¬B) — pure De Morgan — three NANDs make an OR . A chip is, quite literally, a sea of one gate, repeated. Flip A and B and watch the built gates agree with the real ones. (uses A,B above) NAND(A,B) = ¬(A∧B) = 1 Every box is built from NAND alone, and every value matches the genuine gate — proof that De Morgan's duality makes a single component complete. It's why fabs can optimise one cell and tile it billions of times. (NOR is universal for the same reason, by the dual law.) III · Relations, the man, and the honest caveat De Morgan also chafed at Aristotle's ceiling. Term logic could not infer "if all horses are animals, then every head of a horse is a head of an animal" — a step about a relation . De Morgan worked out an early calculus of relations to capture exactly such inferences, the very thing the gate's algebra still couldn't say. He was also a glorious eccentric: first maths professor at the new University College London, who resigned twice over matters of principle; coiner of the term "mathematical induction"; author of the witty A Budget of Paradoxes ; and fond of noting he was x years old in the year x² (he turned 43 in 1849). Gate kept on. The laws bear his name by custom, not by first discovery — William of Ockham and Jean Buridan stated them in the 1300s, and the Stoics had the idea implicitly; De Morgan's contribution was to set them inside the new symbolic algebra of logic, which is where they became a tool rather than an observation. And his relational logic, real as it was, stayed a pointer, not a system — it was Charles Peirce and then Frege who built the full logic of relations and quantifiers. De Morgan opened the door; the next man walks through it. A foundational symmetry, a generous champion of Boole, and a door-opener — honestly all three. AUGUSTUS DE MORGAN · LOGIKĒ VIII · two-layer honest De Morgan's laws verified live (¬(A∧B)=¬A∨¬B, ¬(A∨B)=¬A∧¬B for all A,B) · NAND universality (NOT/AND/OR built from NAND, matching the real gates) the laws predate him (Ockham, Buridan; Stoics implicitly) — he named & algebraised them · logic of relations → completed by Peirce & Frege ← Boole · next → Frege (quantifiers & the logic of relations — modern logic) · Shannon (the gate) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


WILLIAM STANLEY JEVONS

the Boolean cleanup (1866) — the LOGIC PIANO, the first mach

LOGIKĒ · the Boolean cleanup (1866) — William Stanley Jevons, who cleaned up Boole's algebra (the inclusive OR, x+x=x, the 'logical alphabet') and then built the LOGIC PIANO — the first machine that mechanically solved logical problems. Feed it premises and it eliminates every inconsistent combination of terms, displaying the valid conclusions on its face: a literal Boolean computer, decades before electronics. Live: a working Logic Piano you can play. Honest: the machine was real (now in Oxford); Jevons was also a founder of marginal-utility economics. David Lee Wise, two-layer honest.

William Stanley Jevons — the Logic Piano ◄ UD0 ← De Morgan LOGIKĒ · the Boolean cleanup LOGIKĒ · the Boolean cleanup · 1866 William Stanley Jevons 1835 – 1882 · Liverpool & Manchester & London · logician & economist · builder of the Logic Piano [[george-boole]] turned logic into algebra, but left it quirky — uninterpretable terms, an exclusive "or," a division that meant nothing. Jevons cleaned it up : he made "or" inclusive , enshrined the strange law x + x = x , and reduced reasoning to a mechanical method of elimination over a "logical alphabet" of every possible combination of terms. Then he did the thing no one had done: he built a machine to run it . The Logic Piano (1866) — a wooden device with a keyboard — takes in premises and, by knocking out every combination they forbid, displays the valid conclusions on its face . It is the first machine that ever solved a problem in formal logic: a Boolean computer, seventy years before the electronic one. ✓ STRONG A real machine, a real cleanup. The Logic Piano genuinely mechanised Boolean inference (it survives in Oxford's museum), and Jevons's tidying of Boole's algebra — inclusive OR, idempotency, the method of elimination — is sound and standard. ◐ A DEMONSTRATOR Toy-scale, by design. The piano handles a handful of terms — it's a proof-of-concept that logic can be machined, not a general computer. The deep step (Boole's algebra IS switching circuits) waited for [[claude-shannon]]. ◔ LOGIC WAS HIS SIDELINE Better known as an economist. Jevons co-founded the marginal revolution in economics (marginal utility) and named the "Jevons paradox." The Logic Piano is a brilliant footnote to a career mostly spent elsewhere. I · Play the Logic Piano Here is Jevons's method, working. The " logical alphabet " is every possible combination of the terms A, B, C — eight little worlds, one per key. A premise doesn't add information so much as forbid worlds: "All A are B" kills every world where something is A-but-not-B. Switch premises on and watch the contradictory keys go dark; whatever combinations survive are the consistent universe , and the valid conclusion is simply read off what's left. No cleverness, no insight — just elimination : forbid the worlds the premises rule out, and read the answer from the survivors. That is exactly what a truth-table does, and exactly what a logic gate does. ⚑ Jevons turned [[george-boole]]'s algebra into a procedure a machine can run — the first time inference left the human head and entered a device. Set "All A are B" and "All B are C," and the piano derives "All A are C" (Barbara) by itself, mechanically (verified). "I have given much attention to lessening both the manual and mental labour of [reasoning], and I have succeeded in reducing the labour to little more than that of turning keys." — W. S. Jevons II · The cleanup, and the machine that followed TIDYING BOOLE Jevons made "or" inclusive (A or B includes both), kept the idempotent law x + x = x, and recast inference as elimination over the "logical alphabet" — dropping Boole's uninterpretable arithmetic. The form of Boolean logic we teach is closer to his. THE LOGIC PIANO (1866) A keyboard-driven wooden machine that performs that elimination physically. Press the keys for your premises; the inconsistent combinations are hidden; the conclusion stands revealed. The first working logic machine. The Logic Piano sits in a precise spot in this lineage: after [[ramon-llull]]'s combinatorial wheel (which generated combinations) and [[charles-babbage]]'s calculating engines (which crunched numbers ), but before [[claude-shannon]]'s switching circuits — it is the first device to decide a question of logic . It doesn't reason about quantity; it reasons about truth and consistency , by brute elimination. In doing so Jevons proved, in oak and ivory, the thesis the whole rest of computing would confirm: that inference is a mechanical operation, and the head is not its only possible home. ⚑ Press a key, forbid a world — the gate, in its first wooden draft. III · The economist who machined logic Gate kept on. The achievement is real and first : the Logic Piano (1866) is the earliest machine to solve problems of formal logic, it physically works, and one survives in the Oxford Museum of the History of Science; Jevons's cleanup of Boole's algebra — inclusive disjunction, idempotency, the elimination method over the "logical alphabet" — is sound and shaped how Boolean logic is taught. Two honest qualifications. First, scope : the piano is a demonstrator — it handles only a few terms and a fixed routine, proving that logic can be mechanised rather than serving as a general computer; the profound identification of Boolean algebra with switching circuits — the thing that made the universal logic machine possible — is [[claude-shannon]]'s, seventy years later. Second, the man : Jevons is remembered first as an economist , a co-founder of the marginal-utility revolution and author of the "Jevons paradox" (efficiency gains can raise total consumption) — logic was a sideline of a restless polymath who drowned, swimming, at 46. But what a sideline: he took [[george-boole]]'s algebra of thought and built the first object that could think a little of it on its own. ⚑ [[augustus-de-morgan]] ← · next → John Venn (the picture of logic). WILLIAM STANLEY JEVONS · LOGIKĒ · the Boolean cleanup · two-layer honest live Logic Piano (the "logical alphabet" of 8 term-combinations; premises eliminate inconsistent worlds; the survivors are the conclusion — verified All A→B + All B→C derives All A→C, Barbara, mechanically) cleaned up Boole's algebra (inclusive OR, x+x=x, elimination) · the Logic Piano (1866) = the first machine to solve formal logic · co-founder of marginal-utility economics ← De Morgan · next → John Venn (the Venn diagram) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0