# The Map of Ideas > A public map of connected ideas. Each idea is a short note in plain words with its sources, linked to the ideas it relates to. It starts with Gödel's incompleteness theorems and the question of what music is, and grows as people add ideas and links. Every idea has its own page with a plain-text version (add .md to the address). The full text of all public ideas is in one file. ## Start here - [All ideas as a list](https://mapideas.org/i/): every idea, grouped by theme - [Full text of all ideas](https://mapideas.org/llms-full.txt): every public idea in Markdown, with links and sources - [Glossary game](https://mapideas.org/glossary/): wander Wikipedia and keep a glossary - [Sitemap](https://mapideas.org/sitemap.xml) ## Gödel - [Gödel](https://mapideas.org/i/godel.md): In 1931 Gödel showed that a system of rules can be sound and still leave true things unproven, and that it… ## Theorems - [what it says](https://mapideas.org/i/th_says.md): Take any set of rules for proving things in math, as long as the rules are clear enough that a computer could… - [numbering trick](https://mapideas.org/i/th_num.md): Gödel did not turn ordinary language into logic. He started with a small formal language (symbols like 0, +… - [primes in the trick](https://mapideas.org/i/th_primes.md): The recipe for the big number uses primes. For a row of symbols with codes a, b, c you compute 2a × 3b × 5c… - [the sentence](https://mapideas.org/i/th_sent.md): With that machinery Gödel built a sentence that says, in effect, “this sentence cannot be proven in this… - [second theorem](https://mapideas.org/i/th_second.md): The reasoning “if the system is consistent, the sentence can’t be proven” can itself be written down inside… - [the conditions](https://mapideas.org/i/th_cond.md): The theorem is conditional. It needs rules that a computer can check, enough arithmetic (addition and… - [weakest condition?](https://mapideas.org/i/th_q.md): The theorem needs rules a computer can check, enough arithmetic, and consistency. For a mind, each of the… - [seen from outside](https://mapideas.org/i/th_out.md): The system cannot prove the sentence. We can see that it is true, but only from outside, by reasoning about… ## Proof & truth - [what true means](https://mapideas.org/i/tr_true.md): True means true of the actual natural numbers, 0, 1, 2, 3 and so on. A statement like “there are infinitely… - [three words](https://mapideas.org/i/w_three.md): Provable is about one statement: the system can derive it from its rules. True is about one statement: it is… - [the judge](https://mapideas.org/i/w_judge.md): Provable is “the judge ruled on it”. True is “the ruling matches what actually happened”. Consistent is “the… - [complete, incomplete](https://mapideas.org/i/w_comp.md): Complete means every statement you can write in the system’s language gets settled: either the statement or… - [free certainty](https://mapideas.org/i/w_free.md): An inconsistent system proves everything, true and false alike, including “I am consistent”. So a proof of… - [trust](https://mapideas.org/i/w_trust.md): A system that has run for a long time without a contradiction has good reason to expect it is consistent.… - [proof or trust?](https://mapideas.org/i/w_q.md): Mathematicians trust arithmetic because nobody has found a contradiction in centuries. That is evidence of a… - [Tarski](https://mapideas.org/i/tr_tarski.md): Inside the machine you can build “provable”, but you cannot build “true”. Truth always has to be judged from… - [outside is trust](https://mapideas.org/i/tr_trust.md): What we actually do when we say the Gödel sentence is true is assume the system is consistent. That… ## Systems - [what a system is](https://mapideas.org/i/s_sys.md): A system is a rulebook with two parts: a starting list of statements accepted without proof (axioms, like… - [primes: provable](https://mapideas.org/i/s_primes.md): The system proves that there are infinitely many primes, since Euclid’s argument can be written inside it.… - [Skolem, only ×](https://mapideas.org/i/s_skolem.md): This is the theory of whole numbers with multiplication only, no addition. Every number is a unique bundle of… - [why it is safe](https://mapideas.org/i/s_safe.md): Gödel’s theorem has a condition: the system must be strong enough to talk about itself. To build the… - [the threshold](https://mapideas.org/i/s_thr.md): Put addition and multiplication together and you are over it. Peano arithmetic and set theory both are. Above… - [Goldbach](https://mapideas.org/i/s_gold.md): The trouble returns where primes meet addition. Goldbach’s conjecture (every even number is a sum of two… - [programs](https://mapideas.org/i/s_prog.md): Any programming language powerful enough to run arbitrary code can write programs that print their own source… - [fixed means checkable](https://mapideas.org/i/r_fixed.md): “Fixed” means a computer could check it: at any moment there is a clear list of rules and a mechanical way to… - [a rule for rules](https://mapideas.org/i/r_rule.md): If the system changes its rules by a fixed procedure, like “whenever you find a Gödel sentence, add it as a… - [random changes](https://mapideas.org/i/r_rand.md): Once the coin has been flipped and the rule added, that snapshot is just another fixed system. A coin also… - [unchecked system](https://mapideas.org/i/r_none.md): Then Gödel stops applying, because the theorem needs a system whose proofs a computer can check. Such a… - [the triangle](https://mapideas.org/i/r_tri.md): There is one special system that is complete and fully true: take every true statement about numbers as an… - [what escape takes](https://mapideas.org/i/r_escape.md): Two things: the rule-changing cannot be mechanical, and it has to be connected to truth somehow. A person… - [checkable or not](https://mapideas.org/i/c_check.md): If the mechanics are checkable (physics is computable and the brain follows fixed rules), the theorem has… ## Self-reference - [the ladder](https://mapideas.org/i/tr_ladder.md): Each outside is just a stronger system, and that system has its own blind spot, which needs an even stronger… - [three views](https://mapideas.org/i/tr_views.md): Platonism: the top is the actual numbers, which exist apart from any rulebook. The cost is explaining how… - [closed loops](https://mapideas.org/i/tr_loops.md): If the top level loops back to the bottom, the system effectively vouches for its own consistency, and the… - [any top?](https://mapideas.org/i/tr_q1.md): Each outside is a stronger system. Platonism says the top is the numbers themselves. The ladder view says… - [which cost?](https://mapideas.org/i/tr_q2.md): A closed loop gives up consistency (dialetheism), total truth (Kripke’s holes) or the ladder itself… - [Hofstadter](https://mapideas.org/i/c_hof.md): Hofstadter (Gödel, Escher, Bach and I Am a Strange Loop) says self-reference is where a self comes from, so… - [knows its hole](https://mapideas.org/i/c_hole.md): A strong enough system can prove “if I’m consistent, there is a true statement I can’t prove”. So it can know… - [mind contradicts](https://mapideas.org/i/c_contra.md): People contradict themselves all the time, which looks like a reason to say a mind is not a consistent… ## Mind - [Lucas, Penrose](https://mapideas.org/i/c_lucas.md): J. R. Lucas (1961) made the original argument, and Penrose (1989, 1994) sharpened it: we can see that the… - [the replies](https://mapideas.org/i/c_replies.md): Benacerraf and Putnam made the point that we only see the sentence is true if we assume the system is… - [Gödel’s view](https://mapideas.org/i/c_disj.md): He gave a disjunction: either the mind outruns any machine, or there are math problems that are absolutely… - [non-computable mind?](https://mapideas.org/i/r_q.md): A mind that depends on something non-computable would escape the theorem. So far there is no evidence either… - [is it arithmetic?](https://mapideas.org/i/c_q1.md): The theorem needs a system with enough arithmetic. People do arithmetic, which makes it plausible, and it has… - [a test for claims](https://mapideas.org/i/c_test.md): Can the person say what the theorem assumes, and show that their system meets it? Consciousness is a field… - [theorist inside?](https://mapideas.org/i/c_q2.md): The observer who builds a theory is inside the loop too. A model that includes that observer does not exist… ## Physics & reality - [quantum bridge](https://mapideas.org/i/c_quantum.md): The uncertainty principle is a physical limit: a particle does not have a sharp position and momentum at the… - [reaches reality?](https://mapideas.org/i/c_reality.md): Gödel’s theorem is conditional: if a system has fixed rules and can do arithmetic, then it has true… - [gluing obstruction](https://mapideas.org/i/c_glue.md): Local pieces that each make sense but cannot be stitched into one global picture. It is a cousin of Gödel’s… - [gluing for self?](https://mapideas.org/i/c_q3.md): Local pieces that cannot be glued into a global picture are mathematically precise. Whether they describe how… - [a threshold in nature?](https://mapideas.org/i/s_q.md): Gödel needs addition and multiplication together. Whether physical or biological systems have an equivalent… - [numbers out there?](https://mapideas.org/i/tr_q3.md): Platonists say numbers exist apart from any rulebook, and Gödel believed this. Formalists say math is only… ## Trust & doubt - [the asymmetry](https://mapideas.org/i/h_asym.md): A broken system can be certain of itself for free, while a healthy one can only trust. So from inside, total… - [Löb’s theorem](https://mapideas.org/i/h_lob.md): Take a system T that is strong enough to talk about itself, and read its theorems as its beliefs. Löb (1955)… - [escape routes](https://mapideas.org/i/h_escape.md): Partial trust: you trust yourself for restricted classes of statements. Trust the level below: Feferman’s… - [the AI version](https://mapideas.org/i/h_ai.md): An AI that wants to approve a successor as strong as itself runs into Löb. People call it the Löbian obstacle… - [what stays open](https://mapideas.org/i/h_open.md): To make it rigorous for minds or governments, someone has to write down a formal model: an agent is a theory… - [who said it](https://mapideas.org/i/h_said.md): Jacob Bronowski, in The Ascent of Man (the episode “Knowledge or Certainty”), uses Gödel and Heisenberg to… - [the caution](https://mapideas.org/i/h_caution.md): Torkel Franzén, and Sokal and Bricmont (Fashionable Nonsense), point out that Gödel gets borrowed for… - [what is a belief?](https://mapideas.org/i/h_q1.md): In Löb’s setting an agent is a theory and its beliefs are its theorems. Real minds may fit that badly. - [calibrated doubt?](https://mapideas.org/i/h_q2.md): A probabilistic version of self-trust seems to survive if it allows a margin of error. How far that reaches… ## Society - [broken signature](https://mapideas.org/i/p_sig.md): A system that declares itself complete and certain, like an ideology with an answer for everything or a power… - [outside built in](https://mapideas.org/i/p_out.md): Healthy systems build the outside in: courts, a free press, elections, an opposition, ways to amend the… - [Constitution](https://mapideas.org/i/p_const.md): Gödel reportedly found a loophole in the US Constitution that could allow a dictatorship. It fits the… - [argument, no theorem](https://mapideas.org/i/p_arg.md): The political version is different from the math one, because you can argue it and cannot prove it. A… - [designed or emergent?](https://mapideas.org/i/p_q.md): Courts and a free press were designed. Whether doubt can grow inside a system without anyone building it is… ## Reading - [Franzén](https://mapideas.org/i/b_fran.md): The sober guide to what the theorem can and cannot be used for. Start here. - [Hofstadter](https://mapideas.org/i/b_hof.md): The MIU puzzle, tangled hierarchies, and the self as a loop. - [Lucas, Penrose](https://mapideas.org/i/b_lp.md): The argument that minds outrun machines: “Minds, Machines and Gödel”, The Emperor’s New Mind, Shadows of the… - [the replies](https://mapideas.org/i/b_rep.md): The replies, including the point that we only see the sentence is true by assuming consistency. - [Löb, Feferman](https://mapideas.org/i/b_lob.md): Self-trust in provability, and the ladder of stronger and stronger systems. - [Cubitt et al.](https://mapideas.org/i/b_cub.md): Undecidability of the spectral gap, the bridge from Gödel-style limits to quantum physics. - [Bronowski, Popper](https://mapideas.org/i/b_bro.md): The humility-against-certainty theme, from knowledge to politics. Bronowski’s episode is called “Knowledge or… - [Sokal, Bricmont](https://mapideas.org/i/b_sok.md): The warning about borrowing theorems from one field to decorate another. verify details before citing. ## Physics of sound - [Vibration and pitch](https://mapideas.org/i/mu_vibration.md): Every sound starts with something moving back and forth: a violin string, a clarinet reed, the air inside a… - [Sound in the air](https://mapideas.org/i/mu_wave.md): A vibrating string pushes the air next to it, squeezing it and letting it go. These small changes of pressure… - [Strings and pipes](https://mapideas.org/i/mu_string.md): A string is fixed at both ends, so the ends cannot move. The only vibrations that last are the ones that fit… - [The harmonic series](https://mapideas.org/i/mu_harmonics.md): When you play one note on a violin, the string vibrates as a whole and, at the same time, in halves, thirds… - [Timbre](https://mapideas.org/i/mu_timbre.md): Timbre is the colour of a sound. It is why a violin and a flute playing the same A, equally loud, still sound… - [Resonance and formants](https://mapideas.org/i/mu_resonance.md): Every object and every hollow space has frequencies at which it vibrates most easily. These are its… - [Rooms and reverberation](https://mapideas.org/i/mu_rooms.md): A room is part of every performance. The sound reaches you first directly, and then many more times as it… ## Mathematics - [Ratios](https://mapideas.org/i/mu_ratio.md): If you halve a string, it sounds exactly an octave higher. Its frequency has doubled, so the two notes are in… - [Logarithms and cents](https://mapideas.org/i/mu_log.md): Since intervals combine by multiplying, it is hard to compare their sizes directly. Logarithms help, because… - [Fourier's idea](https://mapideas.org/i/mu_fourier.md): In the early 1800s Joseph Fourier showed that any wave that repeats, however complicated its shape, can be… - [Primes and commas](https://mapideas.org/i/mu_primes.md): Every whole number can be written as a product of prime numbers in only one way: 12 is 2 × 2 × 3 and nothing… - [Continued fractions](https://mapideas.org/i/mu_contfrac.md): A continued fraction is a method to find the simplest fractions that come closest to a given number. In… - [Clock arithmetic](https://mapideas.org/i/mu_clock.md): Number the twelve notes of the octave like the hours of a clock, with C at twelve o'clock. After B comes C… ## The ear - [The cochlea](https://mapideas.org/i/mu_cochlea.md): The cochlea is a small coiled tube in the inner ear, about the size of a pea. Inside it runs a membrane that… - [Beats](https://mapideas.org/i/mu_beats.md): When two notes are very close in pitch, for example two violin strings that are almost in tune, their… - [Roughness and consonance](https://mapideas.org/i/mu_roughness.md): When two notes are a little further apart, the beats become too fast to count, and the sound turns grainy or… - [The missing fundamental](https://mapideas.org/i/mu_missing.md): Play a note through a filter that removes its lowest harmonic and keeps harmonics 2 to 6. You might expect… ## Tuning & harmony - [The octave](https://mapideas.org/i/mu_octave.md): Two notes an octave apart, with frequencies in the ratio 2:1, sound so similar that most musical traditions… - [The fifth](https://mapideas.org/i/mu_fifth.md): The fifth, ratio 3:2, is the most consonant interval after the octave. Harmonic 3 of the lower note has the… - [The Pythagorean comma](https://mapideas.org/i/mu_comma.md): Start on a note and go up by twelve pure fifths. You pass through all twelve notes of the piano and arrive… - [Temperament](https://mapideas.org/i/mu_temper.md): A temperament is a decision about where to put the comma. Pythagorean tuning keeps eleven fifths pure and… - [Twelve notes](https://mapideas.org/i/mu_twelve.md): Why does the piano have twelve notes in each octave? If you build notes by going up in fifths, after twelve… - [Pentatonic and diatonic](https://mapideas.org/i/mu_scales.md): If you stop the chain of fifths after five notes, you get the pentatonic scale, the five notes of the black… - [Modes, major and minor](https://mapideas.org/i/mu_modes.md): The seven white keys can give seven different modes, depending on which note feels like home. Starting and… - [Chords and voice leading](https://mapideas.org/i/mu_chords.md): A major chord is already present inside a single note: harmonics 4, 5 and 6 of a low C are C, E and G. When… - [Rhythm and polyrhythm](https://mapideas.org/i/mu_rhythm.md): Rhythm and pitch are closer than they seem. A drum played four times a second is a rhythm. Speed it up to a… - [Notation](https://mapideas.org/i/mu_notation.md): Every way of writing music decides what to keep and what to leave to the performer. The Western staff is a… ## Traditions & arts - [Gamelan and ombak](https://mapideas.org/i/mu_gamelan.md): Gamelan orchestras from Java and Bali are made mostly of bronze metallophones and gongs. Bronze bars do not… - [Rāga and śruti](https://mapideas.org/i/mu_raga.md): A rāga is much more than a scale. It also says which notes are used going up and which going down, which… - [Maqām and makam](https://mapideas.org/i/mu_maqam.md): Arab maqām and Turkish makam are systems of melodic modes, each with its own notes, typical phrases and path… - [Overtone and undertone singing](https://mapideas.org/i/mu_overtone.md): In overtone singing, a singer holds one low note and shapes the mouth and tongue so that one harmonic becomes… - [West African timelines](https://mapideas.org/i/mu_africa.md): In Ewe music from Ghana and Togo, and in many other West African traditions, a bell plays a repeating pattern… - [Andean siku](https://mapideas.org/i/mu_andes.md): The siku is the Andean panpipe. It is divided into two rows of pipes, called ira and arka, and each row is… - [Harmony of the spheres](https://mapideas.org/i/mu_kepler.md): The idea that the sky is a kind of music goes back to the Pythagoreans in ancient Greece. In 1619 Johannes… ## Music - [Music](https://mapideas.org/i/music.md): A way into music through many doors: the physics of sound, the mathematics of ratios, the ear, tuning and…