Strange Loops, or Tangled Hierarchies

Yesterday closed on a ladder rather than a conclusion. A system leaves one sentence undecided; a reasoner standing outside assumes the system is consistent and reads the sentence's truth off in a single step; writing that assumption down produces a larger system with an undecided sentence of its own. The climbing works every time and finishes never. What the last chapter did not supply is a name for the shape of that arrangement, or any way of recognising the same shape somewhere that has nothing to do with arithmetic.

That is today's job, and it is a vocabulary lesson before it is anything else. Chapter XX introduces two terms — Tangled Hierarchy and Strange Loop — and one claim about what every such arrangement needs underneath it. Armed with those, the Gödel construction from Day 03 stops being a clever one-off and becomes an instance of something, and two Escher prints stop being illustrations and become the same object drawn in ink.

Today's boundary is worth stating plainly, because this is the chapter of the book that is most often quoted and least often read carefully. Inside: what a hierarchy is, what it takes to tangle one, three worked cases of the tangling, where G sits among them, and the chapter's claim that below every tangled hierarchy lies a level the tangle cannot reach. Also inside, in its own section: the proposal about minds that the chapter closes on, presented as the proposal it is. Outside: any new mathematics — nothing today needs a construction the last four days did not already supply — and any suggestion that the resemblance between these cases is itself an argument.

The first two boxes are what you already own. The three on the right are today: a shape, its instances, and the thing underneath it that the whole chapter turns on.

Before anything can be called tangled, the untangled case has to be precise enough to be violated, and "levels" has been doing loose work in this course for four days. A hierarchy, in the sense used here, is an arrangement of levels in which each level is described, controlled, or contained by the level above it, and in which the description only ever runs one way. The rules of a game are above its moves: rules decide moves, moves do not touch rules. A programming language is above the program written in it. A sentence about a sentence is above the sentence it is about.

A level-crossing is a single link that runs the wrong way — from a lower level up to the level that was supposed to be above it. Add one to an otherwise ordinary hierarchy and you get a Tangled Hierarchy: an arrangement that still looks like levels everywhere you inspect it locally, but in which following the "is below" relation far enough returns you to where you began. That return trip is the Strange Loop: moving steadily in what feels like one direction through a hierarchy, and arriving back at the starting level.

Three things about that definition deserve emphasis, because each one is a place readers add something Hofstadter did not.

• The loop is a property of the whole, not of any step. Every individual move in a strange loop is an ordinary downward move. Nothing anywhere is paradoxical when examined by itself; the strangeness lives entirely in the closing.
• One edge is enough. The tangling does not require rebuilding the hierarchy. Everything below the loop stays exactly as it was, which is why the shape can appear inside systems that were carefully designed to prevent it.
• Tangled is not the same as contradictory. A tangled hierarchy can be perfectly consistent. Two of today's three cases are.

Both diagrams contain the same three levels and the same two downward edges. The right-hand one has one addition, and that addition is the entire subject of the chapter.

The clearest instance is one you can look at rather than reason about, which is why the chapter opens with pictures. In Escher's 1956 lithograph Print Gallery, a young man stands in a gallery looking at a print. The print shows a harbour town. The town contains a building. The building contains a gallery. The gallery contains a young man looking at a print — and he is the one the picture started with.

Every step in that descent is unremarkable. A picture may contain a town; a town may contain a building; a building may house a gallery; a gallery may hold a man looking at a print. Nothing at any level is impossible, and no single step is the trick. The trick is that the descent closes.

Escher himself is nowhere in that diagram, and that absence is not an oversight. He draws every level and no level draws him. Hold on to that; it becomes the chapter's central claim two sections from now.

Protocol. Start with the innermost level set to land on nothing and step the depth control from three levels to six, watching what the figure reports: an ordinary infinite regress, deeper at every setting, with no loop and nothing strange. Now set the innermost level to land on level 1 and read the loop length. Step the depth control again and record how the loop length moves with it. Then land the innermost level on level 2 and on level 3 in turn, and note that the loop shortens while the levels below it are untouched — evidence for the claim that the tangling is local. Finally, find the one edge the figure draws struck through, and say in your own words why it must be.

Limits. The figure shows a nest with a closing edge, not the lithograph. Escher could not literally draw the print at full size inside itself, and he left a blank patch at the centre of the real print where the construction runs out; nothing about that patch is drawn here. The levels are also discrete and countable, which the actual picture is not — it deforms continuously, and the levels shown are a reader's convenience. Whether the innermost level closes is a control here and was a decision Escher made; the figure shows the consequence of the decision, not a fact about pictures.

If the shape were only about pictures it would be a curiosity, so the chapter's real move is to show the same construction in three settings that appear to share nothing. Each is built the same way: take a hierarchy of two levels that were meant to stay apart, add exactly one device that lets the lower level reach the upper one, and the loop closes.

Two of the three rows above are pictures and sentences, and the third is a theorem — so this is the point at which the chapter either earns its argument or overreaches, and it is worth being slow. In Chapter XIV the coding was a device for a proof; here it is re-described as a level-crossing edge, and nothing about the mathematics changes when it is re-described.

The hierarchy was this. Below: arithmetic, the subject matter — statements about whole numbers, which is all a string of the system is ever officially about. Above: metamathematics, the talk about the system — which strings are axioms, which sequences of strings are derivations, which sentences are theorems. That upper level was supposed to be outside, spoken by mathematicians about the system rather than by the system about itself.

The coding is the edge. Once every string has a number, a claim about numbers can be read as a claim about strings, and the lower level reaches the upper one. G is the sentence that lives on both: an ordinary arithmetical claim about one specific number, and — read through the coding — the claim that a particular string has no derivation. The number and the string are the same object seen twice.

Two edges run between the same pair of boxes, in opposite directions. The downward one was designed in; the upward one arrived with the coding, and was not designed at all — which is the sense in which the tangle was discovered rather than built.

What must not be smuggled in here is that the resemblance does the work. G's undecidability is a theorem, proved by construction, every step checkable; the Epimenides sentence proves nothing, because it says nothing that can be evaluated, and Drawing Hands proves nothing, because it is a drawing of an impossible arrangement rather than an impossible drawing. The shape is a way of seeing three things at once. It is not evidence about any of them.

Grant that the shape exists and one question becomes urgent: if a system can reach up and modify the level above it, what stops the whole arrangement from dissolving into a system with no fixed rules at all? The chapter's answer is its most quotable line and its most load-bearing claim — below every tangled hierarchy lies an inviolate level, a level that the tangle does not reach and cannot modify, and on which the entire loop rests.

The three cases each have one, and naming them side by side makes the pattern hard to miss.

The last movement of the chapter leaves mathematics, and the honest way to read it is as a proposal with a shape rather than as a result. Hofstadter's suggestion is that a brain is a tangled hierarchy of exactly this kind. At the bottom is the neural substrate: cells and signals, obeying physics, modifiable by nothing they represent. Above it is a level of symbols — patterns that stand for things, including patterns that stand for the person having them. His term for that last one is the self-symbol: a symbol in the system that represents the system itself.

The claim is that this arrangement is a strange loop. The symbol level is entirely made of the substrate; the substrate's behaviour is describable in terms of what the symbols mean; and the loop closes when a symbol refers to the very system whose activity constitutes it. On this reading, the sense of an "I" is what a strange loop feels like from inside, and the neural substrate is the inviolate level that makes the tangle possible without making it incoherent.

Two things should be said about this immediately, and the chapter itself says the first. It is a proposal, not a demonstration: nothing in Gödel's theorem implies it, and the mathematical results of the last four days neither support nor refute it. The second is a caution about direction of travel. Because the arithmetic is proved and the proposal is not, the proved part tends to lend its authority to the unproved part when the two are read as one continuous argument. They are not one argument. Hofstadter developed the proposal at book length in I Am a Strange Loop in 2007, which is the right place to evaluate it, and it is evaluated there as philosophy of mind rather than as a corollary of anything proved here.

This is the chapter of the book that travels furthest from its source, and nearly every misuse comes from treating a shared shape as a shared result. The table separates the two.

The course closes here, and what it leaves you with is a small, portable test rather than a body of facts. Given any arrangement described as levels, you can ask three questions in order: does anything run upward, what single device makes that possible, and what is the level the upward edge never touches. You can run that test on Gödel's construction and get the coding, the numbers and the rules of inference. You can run it on a picture and get the pencil and the artist. You can run it on a legal system and get the amendment procedure and Hart's rule of recognition. And you can say, each time, whether the loop you have found is a theorem, a consistent curiosity, or a sentence that fails to say anything.

You also own the boundary. Across five days this course has built a formal system, coded it into arithmetic, watched a sentence about itself appear, watched the repair fail upward forever, and finally named the shape all of that has. The one thing it has not done — and the thing the last section is careful about — is let the proved part underwrite the unproved part. The book itself goes further, into self-reproduction, artificial intelligence and the self, and it goes there openly as speculation. That is where to read next, and it is a better read once the arithmetic underneath it is not being taken on trust.

• [ ] Can you define a tangled hierarchy without using the word "paradox"?
• [ ] Can you name the single edge that tangles Gödel's construction, and say when it entered?
• [ ] Can you say why a strange loop can be perfectly consistent?
• [ ] Can you state the inviolate level for each of the three cases in this chapter?
• [ ] Can you explain what breaks when the last fixed rule level is put in play?
• [ ] Can you say which part of this chapter is proved and which part is proposed?
• [ ] Can you explain why Escher's print, unlike the Epimenides sentence, is not paradoxical?
• [ ] Can you name the assumption underneath "every tangled hierarchy has an inviolate level"?

The chapter is short and the secondary literature is split cleanly between the mathematics and the philosophy, which is itself a useful signal about where the argument changes character.

• Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, Chapter XX, Basic Books, 1979.
• Douglas R. Hofstadter, I Am a Strange Loop, Basic Books, 2007, for the proposal about the self developed at length.
• Douglas R. Hofstadter and Daniel C. Dennett, editors, The Mind's I, Basic Books, 1981, for the same questions posed as a collection of arguments rather than one.
• Bart de Smit and Hendrik W. Lenstra Jr., "The Mathematical Structure of Escher's Print Gallery", Notices of the American Mathematical Society 50(4), April 2003, for the reconstruction of the blank centre.
• H. L. A. Hart, The Concept of Law, Oxford University Press, 1961, for the rule of recognition — the same structural requirement, reached independently.
• Torkel Franzén, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse, A K Peters, 2005, for the misuses this chapter attracts.
• Thomas Bolander, "Self-Reference", Stanford Encyclopedia of Philosophy.

The chapter contributes a vocabulary and one claim, and keeping the two apart is most of the work of reading it well.

• A hierarchy is levels whose description runs one way; a level-crossing is one edge that runs the other way; a tangled hierarchy is what you get after adding one.
• A strange loop is the return trip: moving consistently in one direction through the levels and arriving back where you started.
• One edge is enough, and everything below the loop is unchanged — which is why the shape appears inside systems built to exclude it.
• Tangled is not contradictory: Drawing Hands and Gödel's construction are both consistent, and only the Epimenides sentence fails to say anything.
• Gödel's coding is exactly such an edge, and re-describing it as one changes nothing about the proof from Day 03.
• The chapter claims that below every tangled hierarchy lies an inviolate level the tangle cannot modify — the language, the artist's hand, the numbers and the rules of inference.
• That claim is an argument rather than a theorem, and the parallel with Hart's rule of recognition is a coincidence of structure, not evidence for it.
• The closing proposal — that a brain is such a hierarchy and the self-symbol closes its loop — is philosophy of mind, and the proved mathematics of the earlier chapters neither supports nor refutes it.