Two worlds that both obey the rules

For two thousand years the parallel postulate was suspected of being a theorem. The suspicion was reasonable, the effort was enormous, and it was settled by drawing a picture of somewhere it is false, which is an existence proof of the cheapest kind — one object, exhibited. The same two thousand years produced three other Greek questions about what a construction can reach, and all four were closed in the nineteenth century by the same change of question.

Euclid’s fifth postulate says, in the form Playfair gave it, that through a point not on a given line there is exactly one line parallel to it. In the picture there are many. Everything else Euclid asked for still holds.

What the question was #

Euclid’s Elements opens with five postulates. Four are short and obvious-sounding: a line can be drawn between any two points, a segment can be extended, a circle can be drawn with any centre and radius, all right angles are equal.

The fifth is not like them. In Euclid’s own phrasing it runs to a paragraph and describes two lines cut by a third, converging on the side where the interior angles add to less than two right angles. It reads like a theorem that has been given up on, and Euclid himself seems to have thought so — he avoids using it for the first twenty-eight propositions.

So the programme was obvious: derive the fifth from the other four. Ptolemy tried, and Proclus, and Ibn al-Haytham, and Omar Khayyam, and Saccheri, and Lambert, and Legendre. Every one produced a proof, and every proof turned out to assume something equivalent to what it was proving — that similar triangles of different sizes exist, that three non-collinear points lie on a circle, that a rectangle exists. Saccheri, in 1733, derived a long list of consequences from its denial hoping to reach an absurdity, found none, and concluded that the results were “repugnant to the nature of a straight line”, which was the only thing left to say.

Saccheri had, without knowing it, developed a large part of hyperbolic geometry.

What settles it #

The reason two thousand years of effort produced nothing is that there was nothing to find, and that is what has to be established. An absence of proofs is not an impossibility of proof.

The argument that works has a shape worth stating in general, because it is one of the two or three most useful moves in this field.

A statement $S$ is independent of a set of axioms $A$ when there is a structure in which all of $A$ holds and $S$ holds, and another in which all of $A$ holds and $S$ fails.

That settles it, and the reason is a single observation: a proof is true in every structure where its assumptions are. If $S$ could be derived from $A$, then every structure satisfying $A$ would satisfy $S$. Exhibit one that does not and the derivation cannot exist — not “has not been found”, cannot.

Nothing about this requires knowing anything about proofs. It does not examine derivations, or bound their length, or search. It changes the subject from what can be written down to what is true somewhere, and that is a subject where a single example is conclusive.

The disc, and what its words mean #

The structure in the figure is the Poincaré disc, and the whole of it is a decision about what four words mean.

Point means a point strictly inside the unit disc. The rim is not included.

Line means an arc of a circle that meets the rim at right angles — or a diameter, which is the limiting case.

Between and congruent mean what they mean under a distance that blows up towards the rim, so that the arcs have infinite length and the rim is infinitely far away.

Under those readings, Euclid’s first four postulates all hold. Two points determine exactly one such arc; the arcs extend indefinitely because they never reach the rim; circles of any centre and radius exist; all right angles are equal, because angles in this model are the ordinary Euclidean angles between the arcs.

And the fifth fails, visibly and by a lot. Not by one exceptional line but by infinitely many, since any arc through $P$ that ends on the rim between the base line’s two endpoints misses it.

Putting the two side by side is the whole argument in two pictures. Same four postulates, same question, different answers — so the answer is not a consequence of the four.

Why the arcs are checked and not merely drawn #

There is a specific way a figure like this can be a lie, and it is worth saying what the drawing does about it.

An arc that is nearly orthogonal to the rim looks exactly like one that is orthogonal. A pair of arcs that miss by a hair looks exactly like a pair that meet. A reader cannot tell, and neither can any check that asks whether the labels are legible or whether the ink stays inside the frame.

So the figures do the arithmetic. A circle centred at $c$ with radius $ρ$ is orthogonal to the unit circle exactly when $|c|^2 = 1 + ρ^2$, and every arc drawn asserts that identity to within $10^{-8}$ before it is emitted. Every arc claimed to be parallel has its intersections with the base line computed — a circle-circle intersection, solved exactly — and the figure refuses to draw unless the intersection set inside the disc is empty. The arcs that do meet are drawn differently and their crossings are marked, so the picture states which claim it is making about each one.

That is what makes this a figure rather than an illustration. It is checking the property the essay is about, on the instance it draws, every time it is drawn.

What a model is, said generally #

The disc is a particular thing and the notion behind it is not, so it is worth stating the general version once. A model is small enough to write out in some cases and not in others; where it is, checking every axiom by exhaustion is available and there is nothing left to argue about.

A set of axioms is a set of sentences in some vocabulary — words like point, line, lies on. The sentences do not mean anything by themselves; they are strings with a grammar. A model supplies the meanings: a collection of things to be the points, a collection to be the lines, a relation to be lies on, and so on for every word in the vocabulary. Once that is supplied, every sentence in the vocabulary becomes true or false, and the model satisfies the axioms when all of them come out true.

Two consequences follow immediately and both are the whole of this essay.

A proof is a syntactic object and truth-in-a-model is not. A derivation is a finite arrangement of symbols obeying rules; whether a sentence holds in a model is a question about a structure. The bridge between them is soundness: what is derivable from $A$ holds in every model of $A$. That single sentence is what turns a picture into an impossibility proof.

The vocabulary means nothing until a model says so. Hilbert’s remark that one should be able to say “tables, chairs and beer mugs” in place of “points, lines and planes” is exactly this: the axioms constrain the relations between the words and are silent about what the words are. The disc takes line to mean an arc, and nothing in Euclid’s first four postulates objects, because none of them says what a line is.

That second point is the one that took longest to be accepted, and it is the real content of the nineteenth century’s change of view. Before it, geometry was about space and the axioms were attempts to describe it, so a geometry in which the fifth postulate failed was a description of something false. After it, the axioms are a specification and anything meeting the specification is a model, so a second model is not a rival account of space but a demonstration about the specification.

Angles, and how different this place is #

The disc is not a small perturbation of the plane. Triangles there have angles adding to less than $180°$, always, and the shortfall is not an error term — it is the triangle’s area.

That last identity deserves its own sentence, because it has no Euclidean analogue at all. In the plane, area and angle sum are independent: a triangle can have any area and its angles always come to $180°$. Here the angle sum determines the area completely, so there is no such thing as a large triangle with nearly Euclidean angles, and similar triangles of different sizes do not exist. Two triangles with the same angles are congruent.

Comparing the two triangles explains why nobody noticed for two thousand years. Small figures in this geometry look Euclidean, and the departure grows with size. If the universe, or the page being worked on, is a small patch, the fifth postulate is true to any precision that can be measured, and every consequence of it holds to the same precision. The difference is real and it is invisible locally.

The other denial, which costs more #

The postulate says there is exactly one parallel, and there are two ways to deny it: more than one, or none at all. The disc supplies the first. The second is the sphere, with line read as great circle, where any two great circles meet and nothing is parallel to anything.

The two denials are not on an equal footing, and the asymmetry is the most instructive thing about this whole question.

The sphere is not a model of Euclid’s first four postulates. Two points determine exactly one great circle — unless they are antipodal, in which case infinitely many pass through both, and the first postulate fails outright. Identifying each point with its antipode repairs that and breaks something else: lines now have finite total length, so a segment cannot be extended indefinitely in the sense the second postulate asks for. Whichever repair is chosen, a postulate other than the fifth has been given up.

That is not a defect of the sphere; it is a theorem. Saccheri and Legendre both proved, from the first four postulates alone, that a triangle’s angles can never exceed a straight angle. So the obtuse case is not merely unmodelled — it is refuted, by the four postulates the fifth was supposed to follow from. Saccheri’s long investigation eliminated it correctly and then failed to eliminate the acute case, which is exactly the shape of a man who has proved half a dichotomy and cannot see that the other half is the answer.

So the situation is sharper than the postulate is independent. Given the other four, the angle sum is at most a straight angle, and the only question is whether it can be less. The disc says yes. There is exactly one alternative geometry compatible with the rest of Euclid, not two, and the sphere is a different subject that agrees with Euclid on nothing but the fourth postulate.

That also explains a piece of terminology that otherwise looks arbitrary. Spherical geometry — elliptic, in the usual naming — is studied alongside hyperbolic geometry and is not an independence result about Euclid, because it does not satisfy the axioms the independence would be relative to. Both are geometries; only one is a model.

The lesson generalises past geometry and is worth keeping when reading any independence claim. A structure only settles a question if it satisfies every other axiom, and exhibiting something where the disputed statement fails proves nothing unless the rest of the list has been checked. The disc’s four postulates are verified one at a time above for exactly that reason, and the sphere is what happens when they are not.

What “the axioms are consistent” actually means here #

There is a subtlety in this argument that is worth not glossing over, because it is where the modern subject begins.

The disc shows that the fifth postulate does not follow from the other four provided the disc exists — that is, provided the ordinary plane, out of which it is built, is itself consistent. Everything in the model is made of ordinary Euclidean circles and ordinary real numbers.

So what has been proved is a relative statement: if Euclidean geometry is consistent, so is the geometry that denies the fifth postulate. Beltrami and Klein made exactly this point in the 1860s, and it is the reason the discovery of hyperbolic geometry became a proof rather than remaining a curiosity. Lobachevsky and Bolyai had developed the geometry and could not rule out that a contradiction lay a thousand theorems further on; the model rules it out, by showing that any contradiction there would produce one here.

Nothing in this method ever gives an absolute consistency proof, and by a later result nothing else does either. What it gives is a network: this system is consistent if that one is. That is less than was wanted in 1900 and it is what is available, and it has turned out to be enough for almost everything.

The move, and where else it goes #

Once the pattern is visible it is everywhere.

Independence in set theory. The continuum hypothesis and the axiom of choice are both independent of the usual axioms, and both were settled the same way: Gödel built a structure in 1940 where they hold, Cohen built structures in 1963 where they fail. Cohen’s method — forcing — is enormously harder than drawing a disc, and it is doing the identical job.

Independence in arithmetic. Goodstein’s theorem is true and not provable from the usual axioms of arithmetic, and the proof of the second half is a model-theoretic argument of the same family.

Ordinary mathematics, all the time. The reason a group theorist knows commutativity is not a consequence of the group axioms is that there is a non-abelian group. Nobody calls that an independence proof, and it is one, with the same logic behind it.

The dates, which are the interesting part #

The chronology is worth having because it separates three things that are usually run together.

1829 and 1832. Lobachevsky and Bolyai independently publish developments of the geometry in which the fifth postulate fails. Gauss had reached the same results earlier and did not publish, by his own account because he feared “the outcry of the Boeotians”. What all three had was a body of theorems — an extensive, coherent, non-contradictory-looking subject — and no way to be sure a contradiction was not waiting a thousand theorems further on.

1868. Beltrami exhibits a model. Now the geometry is not merely undisturbed so far; it is as consistent as the ordinary plane, and the two-thousand-year question is closed. Klein and Poincaré give further models over the following fifteen years, of which the disc here is the best known.

The forty-year gap between the two is the gap this essay is about. Developing the geometry was the creative work and it did not settle the question. Building the model was, mathematically, much less: a translation exercise, taking each word to an arc or a distance and checking four postulates. And it settled it completely.

That ordering — the hard creative work first, the decisive small argument afterwards, and the small argument being the one that closes the question — is common enough in this field to be worth expecting. It is the same shape as Cook’s construction for rule 110, fifteen years after the conjecture, and it recurs whenever the thing to be established is an impossibility rather than a fact.

The habit worth taking #

The transferable part is not the disc. It is the change of question.

When a statement resists proof, there are two things it could mean, and they feel identical from the inside: the proof is hard, or the proof does not exist. The second possibility is the one a system can be made to state about itself, and the disc below is the first case where anybody demonstrated it. No amount of further searching distinguishes them, because both look like failing to find a proof. Exhibiting two models settles it in one step, and the manoeuvre is available wherever the axioms are few enough to check — as they are for a finite geometry with seven points, where every model can be written out in full.

What distinguishes them is looking somewhere else entirely. Stop asking whether the statement follows and start asking whether it is true in everything the axioms describe. That question is answered by construction rather than by search, and one construction is conclusive where two thousand years of searching was not.

The lesson, stated as a habit: if a proof will not come, try to build a counterexample to the whole idea of a proof. The two are not alternatives; they are the two possible answers to one question, and only one of them can be established by working harder.

There is a last thing this episode changed, and it is larger than the geometry. Before 1868 an axiom was understood as a self-evident truth about a subject that existed independently of it, and a second geometry was therefore an absurdity — space cannot be two ways at once. Afterwards an axiom is a constraint, a system is whatever satisfies its constraints, and having several models is the ordinary condition rather than a scandal. Every axiom system in modern mathematics is read the second way: the group axioms describe groups, plural, and nobody asks which one is the real one.

That change of reading is what made the rest of this field possible. Asking whether a sentence is provable from a system, rather than whether it is true, requires the system to be an object one can stand outside of and compare things against — and the disc is the first place anybody did that convincingly.