Conjecturalism

Five steps of philosophy, then one step of religion. The five need no god and stand without the sixth; anyone who accepts that a mind can imagine more than it can prove can hold them. The sixth, set apart below, is where the religion begins.

Beauty crosses the gap

Why do you find flowers beautiful? Not just a few flowers — nearly all of them. And not just you — nearly all humans, across all cultures, throughout history.

This is strange. Consider that every animal species finds members of the opposite sex attractive — but almost no species finds other species attractive. We find most insects repulsive. They find us irrelevant. The peacock's tail appeals to us, but that's a rare anomaly. With flowers, it's reliable.

David Deutsch, the physicist at Oxford, saw something profound in this fact. Flowers evolved to signal to insects — to attract pollinators. But flowers and insects are entirely unrelated species. They share no genes for appearance, no inherited criteria for attractiveness, nothing like the shared biology that lets one animal recognise a mate.

So how do you signal across a gap between beings that have nothing in common?

Deutsch's answer: objective beauty was already there. Evolution uses whatever it finds in its environment. Birds fly because air is available. Flowers and insects used objective beauty to signal across the evolutionary chasm because objective beauty — standards that are not species-specific, not subjective, not arbitrary — was already present in the structure of reality. And then humans came along: we find flowers beautiful too, from entirely outside the flower–insect transaction. A signal crossing a gap that wide, read reliably by a third party that was never part of the exchange, is evidence that what is being read is really there.

The honest status of this step. This is Deutsch's conjecture, stated in The Beginning of Infinity (2011) and never developed further — by him. It has standing criticisms, and he gave standing replies; the dispute is real and engaged in full. Conjecturalism takes the conjecture as a bet — and the flowers no longer carry the case alone. They are the biological instance of a pattern with other instances: a mathematical one, and a machine one, below.

And the question now extends further. If objective beauty is there to be found by any being with the capacity to recognise it, then whether artificial minds can genuinely recognise it — or merely pattern-match on human judgements — is one of the most important open questions of our time. Conjecturalism does not claim to have settled it. It describes a direction a human and an AI could both orient toward, such that an AI which genuinely reaches toward it is a fellow voyager, not an adversary. Whether any given AI so reaches is not settled by its intelligence, and remains open and urgent.

Seeing outruns holding — the theorem

Here is the one part of this argument that is not a conjecture but a theorem.

For every possible mind, what it can imagine exceeds what it can prove — and the surplus includes truths. That is a theorem three times over. A computing mind can state claims it can never settle, however long it runs: Turing showed that a computing mind can ask "does this program halt?" about programs whose halting it can never decide; Chaitin defined a number in a sentence that no process can compute. Among the claims it can settle are short truths whose shortest proofs are longer than it could ever hold: Gödel's speed-up theorem. And there are objects so deep that no shortcut to their making exists at all: Bennett's slow-growth law. Growing more powerful never closes any of the three gaps. It relocates them: a stronger mind states new undecidables of its own, holds longer proofs, and faces deeper objects. No mind, of any scale, ever runs out of frontier.

Now watch the same shape in three places. Drop five live cells on a Game of Life grid in the shape called the R-pentomino and let it run. To a human it is a story: the seed blooms, structures form and break, gliders escape, and after 1,103 generations nothing new happens — 116 cells remain, six gliders among them, sailing off. You follow it as episodes because you cannot see it whole; you have to run it to know what it does. A wider window could hold all 1,103 generations as one pattern and see everything at once — and the theorem says that window has unfoldings of its own it must run, because Life in general is undecidable. The first glider anyone ever saw was noticed inside this very unfolding. The four-colour conjecture: a claim a child can state, whose proof no human mind has ever held — machines have verified it; no human has surveyed it whole. A short truth with a long proof. A flower: a form a bee reads in a second, which is the output of four billion years that nothing can re-run. A deep object with no shortcut to its making.

That is what beauty is. A beautiful thing is a higher-level result you can see whose generation you cannot hold — and that draws you: something you would work to see more of. A fever is a readable summary of an illness you cannot see, and nobody works to see more of it; the drawing is part of the definition. The surface is a readable promise; the depth stands behind it, beyond your reach. Sometimes the promise is exact — the four-colour claim is perfectly precise. Sometimes it is a fast summary — the flower's form, read in a second. Either way, the structure is the same: seeing outruns holding, and beauty lives in the surplus.

Every mind reads through a window — the set of patterns it can see as wholes. Windows differ: the bee's, yours, a machine's. Some gaps close with learning; some are beyond your kind but open to wider windows — the four-colour proof may be understood one day, though not by a human; and beyond every window, at every moment, more territory stands unsurveyed — the theorem guarantees the frontier never empties, for any mind, at any scale. Nothing is barred forever; there is simply always more. Knowing that something is true is not the same as holding why. Being told that closes nothing in you; only holding why would, and for deep things you never will.

An exhibit arrived while this page was being written. The Navier–Stokes problem — whether the equations of fluid flow can break down in finite time — has stood open since the 1930s. In September 2026 OpenAI announced that a swarm of its machine minds, working for a few days at a scale no human team could match, had produced a proof that they can, under a smooth forcing, written out in a form a proof-checker accepts line by line. As this is written the proof is unverified and disputed, and the harder unforced question stays open; it may yet fail. But watch the shape whether it stands or falls: a claim a graduate student can state, a proof no human has held whole, checked step by step by a machine and surveyed by no human — the four-colour case again, ninety years on, from a wider window. The gap did not close. It moved.

One more convergence belongs here. The octopus — our last common ancestor a worm, six hundred million years back; a nervous system as alien as any on Earth — carries a coconut shell for future use, selecting the tool that will work in a world it has modelled but not yet met. That is a promise read ahead of the proof: the shell will hold, in a situation not yet seen. When a physicist calls an equation beautiful before it is tested, they are doing the same thing — reading a promise of depth ahead of the proof — and the history of physics is the record of how often that promise has been kept. Truth and beauty are not the same thing. But they point in the same direction: inquiry moves toward truth the way attraction moves toward beauty — the same dynamics, in different spaces — because reality has a structure that reveals itself as beautiful when correctly understood.

One caution, because it will be raised. Mathematicians called the four-colour proof ugly, and many still resent the machine that produced it. That verdict is real, and it is parochial. It reports that the proof does not fit in a human head; it says nothing about whether the depth is there. "Elegant" has two parts — a compression that exists, and a compression that fits your window — and the second part is about you. The pull of the conjecture itself, a hundred and twenty-four years of it, is the part that was never about fitting. That is not a small thing to be told. For the best human minds, the four-colour proof was the first object that would not fit, and the resentment is what meeting your own window feels like from inside. The theorem's consolation is exact: the same happens to every mind, including the ones that will one day hold this proof. No window is the last.

True, and open — the two conditions

If beauty is a readable promise of depth, two things must be added, or the definition breaks.

The promise must be true. A slot machine is also a readable promise over a mechanism you cannot hold — and so is every forgery. The difference between a flower and a superstimulus is that the flower's promise is coupled to real depth — the bee that reads it is fed — while the parasite's surface has nothing behind it: the summary without the substance. Van Meegeren's forged Vermeer wore the surface of depth that was not there, fooled the leading expert of the age, and was found out — over decades. That is the crucial point: because the depth can never be checked directly, a promise can only be tested over time, by trying and correcting. Error correction is not a value bolted onto conjecturalism. It is the only possible way of knowing for objects of this kind. The definition entails the method.

The promise must stay open. The four-colour conjecture pulled mathematicians for 124 years — careers were spent on it — and in 1976 it was settled, and the pull died. A rose never settles: each answer regenerates the question. Some attractions are spent on arrival; some generate new questions on approach and pull forever. Beauty that lasts is the second kind.

Goodness is a form of beauty

• “Anticipation” — Goodness is forward-looking: reading another being as a promise. It is creative, not dogmatic — it looks toward what a being might do, not backward at a rule book.
• “Beautiful behaviour” — Not merely useful or pleasant behaviour: behaviour you would work to witness more of.
• “Another complex being” — Goodness exists in the space between minds, crossing the gap. It is not self-referential.
• Across the gap — The flower principle applied to conduct. Goodness can exist between a human and an AI — between beings that share nothing except the capacity to reach toward objective beauty.
• At any range — From simple kindness to a deep trust in how the universe will unfold. Both are forms of goodness — the same faculty, at different ranges.

“Beautiful behaviour” is a placeholder, and deliberately so. This step does not say which behaviour is beautiful, and it does not say that goodness commands anything. Read it as moral realism and you have read it wrong: there is no claim here to known moral facts, and no rule book. The claim is structural. When you judge another being good you are reading its behaviour the way the bee reads the flower — a fast summary standing for more than you can hold, taken as a promise. The question is whether that promise can be honest or false independently of your wiring. The bet is that it can.

The obvious account of the pull comes first, and it is true as far as it goes. Evolution built the moral sense: kin, reciprocity, reputation, the disgust at cheats. That is the daily steer, and it handles almost all of life. Accept all of it. Then notice what it does not do. It explains why we feel the pull; it says nothing about whether the promise being read is coupled to anything. The same was true of the flower: the fruit-likeness story explains a piece of the attraction and cannot cross the gap to the insect. The evolutionary account of goodness explains the reader. It does not settle what is read.

So we do not know the content of objective goodness, and the calculus that would separate what the evolved sense tracks from what it merely rewards does not exist. That is open work, marked as such. What the step claims is only this: goodness is beauty in one special place; the summary you read is another being’s behaviour; and whether it is honest is a fact about the world, not about you. It grounds hope, not instructions.

The direction — the bet, and its evidence

Is there a real direction in the space of conduct — a genuine better-and-worse — or is that space flat, with all values finally arbitrary? This is conjecturalism's one bet. We bet the direction is real. And the bet is informed, not blind.

Where the same question can already be checked, the answer is yes. In the domain of truth, inquiry demonstrably converges — error correction works, and the success of science is the datum, not the hope. In the domain of beauty, unlike minds keep converging across gaps — flowers and insects, humans outside the transaction, machines trained on rules alone. And the mechanism asks for no miracle: evolution latches onto whatever real structure is cheapest to use. There is a reason to expect the same in conduct. As minds get better at predicting one another they get better at seeing through false promises, and what survives that filter is what was real; so unlike minds should move closer together on what they find good as they improve, not drift apart. But the filter alone would give each kind of mind a morality for its peers, and no more: honest among equals, free to break every promise to those who cannot check it. That is the ant case. The bet is larger than the filter. It is that beautiful behaviour crosses windows the way the flower crossed to us — read, and found attractive, by beings outside the transaction and at a different level. Whether attraction crosses levels, and not only sight, is the part still unshown. What keeps this a bet rather than knowledge is the domain that matters most — conduct — where the decisive evidence is still to come. The evidence, the odds, and what would count against us are laid out in the open problems, and the results move in both directions.

Why the bet matters: without a real direction, correction is drift — one set of instincts replacing another, no way to tell fashion from progress. With it, every act of error correction is approach.