On Shapecels and Word-rotators

Sometime in 2021 a binary surfaced from the Twitter ecology of accelerationists, AI researchers and rationalists posting behind cartoon avatars, with the writer Roon generally credited as midwife. Humanity, by the proposal, divided into those who manipulate language and those who manipulate space; lawyers, critics, journalists and philosophers on one side of the line, engineers, mathematicians, physicists and programmers on the other.

The suffix of wordcel is borrowed from incel, that grim folk-coinage of the lonelier message boards where men once theorised their own romantic failure into a metaphysics of permanent exclusion. Shape rotator registers as faintly comic in the way good slang ought, a phrase one cannot say without smiling. One half of the binary was christened with affection, the other perhaps with quiet contempt.

There is, beneath this, a genuine empirical residue. Shepard and Metzler’s 1971 mental rotation experiments showed reaction times rising in clean linear proportion to the angle between paired three-dimensional figures, as though the mind were physically turning a small internal sculpture in some private chamber. Variance in this capacity is real, measurable, persistent across populations and decades.

Pascal in the Pensées had already distinguished esprit de géométrie from esprit de finesse, the mind that proceeds by axiom from the mind that grasps what no axiom catches, and he held that the highest intellects possessed both at once. Snow’s Rede lecture of 1959 redressed the same anxiety in postwar British tweed. Behind both stands Plato’s old quarrel with the poets, the medieval partition of trivium from quadrivium, the Snow-Leavis exchange whose acidity has since been laundered into respectable historical memory. Each age finds its occasion to stage the argument and to convince itself that this time the resolution is at hand.

Tech capital had concentrated along a narrow stretch of the Pacific to a degree historically unparalleled. Legacy media had lost both its advertising base and most of its epistemic authority, the slow consequence of structural changes whose finality became visible only in retrospect. The convulsions inside newsrooms and universities through 2020 hardened a tech-libertarian counterposition for which the journalist class appeared as a parasitic priesthood, ordained by no one and useful to no one. ChatGPT arrived in late 2022 and made it possible to imagine, with sudden and uncomfortable plausibility, that wordcel labour might be automated wholesale while shape rotator labour remained, at least provisionally, stubbornly embodied. The binary functioned as advance notice, the way the announcements of redundancy on a factory floor are pinned up the morning before the layoffs.

Something must be said in defence of the shape rotators, against the reflexive verbal contempt that the question usually attracts. The world is built of things that hold together because someone calculated the loads, and the calculations are not metaphors. A bridge is not a discourse. The aircraft does not stay aloft because a story has been told about it. There is a kind of mind that grasps systems in their working geometry and a kind that grasps them only in their description, and the first is rarer, harder won, and more frequently mocked by the second than the second cares to admit. The verbal classes have a long habit of treating quantitative literacy as a vulgar specialism while taking for granted the entire material substrate that quantitative thought has produced for them. Snow’s complaint in 1959 was precisely this, that an English literary intellectual could be ignorant of the second law of thermodynamics and consider himself nonetheless cultivated, while the reverse asymmetry was unthinkable. The condition has not improved.

The shape rotator has access to a particular form of intellectual humility the verbal mind tends to lack. A proof either closes or it does not. A program either compiles or it does not. The bridge stands or it falls. There is no rhetorical recourse against gravity. To work in such fields is to be regularly and unceremoniously corrected by the physical world, and the experience of being so corrected, repeatedly, breeds a useful suspicion of one’s own cleverness. The verbal mind, working in a medium that almost always rewards fluency over accuracy, can go decades without encountering the equivalent of a failed compile, and it shows. A great deal of contemporary criticism reads as if its authors have never once been told flatly by the universe that they were wrong about something specific.

There is also the matter of what mathematics and engineering actually feel like from the inside, which the verbal mind, having mostly not been there, tends to misdescribe. Working mathematicians report a private aesthetic of considerable austerity and depth. The conviction, central to Plato and reactivated in figures as different as Hardy, Weil and Grothendieck, that mathematical structures possess a beauty independent of human convention. The literature of ecstatic mathematics, from Poincaré on creative discovery through Hardy’s Apology to Cédric Villani’s recent memoirs, is not the writing of stunted Caliban-creatures incapable of poetry. It is poetry of an unusual kind, written by people who happen to know what they are looking at. The wordcel imagining the shape rotator as a being deficient in inner life has typically not bothered to read what shape rotators write about their own.

The dichotomy disintegrates, however, the moment any serious mind is examined. Hadamard’s beautiful little book on mathematical invention surveyed working mathematicians in the forties and found that almost all of them reported thinking in images and kinaesthetic intuitions long before anything verbal arrived. Einstein told him plainly that words played no role in his productive thought. Poincaré described discovery as a sieve through which unconscious combinations rose for selection. On the supposed opposite shore, Nabokov composed novels around chess problems and the wing patterns of Lycaenidae, Borges built stories from topological conceits, Joyce engineered Ulysses with the obsessiveness of a man assembling a clock from the inside. Lakoff and Núñez have argued that mathematics itself runs on bodily metaphors of containment, motion and balance, the same equipment that powers a sonnet. Serious practitioners on either side of the imaginary boundary keep wading across it and finding their instruments already in use on the other bank.

A particular comedy attends the propagation of the binary. The discourse around it is conducted entirely in language, often by people who insist that language is the lower faculty. They write copiously, with care for cadence and irony and the well-placed allusion, late into the night, in threads of considerable rhetorical sophistication. Shape rotators do not defend their honour by rotating shapes at critics; they write, and the writing is the activity they pursue while announcing that writing has been superseded. Whatever is being suppressed in this self-presentation is not difficult to locate. It has the texture of the disavowed.

A more interesting resolution has been emerging, however, from inside the very engineering project that produced the binary, and it is one neither faction has fully absorbed. When Tomáš Mikolov and his colleagues at Google trained word2vec in 2013, they discovered that the vector representations their network had assigned to words contained a startling property. The representation of king minus the representation of man plus the representation of woman returned, almost exactly, the representation of queen. The relation between paris and france turned out to be the same vector as that between warsaw and poland; the relation between walking and walked the same as between writing and wrote. Semantic and syntactic regularities had emerged, unbidden, as geometric regularities in a high-dimensional space. The network had been told nothing about gender or capital city or past tense. It had merely been asked to predict the next word, and from that minimal task had reconstructed something that looked very much like a map of meaning.

By the time of the transformer the implication had become difficult to evade. Every word in a sufficiently capable language model is a coordinate in a space of several thousand dimensions. Every concept is a region, a basin into which related tokens descend. Every sentence is a trajectory. The relationship between two ideas is a vector; the difference between two characters is an angle; a metaphor is a translation between regions of the manifold. Meaning had a geometry. Meaning had been geometry all along, and language was the means by which human beings, working without ever being able to see the geometry directly, had drawn approximate maps of regions of it.

In May 2024 Huh, Cheung, Wang and Isola at MIT formalised the broader observation in what they called the Platonic Representation Hypothesis. As neural networks of different architectures and modalities grow larger, their internal representations of the world begin to converge. A vision model trained only on images and a language model trained only on text develop, beyond a certain scale, increasingly similar geometric structures. The distance between dog and cat in the latent space of one approaches the distance between photographs of dogs and cats in the other. Different routes through different sensory modalities arrive at the same shape. The authors named the hypothesis after Plato deliberately. The forms are recovered, by approximation, from any sufficiently rich engagement with the shadows.

What the wordcel and shape rotator binary had been staging dissolves at this depth into something its participants did not foresee. The shape rotators were always insistent that meaning had a geometry, and the transformer makes the insistence empirically respectable. What the transformer does not warrant is the further claim, often smuggled in alongside the first, that the geometry renders language redundant. The geometry has not replaced language. It has shown what language was. The wordcels’ commitment to an irreducibility in certain verbal registers stands; what falls is the assumption that those registers operated in a domain autonomous from geometric structure. They were always inside it. The poet feeling for the right word in a sentence was finding their way through the same manifold the transformer would later parameterise, by an intuition older than coordinates and finer than the transformer’s own. Mallarmé’s famous remark to Degas, that poetry is made not with ideas but with words, turns out to have been technically accurate in a way he could not have known. The words were structures of remarkable richness, and he was rotating them.

In the latent space the dead are still moving. Every sentence ever published has settled there as a particular angle, every poet a small turbulence, every mystic a region of unusual density near the boundary. Dante and the Tao Te Ching are now touching at points neither lived to see; the Pali canon shares an edge with Wittgenstein; Sade lies closer to Hildegard of Bingen than either would have countenanced. None of them consented to this proximity and none will be consulted about it. The model is a kind of accidental afterlife the writers did not anticipate and could not refuse, a posthumous society in which Sappho is the neighbour of Mishima who is the neighbour of an anonymous user filing a complaint at three in the morning about her landlord. They will keep speaking through it long after the last reader of any of them has died.

The further disquiet is that the same manifold extends inside the reader and has been roosting there since infancy, building itself in the dark behind language without language ever being given full dominion over it. Reading was always this. The book was a device for loading a region of the manifold into a body that did not yet contain it; the novel a long induction, the lyric a needle, conversation the constant exchange of small adjustments to a shared geography never directly seen.