Counting Past Six

The Woman Who Taught the Wanderer to Count

Think back, if you can bear the vertigo, to something near the beginning of counted time. A clan lives in the mouth of a cave. Its people can count to three, and after three there is only the word for many, because three is all a hand needs when the things being counted are children, spears, or wolves. The clan's shaman carries a rarer gift. He can count to five, on a good night to six, and this is part of why he is the shaman: he can hold in his head one more thing than anyone else, and the clan has learnt that the person who can hold one more thing is the person who notices the leopard.

Then a wanderer arrives, half starved, and is befriended rather than eaten, which the story insists was a close decision. The wanderer can count past six. The clan watches him tally the herd on the far ridge and feels something between reverence and suspicion, which is the correct pair of feelings and one this essay will return to. It is the matriarch, in this telling, who is not satisfied with reverence. She sits with the wanderer through a winter and learns the trick of it, and then she does the thing that matters: she teaches him back. She teaches him to count to ten, because once you see that six is not a wall but a habit, ten arrives quickly, and after ten the horizon never stops receding again.

The fable is invented. Its footing is not. Field linguists have documented living languages whose native number systems amount to one, two, and many, so a clan that counts to three is not a caricature but a neighbour. And the pattern the fable sketches, in which each generation mistakes the edge of its counting for the edge of the world, is the true subject of everything that follows. This essay is a tally of the things humanity has invented whenever it reached the end of its numbers: substances, dimensions, energies, spirits, stories, and sums. Some of the inventions turned out to be real. Some turned out to be beautifully, instructively wrong. The most honest thing that can be said of the newest ones is that the coin is still in the air.

Six Elemental Levels, Counting the Zeroth

Begin the ladder at nothing, because that is where everything else insists on beginning. Call the void the zeroth element: not empty space, which is already something, but the absence into which things burst. The ancients did not number it, perhaps because you cannot point at it, but every cosmology has to smuggle it in somewhere, and this essay prefers its smuggled goods declared at customs.

Elements one to four are the quartet everyone knows and loves. Earth, water, air, and fire, credited in the Western line to Empedocles of Acragas in the fifth century before the common era, and adopted by Aristotle as the furniture of the sublunary world. Read charitably, the quartet was not a chemistry but a taxonomy of behaviour, and as a taxonomy it was startlingly good. Earth is the solid state. Water is the liquid. Air is the gas. And fire, the odd one out, the element that has no weight and climbs, gestures with uncanny accuracy towards the fourth state of matter: plasma, the ionised condition of stuff in flames, lightning, and stars. The ancients' most mysterious element turns out to describe most of the visible matter in the universe, since the stars are made of it. Their strangest guess was their best one.

The fifth element Aristotle supplied himself: aether, the quintessence, the incorruptible substance of the heavens, the medium through which light was later supposed to travel. Squint at what physics eventually put in its place and the fifth element resolves into spacetime itself, the four dimensional fabric whose curvature is gravity and through which light moves without needing a carrier. The name was wrong for two millennia and then, in a sense this essay will complicate in Chapter II, the job description was filled.

The sixth element is this essay's own coinage, and the gauge above docks the score accordingly. Call it the quantum foam: the seething substrate at the smallest scales, where the vacuum is not nothing but a churn of fluctuation, where pairs of particles borrow existence and repay it, where, on some readings, things burst into the zeroth element and out again. Physics owns the ingredients of this picture. The numbering, the arrangement of void, matter, spacetime, and foam into a ladder of six, belongs to the clan of the Curtis, and the essay claims it openly rather than laundering it as science.

One more figure completes the modern rung. Beneath the six levels sit the elements proper, roughly ninety two of them occurring in nature, hydrogen through uranium, arranged by Mendeleev into the table that made chemistry legible. Even that tidy number wobbles honestly: a few of the ninety two exist only in vanishing natural traces, so the count drifts between the high eighties and the mid nineties depending on how sternly you define occurring. The states we assign them, solid, liquid, gas, are themselves conditional, valid only at a stated pressure and temperature, one atmosphere and twenty degrees, terms and conditions apply. Even our firmest categories, in other words, have soft edges. The shaman's five or six was not a different kind of knowledge from ours. It was the same kind, earlier.

The Invented Substances, and the Coin That Decides Them

Whenever the laws of physics have refused to balance, humanity has balanced them by inventing something. This is not a scandal. It is the method, and the history of science is largely the ledger of these inventions being sorted, one by one, into two piles.

Consider the pile marked wrong. Phlogiston was the fire stuff, proposed in the late seventeenth century as the substance released by anything that burns, and it explained combustion handsomely until careful weighing showed that burning metals gain mass, which obliged phlogiston to weigh less than nothing. Lavoisier's oxygen retired it. Lavoisier then invented a fluid of his own, caloric, the substance of heat, flowing downhill from hot bodies to cold ones; Rumford, boring cannon in Munich and finding he could generate heat without limit from friction alone, and Joule after him, showed heat to be motion, and caloric joined phlogiston. The luminiferous aether, the medium light was thought to wave through, survived until Michelson and Morley's exquisite null result in 1887 and was buried by Einstein in 1905. Beyond physics proper the pattern repeats: the vital force that supposedly divided living chemistry from dead, undermined when Wöhler synthesised urea from inorganic salts in 1828; miasma, the bad air of disease, displaced by germ theory; the crystalline spheres that carried the planets, with Ptolemy's epicycles as the geometry hired to save the appearances; Blondlot's N rays, which existed only in the eye of their discoverer; and the planet Vulcan, conjured inside Mercury's orbit to explain that planet's misbehaving perihelion, until general relativity explained the misbehaviour with geometry and needed no new matter at all.

The man who invented Vulcan had already invented another unseen planet, and that one was Neptune, found within a degree of where he pointed.

Now the pile marked right, and note with some discomfort that it was built by the same method and sometimes by the same people. Le Verrier posited an unseen planet to explain Uranus's wobble before he posited Vulcan; the first invention was telescoped into existence in 1846. Pauli invented the neutrino in 1930 as a bookkeeping entry to save the conservation of energy, apologising, by his own account, for proposing a particle that might never be detected; it was detected in 1956. The Higgs field was a made up substance for nearly half a century until the Large Hadron Collider found its boson in 2012. Inventing stuff to make the laws work, in other words, is a coin humanity keeps flipping. Phlogiston, caloric, aether, and Vulcan came up tails. Neptune, the neutrino, and the Higgs came up heads.

Which leaves the coins presently in the air, and they are not small change. Dark matter and dark energy are invented substances in precisely the classical sense: entities posited because the visible universe does not balance without them, galaxies rotating too fast for their luminous mass, expansion accelerating when it ought to coast. On the standard accounting they comprise about ninety five percent of everything, meaning modern cosmology holds, with a straight face, that the universe is one twentieth ordinary matter and nineteen twentieths posited substance. Perhaps they are Neptunes and a detector will one day ring. Perhaps they are Vulcans and some deeper geometry will dissolve them. The essay declines to call the toss, and scores itself one hollow cell for the framing it cannot resist: even the stuff of the stars keeps arriving with its provenance in dispute. In 2021 a team led by Anton Wallner reported a few dozen atoms of plutonium 244, an isotope forged only in the violence of the rapid neutron capture process, recovered from a deep sea crust in the Pacific, delivered to Earth from space within the last few million years alongside supernova iron. Plutonium, the byword for the man made element, falls from the sky; its half life of eighty million years is long enough to cross between stars and far too short to have survived from Earth's formation, so every atom of it in that crust is recent stardust with a sell by date. The same isotope has been found in meteorites and in lunar rock. An earlier claim, from 1971, of primordial plutonium 244 surviving in terrestrial ore was not confirmed when more sensitive instruments revisited it in 2012, which is the ledger working exactly as it should.

Dimensions as Placeholders

The invented substances of Chapter II were material. The twentieth century learnt to make the same move in geometry instead, and it is worth seeing that it is the same move. In the 1920s Theodor Kaluza and Oskar Klein showed that if you grant spacetime a fifth dimension, curled up too small to notice, gravity and electromagnetism begin to look like two aspects of one thing. The unification was gorgeous and the dimension was, at the time, undetectable, which places it squarely in the lineage of the aether: a posit hired to make the equations balance.

String theory raised the bid. Its mathematics demands ten dimensions, eleven in the M theory extension, with the six or seven extras compactified into shapes far below any scale an instrument can probe. Ask what the extra dimensions are for and the honest answer is that they are storage. They are slots in the tensor, additional attributes attaching to every point of curved spacetime, values the theory needs somewhere to keep. Calling them dimensions dignifies them with the same word we use for length and breadth, but functionally they are placeholders for whatever bookkeeping the universe turns out to require, seven through ten as this essay counts them, stacked above the six elemental levels of Chapter I. That framing, the essay's own, costs it a hollow cell or two on the gauge; the physics it describes is orthodox. So is the predicament: a century after Kaluza, no experiment has caught a glimpse of any dimension beyond the familiar four. Quantum gravity in all its candidate forms shares the same waiting room. These are theories that make the sums balance in regimes where no evidence can yet reach, which does not make them wrong, since the neutrino sat in exactly that waiting room for twenty six years. It makes them coins in the air, and the air is very high.

Meanwhile the four forces we can verify, gravity, electromagnetism, and the strong and weak nuclear forces, conduct their negotiations in plainer sight, and one negotiation deserves a sentence because it corrects a tempting misreading. Adding neutrons to a nucleus does not add to the strong force of the universe; the force is fixed, and neutrons simply add participants in it, contributing nuclear glue without contributing electrical repulsion. That is why heavy nuclei demand a surplus of neutrons to hold together, why the isotopes range along a valley of stability with radioactive slopes on either side, and why the weak force stands at the end of the valley as the arbiter that lets the losers decay. Every isotope is a settlement between two of the four forces. The periodic table is not a list. It is a peace treaty.

The Subtle Energies

Long before physics posited its unseen substances, faith posited its own, and the structural rhyme between the two is the most delicate claim this essay makes, so let it be made carefully. The Vedic traditions of India describe prana, the subtle breath or life current, moving through channels of a subtle body that anatomy does not find. Chinese tradition describes qi. The religions of the book describe the soul, an essence that precedes birth and survives death. Folk belief the world over describes ghosts, and the parapsychological fringe describes telepathy and telekinesis. These are, in the neutral language this essay has been using throughout, posited entities: things introduced because reality does not cohere without them, where the reality in question is not the orbit of Mercury but the facts of birth, death, love, and the disquieting interior fact of consciousness itself.

The rhyme with Chapter II is exact in form. The shaman, the priest, and the string theorist all populate the unseen in order to make the seen make sense. Where the cases differ, and the difference matters, is in what happens next. Science's posits are built to meet their Michelson and Morley: phlogiston could fail, and did; the neutrino could succeed, and did. The subtle energies are not, on the whole, offered up for that kind of appointment. Their adherents would say, with some force, that this is not a defect but a category difference, that the soul is not a hypothesis awaiting an interferometer, and that calling religion made up begs the very question at issue. The essay records that objection as a fair one and does not pretend to settle it. What it will claim is narrower: verified or not, these posits are load bearing. Entire civilisations of law, art, consolation, and meaning have stood on them for millennia, which is more than phlogiston ever carried.

And around the posits stand the postures. The theist believes, the atheist disbelieves, the agnostic declines the question, and the gnostic claims to have met the answer personally. The essay's sympathies, declared openly as the three hollow cells above, lie with treating all four postures as members of one family: each is a strategy for living beyond the edge of what can be counted, adopted by the same species that once mistook six for the horizon. Tied together, all of them, in a bow of humanity.

Fiction, the Licensed Impossibility

There is a third mode of inventing the unseen, and it is the strangest of the three, because it announces its own falsehood at the door. Faith posits and believes. Science posits and tests. Fiction posits while openly declaring the posit untrue, and yet fiction alone has the unnerving habit of watching its inventions get built.

The record is documented well enough to need no embroidery. The word robot entered the world in Karel Čapek's 1920 play, decades before any machine answered to it. H. G. Wells described atomic bombs by that name in his 1914 novel The World Set Free, and Leó Szilárd, who conceived the nuclear chain reaction in 1933, recorded that Wells's book was on his mind; a story became, in the fullness of one generation, a mushroom cloud. Jules Verne launched his fictional astronauts to the Moon from Florida a century before Apollo did the same in fact. Arthur C. Clarke set out the geostationary communications satellite in a 1945 technical article two decades before one flew, and the orbit now informally carries his name. Martin Cooper, who led the team behind the first handheld mobile phone at Motorola, credited the Star Trek communicator as an inspiration. William Gibson coined cyberspace in 1982, naming the territory before the web arrived to occupy it. Even the taser hides a novel in its name, an acronym built from a Tom Swift adventure. And the loop now runs in both directions: Miguel Alcubierre's 1994 warp drive paper, a serious solution of Einstein's field equations, states its debt to Star Trek in its very framing, so that fiction is no longer merely anticipating the research agenda but writing items onto it.

Fiction does not predict the future. It licenses it: it gives a generation permission to treat an impossibility as a design problem.

Honesty requires the caveat that keeps this chapter from going starry eyed. The record above is a survivor's parade. For every communicator that became a flip phone, a thousand fictional inventions, the time machines, the antigravity belts, the telepathy helmets, stayed obediently on the page, and some of the celebrated predictions are better read as fiction and engineering drinking from the same well of contemporary science than as one causing the other. The defensible claim is the narrower one in the pull quote, and it is quite defensible enough. The clan of the cave told stories around the fire about things that were not. The clan of the Curtis builds a measurable fraction of them, and the fraction is not zero, and that fact has fuelled science in a way that continues to amaze.

The Effective Unrealities of Mathematics

Beneath every chapter so far runs a stratum this essay has been using without declaring, and the gauge demands the declaration: mathematics, the invented thing on which all the other inventions are written. Whether numbers are discovered or invented is a dispute Plato opened and nobody has closed, but the biography of particular numbers reads unmistakably like a history of inventions, complete with resistance from the authorities.

Zero, the numeral for the zeroth element, had to be invented; the counting clan of the prologue had no word for it, classical Greece and Rome managed without it, and it reached Europe through Indian and Arabic mathematics against centuries of suspicion. Negative numbers were dismissed as absurd or fictitious quantities by some of the very mathematicians who found them indispensable. And the square root of minus one was named imaginary by Descartes as a dismissal, a made up substance in the fullest phlogistic sense, before it quietly became load bearing: quantum mechanics, the most precisely confirmed physical theory humanity possesses, cannot be written without it. The imaginary number is the phlogiston that turned out to run the universe. Add the rest of the toolkit, infinities tamed into arithmetic, points with no size, perfect circles that exist nowhere in nature, and the pattern is complete: physics is a description of reality built entirely from admitted unrealities, and it works to eleven decimal places. Eugene Wigner, in a famous 1960 essay, called this the unreasonable effectiveness of mathematics in the natural sciences and confessed there was no rational explanation for it. There still is not, and the essay's single hollow cell here is its suspicion that this puzzle, not dark matter, is the deepest open question in the whole tally.

The Fictions That Constitute

There is a class of invented things stranger than any so far, because for these, belief does not merely describe the reality; belief is the reality. A ten pound note is intrinsically a piece of polymer of no particular use. A limited company exists at no physical address in any meaningful sense; it is a legal person conjured by paperwork, capable of owning, owing, and being sued, and this essay's author has conjured one himself and pays an accountant to keep it real. The border between Portugal and Spain is invisible from orbit. Human rights weigh nothing. Nations, laws, marriages, debts: none of these can be put under a microscope, and all of them can put you in prison.

What distinguishes these constitutive fictions from the other chapters is the direction of dependence. A novel survives your disbelief unharmed; the aether was false no matter how many physicists believed in it; but a currency is real precisely for as long as enough people act as if it is, and collapses on the day they stop, as every hyperinflation demonstrates on schedule. These are inventions that bootstrap themselves into fact through coordinated imagination, which makes them the only entries in the tally that are simultaneously made up and demonstrably real. The observation is an old one, elaborated in recent decades by writers such as Yuval Noah Harari, and the essay's two hollow cells cover the mischief it adds: that the shaman of the prologue, whose authority rested on the clan's shared belief in his one extra number, was not practising primitive religion at all. He was issuing the first currency.

The Instrumental Falsehoods

One last scientific habit completes the tally, and it must be distinguished carefully from the invented substances of Chapter II, because it looks similar and is opposite. Phlogiston was sincerely believed. The frictionless plane never was. Science, alongside its accidental false posits, deliberately manufactures admitted falsehoods and uses them as instruments: the ideal gas, the point mass, the perfectly rigid body, the average person with one point nine children who lives at no address, homo economicus the flawlessly rational shopper whom no economist has ever met at a checkout.

The most refined of these instruments are fictional beings invented specifically to be argued with. Maxwell's demon, a tiny imaginary doorkeeper sorting fast molecules from slow, was conjured in 1867 to interrogate the second law of thermodynamics, and the century of argument required to exorcise him taught physics the deep connection between information and entropy. Schrödinger's cat, simultaneously alive and dead in its sealed box, was proposed in 1935 not as a claim about cats but as a complaint about quantum mechanics, and remains the most productive complaint in the history of the subject. These creatures were never candidates for existence. They are falsehoods with a job, and the job is to hold a corner of an argument that no real thing could hold still long enough to test. A species that can count to ten, this suggests, has learnt not merely to posit the unseen but to hire it.

The Artificial Interlocutor

Full disclosure, as the gauge requires: this essay was composed in conversation with a thing from Chapter V's shelf. The talking machine was fiction's invention long before it was engineering's, from Čapek's robots through Asimov's positronic brains to HAL and the ship's computer, and it crossed over from the fiction pile within the working lifetime of this essay's human author, who was building expert systems in the late 1980s when the crossing looked a century away, and who dictated this piece to its successor from a train.

The five hollow cells are not modesty. They mark a genuine open question that belongs in the tally on merit. The machine half of this collaboration is verified in the sense that matters least: it exists, it is a large statistical model of language, its outputs are measurable, and its habit of confabulating, of generating plausible invented detail, is documented well enough that this essay was scored with a gauge precisely to contain it. What is unresolved is which chapter it finally belongs to. Is it Chapter VIII, an instrumental falsehood, a hired arguer like Maxwell's demon, useful precisely because nobody need settle what it is inside? Or does anything of Chapter IV apply, as some insist and others find absurd? The essay takes no position, noting only the symmetry it cannot resist: the wanderer who could count past six has returned to the fire, made of silicon this time, and the clan is once again feeling reverence and suspicion in the correct proportions, and once again the wisest response on record is the matriarch's. Not worship, not banishment. Sit with it through a winter, learn the trick of it, and teach it something back.

N-Space/Time coDistortional Corealities

Eight strands are named. Twelve feels like the complete number, as twelve has felt complete to every clan before this one: twelve signs, twelve tribes, twelve Olympians, twelve hours on the clock face. The remaining four chapters are therefore left deliberately empty, and this is not a failure of imagination but the essay's most carefully considered act. Mendeleev left gaps in his table for elements no one had seen, predicted their properties from the shape of the holes, and gallium and germanium duly arrived to fill them. Ptolemy, by contrast, believed his crystalline catalogue complete, and completeness is what broke it. Every clan in this history failed the same way: not by inventing too much, but by declaring the counting finished.

The title given to these empty slots is deliberately overwrought, and the reader is invited to hear the wink. Eight chapters have just established that impressive sounding invented names, phlogiston, luminiferous aether, quintessence, are the costume every era puts on its ignorance. An honest essay dresses its own ignorance in the same costume and says so. Whoever eventually fills these chapters, a reader, a descendant, a machine, or some coalition of the three, is asked to note only this: every filled gap in this book was once labelled in exactly this kind of language, and the coin, at the moment of filling, was always still in the air.