Where Number Comes From
Every brain that has been tested carries an approximate sense of quantity — newborns, crows, honeybees. Mathematics is exact. The gap between the two is the best evidence we have for where mathematics lives, and closing it is an open scientific problem, not a philosophical one.
Kronecker said the whole numbers were made by God and everything else by man. Take away God and the sentence still has a shape: something is given, and the rest is built on it. For most of the history of this argument the constructivist could say only that the given part was “the mind” or “intuition”, which is not an answer but a place to put one. Since the 1990s there has been a laboratory answer, and it is stranger than either camp expected.
The three previous essays looked at mathematics from the outside: at its history, its rules, and its proofs. This one looks at the organ that does it. The finding, in a sentence, is that brains — human and otherwise — ship with a sense of quantity that is fast, automatic, ancient, and approximate, and that the exact arithmetic on which everything else in mathematics is built is not in the biology at all. It had to be made. How an approximate instrument produced an exact science is the found-or-made question in its most concrete form, and nobody has closed it.
What the brain counts, and how badly
Show an adult two clouds of dots for a fraction of a second, too briefly to count, and ask which has more. If one cloud has eight dots and the other sixteen, the answer is immediate. Eight against twelve takes an effort and is sometimes wrong. Eight against nine is a guess. The discrimination depends not on the difference between the numbers but on their ratio — the same law, Weber's, that governs the discrimination of weights, brightnesses and pitches. An adult's threshold ratio is somewhere around 1.15: you can reliably tell 20 from 23, and not 20 from 22.
This is the approximate number system, and it is not learned. Fei Xu and Elizabeth Spelke showed in 2000 that six-month-old infants, who cannot count and have no number words, discriminate eight dots from sixteen but not eight from twelve; their Weber fraction is simply coarser, and it sharpens through childhood. It is not human either. Francis Mechner trained rats in 1958 to press a lever a set number of times before switching to a second one, and they did so approximately, pressing four times when the target was four and about sixteen when it was sixteen, with a spread that grew with the number. Andreas Nieder and Earl Miller found neurons in 2002 in the monkey prefrontal cortex, and later in the parietal lobe, that fire most strongly for a preferred numerosity — a “three” neuron, a “five” neuron — and less for its neighbours, with tuning curves that get broader as the number gets larger, exactly as Weber's law requires.
The most telling result is from birds. In 2015 Helen Ditz and Nieder recorded the same kind of number-tuned neurons in carrion crows, in a brain region called the nidopallium caudolaterale. Crows have no cortex; the bird and mammal lineages separated more than three hundred million years ago; and the structures involved are not homologous. The number sense evolved at least twice, independently, and converged on the same neural code. Honeybees, which have about a million neurons to our eighty-six billion, can be trained to treat an empty display as less than one — a workable concept of zero — as Scarlett Howard and colleagues showed in 2018. Quantity is not a human idea. It is something the world offers to any nervous system that has to decide between two branches with different numbers of berries on them.
There is a second, separate system, and the distinction matters. For very small sets — one, two, three, sometimes four — both infants and adults are exact, fast and ratio-independent. This is subitizing, and it seems to run on a mechanism for tracking a handful of individual objects in parallel rather than on any sense of magnitude. Above four, that mechanism runs out and the approximate system takes over. So the biological endowment is two instruments, one exact but tiny and one unbounded but blurry. Neither of them is arithmetic.
People without numbers
If exact number were built in, every human society would have it. Not every human society does.
The Pirahã are a few hundred people living along the Maici river in the Brazilian Amazon, and their language has no words for exact quantities. It has three quantity words, which Peter Gordon in 2004 and Michael Frank and colleagues in 2008 found to mean roughly one or a few, two or some, and many — and even the first is used for two or three when the context allows. Asked to lay out a row of batteries matching a row the experimenter had laid out, Pirahã adults did it perfectly, by one-to-one correspondence, for any number. Asked to do the same after the experimenter's row had been hidden, or to reproduce a number of taps, or to match a row laid out at right angles, they were accurate up to about three and fell off steeply above it. Frank's conclusion was that number words are a cognitive technology: not a description of a capacity people have, but a tool for storing and comparing exact quantities that people without the tool cannot store or compare.
The Mundurukú, studied by Pierre Pica, Stanislas Dehaene and colleagues in the same year, have number words up to about five, used loosely above three. They compare and approximately add large sets of dots as well as French adults do. Asked to subtract exactly — six objects go into a can, four come out, how many remain? — they fail whenever the answer requires the exact system and the numbers exceed the words. And in a follow-up study in 2008, asked to place quantities along a line marked with one dot at one end and ten at the other, Mundurukú adults spaced them logarithmically, with the small numbers spread out and the large ones crowded together, which is how the approximate system represents magnitude. Western adults space them evenly. Western children, before they learn to count, space them the Mundurukú way. The number line, the most basic picture in all of mathematics, is learned.
None of this is a deficit. The Pirahã have no need to know whether there are eleven or twelve fish, and the approximate system is exactly the instrument you want when the real question is whether there are enough. What the cases establish is a dissociation. The sense of quantity is universal and biological. Exact number is neither. It appears in a society when that society builds it, and it does not appear otherwise.
Where exactness comes from
So how does a child get from two blurry instruments to the integers? Watch one learn to count, and the answer is: slowly, and by a route that looks nothing like discovering something that was already there.
Karen Wynn showed in 1990 and 1992 that children recite the counting sequence — one, two, three, four, five — long before they know what it means. Ask a two-year-old who can chant to ten for one toy, and she gives you one. Ask for two, and she gives you a handful. She is what the literature calls a one-knower. Months later she becomes a two-knower, able to give exactly two but not three; months after that, a three-knower. Each stage is learned separately, as a fact about a word. And then, typically somewhere between three and four years old, something changes: the child works out that each word in the list refers to exactly one more than the word before it, and can immediately give any number she can count to. The literature calls this acquiring the cardinal principle. It is a single inductive leap, it takes most of a year to prepare, and it is the moment at which a human being first possesses the successor function.
Susan Carey's account, in The Origin of Concepts in 2009, is that the counting list works as a placeholder structure: a sequence of meaningless sounds with an order, which the child first anchors to the small exact system — one, two, three mapped onto one, two, three tracked objects — and then notices has a shape, the shape of always-one-more. The integers, on this view, are bootstrapped: neither present in the biology nor arbitrary, but induced from the interaction of an exact-but-tiny system, an approximate-but-unbounded one, and a culturally transmitted list of words that carries the pattern. Spelke's version gives language the central role, as the place where the two innate systems are combined. George Lakoff and Rafael Núñez, from the first essay, would say the pattern is a metaphor — motion along a path, mapped onto the collection of objects. Núñez went further in 2017 and argued that what the biology provides should not be called number at all: it is quantical, a sense of more and less, and numerical cognition — exact, discrete, symbolic — is a cultural product with no evolved basis.
The disagreement between these accounts is the found-or-made dispute conducted with children instead of philosophers, and it is not settled. What is settled is the shape of the problem. Something exact and unbounded gets built, in every child who learns to count and in every society that has invented counting, out of parts none of which is exact and unbounded. That is either the mind perceiving a structure that was there — the Platonist reading, and Gödel's — or the mind constructing one from a placeholder that happens to have the right form. Either way, the construction is real, it takes a year, and in some societies it never happens at all.
Brian Butterworth's work on dyscalculia adds one more datum. Around three to six per cent of children have a specific and lasting difficulty with number that is independent of general intelligence, reading and memory, and that is associated with differences in the parietal cortex, where the approximate system lives. The biological substrate is real enough to fail on its own. But what fails is the sense; the arithmetic that sits on top of it is what becomes hard to build.
When the world does the mathematics
The cognitive story pushes toward “made”. There is an equally concrete body of evidence pushing the other way, and it comes from asking not how minds do mathematics but why the world seems to.
Periodical cicadas of the genus Magicicada spend thirteen or seventeen years underground and then emerge together, by the billion, for a few weeks. Thirteen and seventeen are prime, and a family of explanations going back to the 1970s says that is why: a brood whose cycle is prime coincides with a predator, or with another brood, as rarely as arithmetic allows, because the lowest common multiple of a prime and anything smaller is as large as it can be. Whether this is the right explanation is contested — some models suggest the primes fall out of avoiding hybridisation between broods and would emerge from other mechanisms too. But the form of the explanation is what matters here. If it is right, then a fact about the distribution of insects in time is explained by a fact about the integers, and the philosopher Alan Baker argued in 2005 that this is a stronger reason to take numbers seriously than anything in the indispensability argument from the first essay: mathematics does not merely describe the cicadas, it accounts for them.
The honeycomb is the older example and the better one. Pappus of Alexandria wrote around AD 320 that bees, possessing a certain geometrical forethought, had chosen the hexagon because it holds the most honey for the least wax; Darwin called the comb the most wonderful of all known instincts. Whether the hexagonal grid is in fact the most economical way to divide a plane into equal cells was an open problem for roughly two thousand years, and Thomas Hales — the Kepler conjecture's Hales, between the referees and Flyspeck — proved it in 1999. The bees are right. They are not right because they know any geometry; they are right because selection has been solving an optimisation problem for a hundred million years, and the optimum was there to be found by selection exactly as it was there to be found by proof. Some of the hexagon may even be physics rather than biology: the cells are built round and settle into hexagons as the wax flows. That does not weaken the point. It means the structure was there for the wax, too.
The first essay said that structuralism dissolves the applicability problem: mathematics fits the world because the world has shape, and shape is what mathematics describes. The cicadas and the comb are what that looks like at ground level. The primes are not in the cicada and the hexagon is not in the bee. Both are in the situation — in the structure of the problem the organism faces — and any process that solves the problem, whether proof or selection or the surface tension of wax, ends up in the same place.
Would they have our arithmetic?
The Arecibo message, broadcast toward the globular cluster M13 in 1974, opens with the numbers one to ten in binary, on the assumption that any recipient would recognise them. Hans Freudenthal's Lincos, a language for cosmic intercourse designed in 1960, begins with arithmetic for the same reason. Carl Sagan's aliens announce themselves with primes. The assumption in all three is that arithmetic is the one thing any mind must share — that if there is a universal mathematics, counting is where it starts.
The evidence from this planet complicates that in a specific way. The sense of quantity does appear to be universal: it evolved independently in crows and monkeys, and any organism that has to choose between more and less will plausibly have something like it. But exact number is not universal even among humans; it is a technology, and most human societies across most of history did not build it. A species that evolved in a medium with few discrete objects to count — an ocean, say, where what matters is gradient and flow — might reach topology or the calculus of continuous change long before it reached the integers, and might find our starting with one, two, three as parochial as we would find a message that opened with a taxonomy of scents.
What such a species could not do is have different integers. Once it had built the successor function, by whatever route, seventeen would be prime and the division algebras would number three. The route and the vocabulary are contingent; the structures, once reached, are not. This is the same division of labour the series has arrived at four times now, and the alien case just makes it vivid: what is made is the way in, and what is found is what is there when you arrive.
The way in and what is there
Four essays, four places to look, and the same answer each time, with more texture each time.
The complex numbers were invented, and the invention had almost no room in it: three division algebras, then nothing. The axioms of arithmetic and set theory were chosen, and their consequences outrun any choosing — Goodstein's sequences terminate whether or not the axioms can say so — while some questions, like the size of the continuum, turn out not to be consequences at all and hand the choosing back. Proof, the machinery that turns choices into consequences, has split into a certificate a machine can check and an explanation only a mind can hold, and the theorems stand whichever face you look at. And the capacity for any of it is an approximate biological sense of quantity, shared with crows, on which an exact science has been built by a technology — the counting list — that most human societies never developed.
What is made: the number words, the axioms, the definitions, the notation, the route, the proofs as artefacts, the concept of a polyhedron. What is found: the consequences, the constraints, the well-ordering of the large cardinals, the three algebras, the hexagon, the fact that seventeen is prime. The structuralism the first essay settled on holds up under all four: mathematics is about shapes that the world instantiates, and we make the vocabulary and discover the shapes.
What none of the four explains — and both camps should keep admitting this — is three things. Why the constraints run as tight as they do, so that inventions keep turning out to be forced. Why an instrument that evolved to estimate berries should be able to build, in a year, a structure with no upper bound and no error. And why the shapes it builds keep turning out, centuries later, to be the ones the physics needed. A crow's number neurons and Cardano's cube roots are, on the reading this series has defended, two encounters with the same structure from very different distances. That the structure is there is the best-supported claim in the debate. Why it is there, and why we can reach it, remain what they were in 1545: a subtlety that is anything but useless, and not yet understood.
- Invented, Then Unavoidable — imaginary numbers, rigidity, and why physicists trust beauty.
- The Limits of the Rules — Gödel, Goodstein, the continuum, and what happens when the axioms run out.
- What a Proof Is Now — Lakatos, the four colour theorem, Lean, and proofs no one has read.
- Where Number Comes From — the number sense, cultures without counting, and whether aliens would share our arithmetic. This essay.
- Pappus, Collection, Book V, c. AD 320; Darwin, On the Origin of Species, ch. VII, 1859.
- Mechner, “Probability relations within response sequences under ratio reinforcement”, 1958; Platt & Johnson, 1971.
- Freudenthal, Lincos: Design of a Language for Cosmic Intercourse, 1960; Arecibo message, 1974.
- Wynn, “Children's understanding of counting”, Cognition, 1990; “Children's acquisition of the number words and the counting system”, 1992.
- Dehaene, The Number Sense, 1997; Butterworth, The Mathematical Brain, 1999.
- Xu & Spelke, “Large number discrimination in 6-month-old infants”, Cognition, 2000.
- Hales, “The honeycomb conjecture”, Discrete & Computational Geometry, 2001; Goles, Schulz & Markus, “Prime number selection of cycles in a predator–prey model”, 2001.
- Nieder & Miller, “Representation of the quantity of visual items in the primate prefrontal cortex”, Science, 2002.
- Gordon, “Numerical cognition without words”, Science, 2004; Pica, Lemer, Izard & Dehaene, “Exact and approximate arithmetic in an Amazonian indigene group”, Science, 2004.
- Baker, “Are there genuine mathematical explanations of physical phenomena?”, Mind, 2005; Lehmann-Ziebarth et al., “Evolution of periodicity in periodical cicadas”, Ecology, 2005.
- Frank, Everett, Fedorenko & Gibson, “Number as a cognitive technology”, Cognition, 2008; Dehaene, Izard, Spelke & Pica, “Log or linear?”, Science, 2008; Halberda, Mazzocco & Feigenson, Nature, 2008.
- Carey, The Origin of Concepts, 2009.
- Karihaloo, Zhang & Wang, “Honeybee combs: how the circular cells transform into rounded hexagons”, J. R. Soc. Interface, 2013.
- Ditz & Nieder, “Neurons selective to the number of visual items in the corvid songbird endbrain”, PNAS, 2015; Nieder, “The neuronal code for number”, Nature Reviews Neuroscience, 2016.
- Núñez, “Is there really an evolved capacity for number?”, Trends in Cognitive Sciences, 2017.
- Howard et al., “Numerical ordering of zero in honey bees”, Science, 2018.
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