The Bus at Coutances
Ask working mathematicians and physicists what is in their heads while they work, and almost none say words or equations. They say pictures, shapes, a sense of something moving. The symbols come afterwards. What that tells us about mathematics depends on what happens to the pictures when you take them away — and there are people to ask.
In the summer of 1880 Henri Poincaré spent a fortnight trying to prove that a certain class of functions could not exist. Every evening he sat down, tried combinations, and got nowhere. One night he drank black coffee, could not sleep, and felt the ideas “rise in crowds” and collide until pairs of them locked together; by morning he had established the existence of the functions he had been trying to rule out. Then he left Caen on a geological excursion, forgot about mathematics, and at Coutances put his foot on the step of an omnibus. “At the moment when I put my foot on the step,” he wrote, “the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry.” He did not verify it. He went on with the conversation he had been having. He was certain.
Poincaré told this story in a lecture to psychologists in 1908, and it has been the standard anecdote about mathematical thinking ever since. It is easy to read as a story about genius, and it is not; it is a story about mechanism. It says that the work was done somewhere he could not see, that it surfaced as a shape rather than a sentence, and that the feeling of certainty arrived before the checking. Four essays in this series have asked whether mathematics is found or made by looking at its history, its rules, its proofs, and its biological roots. This one looks at the process from inside — at what researchers say happens in their heads, at what the laboratory has been able to confirm, and at the awkward evidence from people who do mathematics with no pictures at all.
What they say it feels like
In 1945 Jacques Hadamard, who had proved the prime number theorem half a century earlier, published a short book called The Psychology of Invention in the Mathematical Field. Much of it is a survey. He had written to mathematicians and physicists in America asking, in effect, what was in their minds while they worked, and the answers were unexpectedly uniform: almost nobody thought in words. The mental furniture was images — vague, spatial, sometimes kinetic — and words were fetched at the end, to communicate.
The most famous reply was Einstein's. “The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought,” he wrote. The elements were “certain signs and more or less clear images which can be voluntarily reproduced and combined”, and this “combinatory play” was the essential feature of productive thought; the elements were, in his case, “of visual and some of muscular type”, and conventional words had to be sought “laboriously” in a second stage.
Hadamard turned the question on himself and reported what he saw when he followed Euclid's proof that there are infinitely many primes. He did not see the proof. He saw a confused mass (all the primes up to some point), then a second point some way off (the product of them all), then a point between the two (the product plus one) — a sequence of spots and distances that carried the logic without containing it. He called these images vague and insisted they were indispensable: they kept the whole argument in view at once, which the words could not do.
Poincaré's own lecture had already supplied the other half. Invention, he said, has stages: conscious preparation, which fails; an interval of unconscious work, which is invisible; a sudden illumination, which feels certain; and a period of conscious verification, which is necessary because the illumination is sometimes wrong. And the unconscious does not try everything — it selects, and what it selects by is an aesthetic feeling, the sense of harmony and elegance that is the subject of the first essay in this series. Beauty, on Poincaré's account, is not a judgment made about finished mathematics; it is the filter that decides which combinations reach consciousness at all.
Seven ways to think of one thing
The pictures are not one picture. In the 1994 essay quoted in the third part of this series, William Thurston listed the ways he could think about a derivative, and the list is worth reproducing because it is the clearest statement in print of what mathematical understanding consists of. The derivative is: (1) infinitesimal, the ratio of a tiny change in output to a tiny change in input; (2) symbolic, the thing you get by applying the rules for polynomials and exponentials and compositions; (3) logical, the epsilon-delta condition; (4) geometric, the slope of the tangent line; (5) a rate, the instantaneous speed; (6) an approximation, the best linear approximation to the function near a point; (7) microscopic, the limit of what you see as you magnify the graph. He added, as item thirty-seven, a definition in terms of Lagrangian sections of the cotangent bundle, which is what the derivative looks like from a great height.
These are not seven descriptions of one concept, Thurston said; they are seven different ways of thinking, each carrying different intuitions and different possible moves, and a mathematician who understands derivatives is one who can switch between them without noticing. The list explains something about why mathematics is hard to teach and hard to talk about: the words “the derivative” are the label on the whole bundle, and the bundle is what the expert has.
Terence Tao describes the acquisition of this in three stages. In the pre-rigorous stage one works by intuition, examples and hand-waving, and is often wrong. In the rigorous stage, imposed by graduate training, one learns to distrust intuition and prove everything, and is slow. In the post-rigorous stage, which not everyone reaches, the intuition has been rebuilt on top of the rigour and one can again move fast, because the pictures now know where the traps are. The goal of the middle stage, on this account, is not to replace the pictures but to make them trustworthy.
Two more images from working mathematicians describe the same thing from further away. Andrew Wiles, on the seven years he spent on Fermat's Last Theorem: you enter a dark mansion, feel your way around the furniture of the first room for months, eventually find the light switch, and then move to the next dark room. Alexander Grothendieck, on his own method: there is the approach to a nut that takes a hammer and chisel to it, and there is the approach that puts the nut in water and waits, weeks or months, until the shell softens and opens under the pressure of a hand — the rising sea, in which the theory is raised until the problem is submerged and dissolves. Both are descriptions of what it is like to be in the incubation stage for a very long time.
Birds and frogs
The same subject, approached by different people, is approached differently, and the difference has been noticed often enough to be real. Poincaré divided mathematicians into logicians and intuitives, and named names: Weierstrass and Méray on one side, Riemann and Klein on the other. Timothy Gowers, in a 2000 essay called “The Two Cultures of Mathematics”, drew the line between theory-builders, for whom the point is the structure and the problems are its test cases, and problem-solvers, for whom the point is the problem and the theory is a tool. Freeman Dyson, in 2009, called them birds and frogs: birds fly high and see the connections between distant parts of the landscape; frogs live in the mud and see the details of one pond.
The division is a caricature, and everyone who uses it says so. Its interest is that it survives. It is stable across a century, across countries and subfields, and across people describing themselves and describing others. Whether it is a cognitive difference — something like the visual-versus-verbal split that Hadamard's survey found not to exist — or a cultural one, learned from a supervisor and reinforced by a subfield, nobody has established. Gowers's essay makes a point that is easy to miss: the two cultures do not merely differ in taste, they differ in what they think mathematics is, with theory-builders inclined to see it as a landscape to be mapped and problem-solvers as a set of things to be done. It is the found-or-made argument, conducted in temperament.
The picture before the equation
Physics has kept a more honest record than mathematics of the picture coming first, because physicists have less reason to hide it. Michael Faraday had no mathematics beyond arithmetic, and thought about electricity and magnetism entirely in terms of lines of force filling space, which he could see and could draw. When Maxwell set out in the 1860s to make the theory mathematical he began by translating Faraday's lines into equations, and wrote in the preface to his Treatise that as he did so he perceived that Faraday's way of conceiving the phenomena “was also a mathematical one, though not exhibited in the conventional form of mathematical symbols”. The field, the central concept of physics for the next century, was a picture before it was a formalism.
Einstein's habit of thinking in staged scenes is well known: chasing a beam of light at sixteen and asking what he would see; the man falling from a roof who feels no gravity, which he called the happiest thought of his life; the lift, the train, the clocks. What is less often said is that these were not illustrations for the public. They were the working method, and the letter to Hadamard describes the same thing from inside: images, muscular as well as visual, combined in play, with the equations sought afterwards and with effort.
Richard Feynman's diagrams are the case in which the private picture became the public formalism. He introduced them at a meeting in the Poconos in 1948 as a way of keeping track of the terms in a calculation that was otherwise unmanageable, and they were at first understood by almost nobody; it took Freeman Dyson, the following year, to show that they were equivalent to the more respectable algebra of Schwinger and Tomonaga. David Kaiser's history of what happened next, Drawing Theories Apart, is a study of how a way of thinking travels: the diagrams spread not through the papers but through people, postdocs who had learned them at Feynman's or Dyson's elbow and carried them to the next department. A picture that had been one man's scratch-work became, within a decade, the language in which a field thought. The diagrams deserve more than a paragraph, and a companion essay gives them one: what they compute, what they falsely suggest, and the discovery since 1986 that they were hiding a simpler object all along.
The same happened to spacetime. Minkowski's 1908 lecture turned special relativity into geometry — a diagram with a light-cone in it — and Einstein, who had initially dismissed the reformulation as superfluous learnedness, needed exactly that geometry for general relativity. Half a century later Roger Penrose drew diagrams that fold an infinite spacetime onto a finite page, and they became the standard tool for reasoning about black holes and horizons. In each case the picture was not a summary of the mathematics. It was the place the mathematics was done.
What can be measured
Most of this is testimony, and testimony about one's own mind is unreliable in known ways. Some of it, though, has been checked.
Incubation is real. Poincaré's bus is an anecdote; the effect is not. A 2009 meta-analysis by Ut Na Sio and Thomas Ormerod, pooling more than a hundred experiments, found that setting a problem aside and returning to it produces a reliable improvement over continuous work, that the improvement is larger for problems on which the solver has reached an impasse, and that it is larger still when the interval is filled with an undemanding task rather than with rest or with hard work on something else. A 2004 study by Ullrich Wagner and colleagues found that a night's sleep more than doubled the proportion of subjects who noticed a hidden shortcut in a numerical task they had practised the evening before. Nobody has watched the unconscious work. Its output has been counted.
Expertise changes perception, not just knowledge. Adriaan de Groot, in studies begun in the 1940s, showed chess masters positions for a few seconds and asked them to reconstruct the board; they did so almost perfectly, where novices managed a handful of pieces. Chase and Simon showed in 1973 that the advantage disappears for random arrangements of pieces: the master is not remembering more, but perceiving the board in larger meaningful units — chunks — and there are only a few of them to remember. The same holds for mathematicians reading formulae and physicists reading diagrams. What the expert sees when looking at a page is different from what the novice sees, and it is that difference, not any difference in reasoning, that carries most of the skill. Thurston's seven derivatives are seven chunkings.
Imagery is a format, not a metaphor. Roger Shepard and Jacqueline Metzler, in 1971, showed subjects pairs of drawings of three-dimensional block shapes and asked whether they were the same object rotated. The time taken to answer rose in a straight line with the angle of rotation, at a rate of about sixty degrees per second. People were not inferring the answer; they were turning the object in their heads, at a speed, and the turning took time in proportion to the distance. Whatever mental images are, they behave like things that occupy space.
The body is in the loop. Susan Goldin-Meadow's work on gesture found that children solving arithmetic problems frequently produce, with their hands, a strategy that is correct while the strategy they state aloud is wrong, and that this mismatch predicts which children are about to learn. Adult mathematicians gesture constantly when explaining, and the gestures encode the spatial content — the direction of a map, the collapsing of a dimension — that the words leave out. Einstein's “muscular” elements have a literature.
Mathematics without pictures
If the testimony is right and mathematical thought is made of images, then people without images should not be able to do it. They can, and this is the most useful fact in the whole subject.
In 2015 the neurologist Adam Zeman and colleagues described a group of people who reported having no visual imagery at all — who could not picture a face, a beach, or the room they had just left — and gave the condition a name, aphantasia. It is lifelong, it affects perhaps two to four per cent of people, and most of them had assumed until adulthood that “picturing” something was a figure of speech. A follow-up study in 2020 with several thousand respondents found that aphantasics are, if anything, over-represented in scientific and mathematical occupations, and people with unusually vivid imagery over-represented in the arts. Whatever Einstein's “more or less clear images” were, a meaningful fraction of working scientists do not have them and work anyway.
The older evidence is from mathematicians who were blind. Nicholas Saunderson, blind from infancy, held Newton's chair at Cambridge in the 1710s and taught optics. Leonhard Euler lost his sight and produced roughly half his work afterwards, dictating. Lev Pontryagin was blinded at fourteen and became one of the great topologists of the twentieth century. Bernard Morin, blind from the age of six, found in the 1970s the crucial intermediate stage in turning a sphere inside out — a problem so resistant to visualisation that when Stephen Smale proved it could be done, in 1958, nobody could at first see how. Allyn Jackson's 2002 survey of blind mathematicians noticed that they cluster, disproportionately, in geometry. Morin's own explanation was that sighted people are misled by the surface of things, and that his mental model of a shape was built from the inside, all at once, rather than from a series of views.
Put the two facts together and the testimony has to be re-read. What Hadamard's respondents called images, and what Einstein called visual, is evidently not tied to the visual system, since it survives the loss of that system and its absence in people who never had it. It looks instead like something spatial and structural — a representation of things and their relations that can be reported as a picture by people who have pictures, and reported some other way by people who do not. Morin's account of the sphere is the clearest description available of that underlying format, precisely because he had no picture to confuse it with.
Whether a picture can be a proof
Mathematics has argued with itself about this for as long as it has existed. Euclid's diagrams are part of his proofs, not decoration; several of his arguments are invalid without the figure, and it took until the nineteenth century for anyone to mind. Then the nineteenth century minded a great deal. The programme associated with Weierstrass replaced geometric intuition in analysis with epsilons and deltas, partly because the intuition had produced false theorems — every continuous function had “obviously” been differentiable almost everywhere, until Weierstrass wrote down one that was differentiable nowhere. The lesson drawn was that the picture lies.
The reaction had a reaction. Hilbert, whose formalism this series met in its second part, published in 1932 with Stefan Cohn-Vossen a book called Anschauliche Geometrie, “intuitive geometry”, that was almost entirely pictures, and said in the preface that intuition had been unjustly neglected. Bourbaki, a generation later, produced thousands of pages containing practically no diagrams at all. Roger Nelsen's Proofs Without Words, from 1993, collects arguments that are nothing but diagrams and lets the reader decide whether they are proofs. And in the last fifty years a strand of mathematics has settled the question by fiat: the commutative diagrams of category theory, and the string diagrams that grew out of them, are not pictures of the rigorous objects. They are the rigorous objects, with a formal grammar and a theorem saying that manipulating the picture is a valid inference.
The pattern is the one from the third essay, seen from the other side. Proof as certificate distrusts the picture, for good reason: pictures have lied. Proof as explanation cannot do without it, for equally good reason: nobody understands a page of epsilons. The formalisms of the last century were built to be picture-proof, and the working mathematician's response has been to keep the pictures and use the formalism to check them — Tao's post-rigorous stage, institutionalised.
The shape before the sentence
For the question this series is about, the evidence from inside the head points, with unusual consistency, in one direction. Mathematicians and physicists report thinking in structures — spatial, relational, kinetic — before and beneath the symbols. The laboratory confirms that imagery behaves like a genuine format, that expertise is a change in what one perceives, and that the unconscious does measurable work between sessions. And the people who lack visual imagery altogether, or lost their sight, show that the underlying format is not visual but structural: Morin turned the sphere inside out without ever having seen one.
That is structuralism's home ground. If what the mind grasps first is shape and relation, and the symbols are a made vocabulary fetched afterwards to fix and communicate what was grasped, then the phenomenology matches the metaphysics the first essay settled on: the structures are found, the language for them is made, and the recurrent report of “finding” is the experience of the first stage, before the second has begun. The constructivist can still say the structures are built by the mind rather than met by it. What the constructivist can no longer easily say is that mathematics is the symbol-game, because the people who play it best report that the symbols are the last thing to arrive.
It also complicates the third essay. A Lean certificate is exactly the object Hadamard's respondents said they did not think with. If the certificate becomes the only thing that counts, mathematics will have been formally verified into a shape that none of its practitioners recognise as thinking. Thurston's worry, restated: the danger is not machines that prove, but a practice that forgets the rooms in the dark mansion were found by feel.
There is a practical residue, and Poincaré and Pólya and Tao all give the same advice, which is a sign that it is true. Keep several representations of anything you are working on, and switch. Keep examples in your pocket, since the picture is built from them. Work hard, then stop; the effect of stopping is measurable. And distrust the certainty on the step of the bus just enough to check it afterwards, because the same mechanism that found the Fuchsian functions has, in every mathematician's life, also found things that were not there.
Where this could go
The brain doing mathematics. Marie Amalric and Stanislas Dehaene put professional mathematicians in a scanner in 2016 and found that high-level mathematical statements — in algebra, analysis, topology — activate the same parietal and prefrontal regions as elementary number and space, and not the language areas. That is the fourth essay's number sense and this essay's spatial format meeting in one result, and it has barely been followed up. A piece on what the imaging actually shows, and what it cannot, would close the loop between Parts IV and V.
Aphantasia, surveyed properly. Zeman's 2020 finding that aphantasics lean toward science is a self-selected sample. Nobody has systematically asked mathematicians about imagery, subfield and working style together. It would be a cheap study and it would test the birds-and-frogs hypothesis, Hadamard's survey, and Morin's geometry claim at once.
Notation as a tool of thought. Leibniz designed his calculus notation so that the symbols would do the thinking; Kenneth Iverson's 1979 Turing lecture made the same argument for programming languages. A history of what good notation has made easy and what bad notation has made invisible — Roman numerals, Newton's dots against Leibniz's d, Dirac's brackets, Feynman's diagrams, Penrose's tensor pictures — would be the made half of this essay's story told on its own. The Feynman-diagram essay is a first instalment.
The process, observed. Almost everything here is retrospective testimony. Leone Burton interviewed seventy research mathematicians in the early 2000s about how they worked; a handful of studies have used diaries or think-aloud protocols. A modern version — instrumented, over months, with the blackboard photographed daily — does not exist, and the tools to do it now do.
Machines that prove without pictures. The AI provers of the third essay have no imagery and no incubation. They are, in this essay's terms, pure symbol-players, and they now reach gold-medal level. Either the pictures were never necessary, or the machines are doing something structurally equivalent in a form we do not recognise, or the problems they solve are the ones for which pictures were dispensable. Each of the three would say something different about what mathematics is, and the experiment that separates them has not been designed.
- 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.
- The Bus at Coutances — how mathematicians and physicists actually think: images, incubation, birds and frogs, and minds without pictures. This essay.
- Poincaré, “L'invention mathématique”, in Science et méthode, 1908; “L'intuition et la logique en mathématiques”, in La valeur de la science, 1905.
- Maxwell, A Treatise on Electricity and Magnetism, preface, 1873.
- Minkowski, “Raum und Zeit”, 1908.
- Hilbert & Cohn-Vossen, Anschauliche Geometrie, 1932.
- Hadamard, The Psychology of Invention in the Mathematical Field, 1945 (including Einstein's letter).
- Pólya, How to Solve It, 1945.
- De Groot, Thought and Choice in Chess, 1946 (English edition 1965); Chase & Simon, “Perception in chess”, Cognitive Psychology, 1973.
- Einstein, “Autobiographical Notes”, 1949.
- Shepard & Metzler, “Mental rotation of three-dimensional objects”, Science, 1971.
- Grothendieck, Récoltes et semailles, 1985–86; McLarty, “The rising sea”, 2003.
- Church & Goldin-Meadow, “The mismatch between gesture and speech as an index of transitional knowledge”, Cognition, 1986; Perry, Church & Goldin-Meadow, “Transitional knowledge in the acquisition of concepts”, 1988; Goldin-Meadow, Hearing Gesture, 2003.
- Nelsen, Proofs Without Words, 1993.
- Thurston, “On proof and progress in mathematics”, Bulletin of the AMS, 1994.
- Gowers, “The Two Cultures of Mathematics”, in Mathematics: Frontiers and Perspectives, 2000.
- Jackson, “The World of Blind Mathematicians”, Notices of the AMS, 2002.
- Wagner, Gais, Haider, Verleger & Born, “Sleep inspires insight”, Nature, 2004.
- Kaiser, Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics, 2005.
- Tao, “There's more to mathematics than rigour and proofs”, 2007.
- Dyson, “Birds and Frogs”, Notices of the AMS, 2009.
- Sio & Ormerod, “Does incubation enhance problem solving? A meta-analytic review”, Psychological Bulletin, 2009.
- Zeman, Dewar & Della Sala, “Lives without imagery — congenital aphantasia”, Cortex, 2015; Zeman et al., “Phantasia — the psychological significance of lifelong visual imagery vividness extremes”, Cortex, 2020.
- Amalric & Dehaene, “Origins of the brain networks for advanced mathematics in expert mathematicians”, PNAS, 2016.
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