The Birth of the Notation: Why Symbolic Innovation Precedes Experimentation

I. A claim about the order of events

There is a familiar picture of how a science advances. Someone performs a decisive experiment, or makes a decisive observation, and the world yields up a fact that the old theory cannot accommodate. The anomaly accumulates, the framework strains, and eventually a new theory replaces the old one, better fitted to what has been seen. On this picture the engine of scientific change is the encounter with nature. Experiment and observation are where the action is, and the symbolic apparatus of a science, its notations, diagrams, and formalisms, is the passive record in which the results of that encounter are written down. First the finding, then the writing.

This article defends a reversal of that order. The claim is that the major transformations in the history of science were enabled, again and again, not by a new encounter with nature but by a new way of symbolising, and that the symbolic innovation typically came first and made the decisive findings possible rather than merely recording them afterward. Alchemy did not become chemistry because someone finally looked hard enough at a reaction. It became chemistry when a notation was invented that could represent transformations of matter without the ordinary-language vocabulary of fire and spirit and affinity in which alchemy had been trapped. Anatomy was not revolutionised by the accumulation of more dissections but by a new form of visual representation that could carry the structure of the body in a way words never could. And the physics of Newton did not begin with the falling apple. It began, in a strict and documentable sense, with a new mathematical notation that Newton had to build before the physics he is remembered for became thinkable. The notation came first.

A thesis of this shape carries an obvious danger, and it is best to name it at once, because the value of the argument depends entirely on not succumbing to it. Every scientific advance involves symbols somewhere, since science is a symbolic practice through and through, and so it is trivially easy to point at any transformation and find a notation somewhere in its vicinity and declare the notation the cause. That would be an empty claim, a skeleton key that opens every door and therefore explains nothing. The argument here is not that. It is a claim about a specific and identifiable kind of event, the innovation of a new symbolic technology that does a particular kind of work, and about the ordering and dependence relations between that event and the discoveries that follow it. It is a claim that can be wrong, and I will say plainly what would make it wrong. If the major transformations of a science regularly occurred without any prior or concurrent innovation in its symbolic technology, drawing only on notations already long available, the thesis would be refuted. The interest of the thesis lies precisely in the fact that this does not seem to be what happened.

The argument draws on an account of symbolisation that must be sketched before the cases can be read, because the cases are unintelligible without it. That account holds that symbolic systems are not all of one kind, and in particular that there is a deep difference between ordinary language and the specialised notations that mature sciences construct. Ordinary language is permanently coupled to lived experience, drawing its meaning from the four registers of embodied, social, material, and dwelt coordination in which human life is conducted, and it can never fully detach from them. This coupling is its great strength for the conduct of life and its great weakness for a certain kind of precise inference, because its terms carry the whole sediment of lived association, the connotations and analogies and evaluative tilts that make them rich and make them imprecise. A specialised notation constructs a symbolic domain whose terms mean what they mean by relations internal to the system rather than by their coupling to lived experience, and thereby buys a kind of inferential power ordinary language cannot sustain. This process, the construction of a symbolic domain whose inference no longer depends on the lived associations of ordinary language, I will call demediation, and the central claim of this article is that scientific revolutions are, very often, demediation events. Demediation, it will emerge, is necessary for the transformative power this article describes but not, by itself, sufficient, and two further conditions complete the account.

II. What a notation does that an experiment cannot

To see why symbolic innovation should have the priority claimed for it, one has to see what a new notation does, and to see that one has to give up the idea that a notation is a mere recording device. A recording device is neutral with respect to what it records. It adds nothing and subtracts nothing; it simply preserves. If notations were like this, they could not be the engines of anything, and the traditional picture in which they passively register the results of experiment would be correct. But notations are not neutral, and their non-neutrality is the whole of the matter.

A notation does three things that no experiment and no observation can do, and each of them is a precondition of the kind of inference that a science needs. The first is that a notation selects. It fixes what shall count as a variable and what shall be ignored, what dimensions of a phenomenon shall be representable and what dimensions shall drop out of view. A notation is a decision about what matters, made in advance of any particular use, and built into the very form of the symbols. When a phenomenon is brought under a notation, it is stripped of everything the notation cannot represent, and this stripping, far from being a loss, is the condition of the notation's power, because it is only by ignoring almost everything that a notation can render the remainder tractable. The second thing a notation does is that it makes its objects manipulable. Once a phenomenon is represented in a notation, one can operate on the representation by the internal rules of the notation, transforming, combining, and deriving, and these operations proceed within the symbolic domain without any further reference to the phenomenon. This is what allows inference to run ahead of observation, to reach conclusions that no one has seen and that the notation nonetheless licenses. The third thing a notation does is that it makes its results transmissible and stable, so that they can be carried across persons and across time without the degradation that afflicts anything communicated in the loose and lived medium of ordinary language.

Put these three together and the priority of notation over experiment becomes intelligible rather than paradoxical. An experiment can only tell you about the phenomenon under the description you bring to it, and the description is supplied by the available symbolic technology. Where the symbolic technology is impoverished, the experiment can yield only impoverished findings, because there is no way to represent what would be needed to see more. The reason a science can look at a phenomenon for centuries and see nothing decisive, and then, quite suddenly, see everything, is very often that a new notation has arrived that permits the phenomenon to be represented in a way that makes the decisive relations visible and manipulable for the first time. The phenomenon did not change. The notation changed, and with it what could be seen in the phenomenon and inferred from it. This is why the notation can come first. It is not that the notation predicts the finding before the finding is made. It is that the notation constitutes the space within which that kind of finding can be made at all, and until the space exists the finding is not merely undiscovered but unthinkable.

There is a further point, and it is the one that most sharply distinguishes this account from the traditional picture. The impoverishment that a new notation overcomes is very often an impoverishment imposed by ordinary language itself. A pre-scientific field is typically not a field without symbols. It is a field whose symbols are those of ordinary language, and whose theorising is therefore conducted in terms that carry the whole load of lived association, terms like fire and spirit and humour and force and melancholy, each of which means what it means by its coupling to embodied, social, material, and dwelt experience. Such a field is not held back by a lack of observation. It is held back by the coupling of its vocabulary to lived experience, which is exactly the coupling that makes the vocabulary unfit for the kind of internally governed inference that a mature science requires. The decisive innovation, when it comes, is not a new fact. It is a notation that cuts the vocabulary free of its lived coupling, that replaces the ordinary-language term saturated with association by a symbol that means what it means by its place in a formal system. That is demediation, and the history of science is in large part the history of one field after another achieving it.

There is a way of putting the three achievements of the previous section together that will matter for everything that follows. To select, to make manipulable, and to make transmissible is, at bottom, to hold a phenomenon still. A notation is a stabilisation in advance, a fixing of the object into a determinate form that can be read at once, operated on by rules internal to the system, and carried unchanged across persons and time. This is what a notation does that mere description cannot, and it raises two further questions that the historical cases, on their own, cannot answer. What kind of symbol is capable of holding a phenomenon still, as opposed to merely describing or recounting it? And what kind of phenomenon can be held still at all, without the holding falsifying what it holds? The next two sections take up these questions in turn, because the cases that follow, chemistry, anatomy, Newton, and finally the hard case of psychiatry, cannot be properly read without them.

III. Two kinds of symbol: narration and notation

A symbol system, in the broadest sense of the term, does one of two things. It describes, or it coordinates. These are not two degrees of the same activity but two different relations to what is symbolised, and the difference between them is the difference between narration and notation.

A descriptive or narrative symbol depicts or recounts. It refers outward to something and carries an account of it, and its elements are ordered by the order of the telling, by what came first and what follows, by what was seen and in what sequence. Ordinary language, in its normal use, is narrative in exactly this sense. A sentence unfolds in time, one word after another, and its meaning is assembled by a hearer or reader who follows the unfolding and reconstructs, from the sequence of words, the state of affairs the words are about. A photograph is narrative in the same structural sense, despite being visual rather than verbal. It is a trace of a single moment from a single vantage, coupled to what was in front of the lens at that instant, and it carries no internal grammar beyond what the eye already knows how to read. To describe a photograph in words or to look at it directly are, for the purposes of this argument, the same kind of symbolic act, an act of depiction or recounting that refers outward and stops there.

A coordinative or notational symbol does something categorically different. It does not merely refer outward to what it represents. It fixes the elements of a domain into a structure governed by relations internal to the notation, so that operating on the structure by the notation's own rules yields conclusions that were not already given in whatever prompted the notation to be built. A diagram is coordinative in this sense whenever its lines and positions are not a picture of how something looks but a schema of how its parts relate, a code that must be learned before the diagram can be read, and that licenses inference once it is learned. A page of mathematics is coordinative in the purest form the human mind has built, its symbols meaning what they mean entirely by their combinatorial relations to the other symbols of the system, answering to nothing but the rules of the system itself until the moment its conclusions are tested against the world. A musical score is coordinative for the same structural reason, despite bearing no representational resemblance to what it notates. A sequence of marks on a staff fixes pitch and duration and their relations to one another in a form that can be operated on, transposed, inverted, read by an instrument that has never heard the piece, and reconstructed identically by any reader who knows the code.

The distinction is not, it must be stressed, a distinction between the verbal and the visual, or between the informal and the formal. Ordinary language can be narrative or, in its rarer technical uses, can approach the coordinative, as when a legal contract fixes the meaning of its terms by internal cross-reference rather than by their ordinary sense, or when a piece of formal logic is written out in words that function exactly as the symbols of an algebra would. And the visual can be narrative, as the photograph is, or coordinative, as the diagram is, and the same image can shift from one register to the other depending on how it is used. What makes a symbol coordinative is not its medium but its structure, whether it fixes its elements into a system of internal relations that can be operated on, or whether it merely carries an account of something outward and leaves the relations to be reconstructed, case by case, by whoever follows the account.

This distinction reorganises the claim of this article. Demediation, the severing of a vocabulary from its coupling to lived experience, is a necessary condition for a symbol to become coordinative, because a term still governed by its lived associations cannot be fixed into a system of purely internal relations, its meaning keeps leaking back to the life-world that gave it sense. But demediation alone does not make a symbol coordinative. A narrative can be demediated, cut free of folk vocabulary and conducted in careful, technical, disciplined prose, and remain narrative in form, unfolding in time, referring outward, without ever becoming a structure that can be operated on by internal rules. Such a narrative is a real achievement, and it is not what this article means by a notation. The transformative power this article has been describing belongs to the further and rarer achievement in which a demediated vocabulary is also organised coordinatively, fixed into a system of internal relations rather than merely freed from lived coupling. Chemistry's formulae, anatomy's diagrams, and Newton's fluxions are demediated in the way the article's first sections describe, and they are coordinative in the way this section has just described, and it is the combination of the two that gives them their power. A vocabulary can achieve one without the other, and when it does, the transformative power does not follow.

IV. The general law: why coordinative fit requires nonrecursivity

Even a coordinative symbol, fixed into a structure of internal relations and answerable to nothing but its own rules, does not automatically do useful work in the world. A perfectly coherent notation can be built and can fail utterly to track anything real, a game whose moves are all legal and whose results mean nothing, because the domain it was built to represent does not have the structure the notation assumes. What a working notation needs, beyond its own internal coherence, is fit. Its internal relations must track relations the phenomenon actually has, at the scale where the phenomenon is engaged. I will call this mesocosmic fit, fit at the ordinary scale of bodies and processes and lives, neither the scale of the very small nor of the very large. Newton's notation had it. Chemistry's notation had it. A notation can be coherent without having it, and coherence without fit is the fate of every notation that mistakes its own internal consistency for an achievement about the world.

This yields the strong claim toward which this article has been building. An experimental science, in the transformative sense under discussion, requires as an ontological precondition that its central phenomena be nonrecursive, or be reducible without essential loss to nonrecursive components, because only nonrecursive phenomena can be held still by the coordinative notation an experimental science needs in order to select, manipulate, and predict. This is not a claim that recursive phenomena cannot be known, studied, or understood. The narrative disciplines, history, ethnography, clinical case work, know a great deal about exactly such phenomena, and know it well. It is a claim that the specific and rare achievement this article has been calling transformative discovery, the achievement of chemistry finding its formula and Newton finding his fluxions, depends on a fit that recursive phenomena cannot supply, whatever symbolic ingenuity is brought to bear on them.

Under what condition can fit be had at all. The answer is that mesocosmic fit is available only for phenomena of a particular kind, and unavailable, not as a matter of present difficulty but as a matter of what a notation and what such a phenomenon respectively are, for phenomena of another kind. The kind for which fit is available I will call nonrecursive. A nonrecursive phenomenon does not respond to being coordinated with, described, measured, or symbolised. Gravity, the composition of a mineral, the course of a blood vessel, the orbit of a planet, the rate at which a substance decays, these do what they do whether or not anyone represents them, and the representing does not reach back to alter them. A nonrecursive phenomenon holds still under symbolisation, coupled to the symbol in one direction only, from world to symbol and never back, and it is exactly this holding still that a coordinative notation requires, because a notation works by fixing its object into a determinate form, and only an object that does not move in response to being fixed will stay fitted once fixed.

Set against this are two further kinds of phenomenon for which fit is not available. A selfrecursive phenomenon responds to its own ongoing states, feeding back upon itself so that its later condition is shaped by its earlier one in a continuing loop, as a wound's inflammation heightens the sensitivity that then worsens the inflammation, or as a market's falling price triggers selling that drives the price further down. Such a phenomenon does not hold still, because it is always transforming itself, and a notation that fixes it into a state captures only a slice of a movement whose nature is to be moving. An interrecursive phenomenon is stranger still, arising where two or more systems respond to each other such that the situation they are jointly in is genuinely constituted, not merely influenced, by their responding, as a conversation is constituted by an exchange neither party could have scripted in advance. An interrecursive phenomenon has no determinate state waiting to be represented, because the state does not exist until the reciprocal exchange brings it into being, and the exchange includes the act of representing itself, so to symbolise the phenomenon is to intervene in the very coupling that constitutes it. selfrecursive and interrecursive phenomena are, for this reason, not merely hard to notate. They cannot be notated in the strict sense this article has defined, because a notation must hold its object still to operate on it, and holding still is precisely what these phenomena, by their nature, do not do and cannot be made to do by any refinement of the notation attempting to hold them.

This is a conceptual claim and not an empirical wager about the limits of present ingenuity, and it is worth being exact about why. A notation is a stabilisation in advance. A recursive phenomenon, selfrecursive or interrecursive, has no state prior to or independent of the ongoing process that constitutes it. To ask for a notation of such a phenomenon is therefore to ask for something to be stabilised in advance of a process whose defining feature is that it has not yet settled into a stable state, and could not be made to settle without ceasing to be the process in question. No future cleverness closes this gap, because the gap is not a gap in cleverness. It is the relation between freezing and moving, and one cannot freeze, without falsifying it, a thing whose being lies in not being frozen.

The claim needs one guard against overreach, because without it the claim would prove too much and would be false. A recursive process throws off nonrecursive residues, and these can be notated perfectly well. A conversation can be transcribed once it is over, and the transcript, the settled words in their settled order, is a coherent and often a useful notation. What the transcript notates, however, is not the conversation as recursive, the live and mutual constituting of each utterance by what preceded it, but the residue the conversation leaves behind once it has stopped being recursive, once the exchange has closed and the words have come to rest. Every recursive process leaves such residues, and a great deal of the coordinative symbolisation that surrounds recursive domains, case notes, transcripts, box scores, financial statements, is notation of exactly this residual kind, valuable and coherent and entirely compatible with the claim of this section, because what such notation fits is not the recursive process itself but its nonrecursive trace.

From this general law follows a test simple enough to be applied by anyone, in advance of building anything, to any phenomenon under investigation. If a good diagram of the phenomenon can be drawn, one that fixes its elements and their relations in a form that can be read at once and reasoned from, then the phenomenon is nonrecursive, or is being engaged through a residue that is. If no amount of skill or care produces a good diagram, that is powerful evidence, though not on a single failed attempt a proof, that the phenomenon is selfrecursive or interrecursive, and that no notation will succeed where the diagram has failed. The test is a heuristic, since a single failure invites trying harder before it counts as evidence of an unnotatable object, but a phenomenon that resists diagramming across many independent and skilled attempts over a long period, in the way the hard case of this article will show mental suffering to have resisted it, is a phenomenon the general law says should resist it.

There is a symbolic form adequate to such phenomena, and naming it prevents the general law from reading as a counsel of silence about them. This is narration, a form that unfolds in time and follows a process as it develops rather than fixing it into a simultaneous structure. Because narration is itself temporal it can follow a selfrecursive loop through its turns or an interrecursive exchange through its constitution without needing to freeze either one, though what it yields is not fit in the predictive sense but understanding through following, an adequacy that is real and that no notation of a recursive phenomenon could achieve, because such a notation does not exist to achieve it or fail at it. Recursive phenomena can be narrated, even where they cannot be notated, and the case history, the ethnographic account, the conversation followed rather than mined for its residues, are the mature symbolic technologies for engaging what a diagram cannot hold.

V. Alchemy into chemistry: the severing of the qualitative word

The transformation of alchemy into chemistry is the clearest case, because the before and after are so sharply distinguished by exactly the feature the thesis predicts, and because the traditional story of chemistry's birth so badly misdescribes what happened. The traditional story is that chemistry emerged when alchemy's mysticism and secrecy gave way to careful experiment and quantitative measurement, when the balance replaced the incantation. There is truth in this, but it locates the decisive change in the wrong place. Alchemists did experiment, sometimes with great care, and weighing was not unknown to them. What alchemy lacked was not experiment and not measurement. It was a notation.

Consider the vocabulary in which alchemy conducted its theorising. It spoke of the four elements, of the three principles, of spirits and souls and bodies, of the marriage of sulphur and mercury, of substances that were noble or base, volatile or fixed, hot or cold, moist or dry. Every one of these terms is an ordinary-language term, coupled to lived experience through and through. Hot and cold are felt on the skin; noble and base are borrowed from the register of social rank; spirit and soul from the register of animate life; marriage from the register of human union. The alchemist theorised transformation of matter in a vocabulary whose every word carried the sediment of embodied, social, and dwelt association, and this is precisely why alchemy could not become a science of matter, however much its practitioners observed and weighed. The vocabulary could not sustain the kind of inference that a science of matter requires, because its terms meant what they meant by their coupling to lived experience rather than by any relations internal to a system of matter. When an alchemist reasoned that a volatile spirit would flee the fire because it was of an airy and ascending nature, the reasoning ran on the lived associations of the words, on what spirit and airy and ascending mean in the life-world, and no discipline of observation could correct an inference conducted in such terms, because the terms themselves imported conclusions that had nothing to do with matter.

What broke this was a notation, and the notation did exactly the work the thesis predicts. The reform of chemical language in the late eighteenth century, and then decisively the introduction of a symbolic notation for the composition of substances in the early nineteenth, replaced the qualitative ordinary-language vocabulary with a system of symbols that meant what they meant by their place in the notation. A substance was no longer noble or base, airy or fixed. It was a determinate composition of elements represented by letters and proportions, and the transformations it could undergo were represented by operations on those symbols. The critical feature of the new notation is that its terms had been cut free of lived coupling. The symbol for an element did not carry the sediment of social rank or animate life or bodily sensation. It meant what it meant by its combinatorial relations with the other symbols of the system, by what it could combine with and in what proportions and with what conservation. This is demediation in the exact sense, and it is coordinative in the further sense the previous section has defined, since the symbols are fixed into a structure of internal relations rather than merely freed from lived association. The qualitative word, saturated with lived association, was severed, and in its place stood a symbol governed by relations internal to the notation.

Once this severing had occurred, the discoveries followed, and they followed because the notation made them possible and not the other way around. Conservation of mass, definite proportions, the systematic prediction of reactions, the whole edifice of compositional chemistry became thinkable only within the notation, because only within the notation could the relevant relations be represented and manipulated. It is not that chemists first discovered conservation of mass and then wrote it in symbols. It is that the symbols, by representing substances as determinate compositions that could be tracked through transformations, made the conservation visible as a feature of the representation, and thereby as a hypothesis about the world that could be tested and confirmed. The balance had been available for a very long time. What had been missing was the notation that made weighing into evidence for something, that gave the numbers on the balance a place in a system of inference. The traditional story credits the balance. The balance was necessary, but it was inert until the notation gave it something to mean. Alchemy had the experiments. It lacked the symbols that would let the experiments add up to a science, and when the symbols arrived the science arrived with them.

Chemistry's notation passes the test proposed above with room to spare. A reaction can be diagrammed, its reactants and products fixed on either side of an arrow, their proportions balanced, the whole transformation held still in a form that can be read at once and manipulated by the rules of the notation. This is possible because the composition of a substance is a nonrecursive fact. A quantity of iron does not alter its combining proportions because a chemist has represented them, and this is exactly the condition the general law requires. Alchemy's qualitative vocabulary failed not only because it remained coupled to lived association but because, even had it been refined indefinitely, spirit and affinity are not terms that fix into a diagram; they narrate a transformation without coordinating it. The chemical formula succeeded because its object cooperated, holding still under a notation built to hold it, and this is the deeper reason the balance, inert for so long, at last had something to mean.

VI. Anatomy and the coming of the decisive image

The case of anatomy shows that the symbolic innovation need not be a formal notation of the algebraic kind, and that the argument is not a claim about mathematics in particular but about symbolisation in general. Anatomy was transformed, in the sixteenth century, by an innovation in visual representation, and the transformation illustrates the same priority of the symbolic over the observational, with a difference that is itself instructive.

Before the innovation, anatomical knowledge was carried very largely in words, in the descriptions handed down from antiquity and copied and recopied through the medieval period. Where images existed they were schematic, diagrammatic in a crude sense, often arranged to display a doctrine rather than a body, and they did not attempt to carry the actual spatial structure of what dissection revealed. The knowledge was, in the terms of this article, coupled to ordinary language and to the authority of transmitted texts, and this coupling had a remarkable consequence that is worth dwelling on, because it shows how a symbolic impoverishment can defeat even direct observation. Anatomists dissected, and they saw what was before them, and yet for a very long time they reconciled what they saw with the descriptions handed down, even where the two conflicted, because the transmitted word carried an authority that the seen body did not. Observation was subordinated to the available symbolic form, which was the authoritative text, and so dissection could proceed for generations without overturning the doctrine it contradicted. Here again the lesson is that observation is not decisive on its own. It is decisive only when there is a symbolic form capable of carrying what is observed against the weight of what has been said.

The innovation that changed this was a new kind of anatomical image, one that attempted to carry the spatial structure of the dissected body with a fidelity and a systematic completeness that words could not match and that earlier images had not attempted. The great anatomical works of the sixteenth century are remembered for their illustrations, and rightly, but the point is not that the illustrations were beautiful or that they popularised anatomy. The point is that the image is a symbolic technology with affordances that ordinary language lacks, and specifically that a well-constructed anatomical image can carry spatial relations, relative positions, and structural connections in a form that the eye can take in at once and that language can only enumerate slowly and partially. The image demediates in its own way. It cuts the representation of bodily structure free from the linear, enumerative, association-laden medium of words, and gives it a form governed by the spatial relations it depicts. A drawing of the course of a vessel through the body carries that course as a spatial fact, not as a sequence of words each of which must be coupled to lived meaning and reassembled by the reader into a structure. The structure is present in the representation.

With this new symbolic form available, observation was at last able to overturn authority, because there was now a way to carry what was seen in a form that could stand against what had been said. The decisive anatomical corrections of the period, the demonstration that the transmitted descriptions were wrong about this structure or that, were not the fruit of dissections more careful than any before. Dissections had been careful before. They were the fruit of a symbolic technology that could carry the results of dissection in a form that made the errors of the received account undeniable, because the error was now visible as a discrepancy between two images rather than negotiable as a matter of interpretation between the seen body and the authoritative word. The image made observation decisive. Before the image, observation had been endlessly subordinated to the text. This is the same pattern as in the chemical case, differing only in that the symbolic innovation was pictorial rather than algebraic. In both cases a field that had been observing for a long time without decisive result was transformed by a symbolic technology that gave observation something it had lacked, a form in which its results could be carried against the coupling of the old vocabulary to lived meaning and transmitted authority.

The anatomical case also brings out a distinction that matters for everything that follows, and the classification needs to be made carefully now that the difference between narration and notation has been set out. A photograph of a dissected body would be a narrative symbol in the strict sense of this article, a trace of one body from one angle at one moment, coupled outward to what was in front of the instrument and carrying no code beyond what the eye already knows. The great anatomical images of the sixteenth century are not this. They are idealised, labelled, and schematised, constructed precisely so that a vessel's course or a muscle's attachment can be traced and consulted and reasoned about as a general structural claim rather than looked at as a record of one cadaver. This is coordinative work in the sense this article has defined it, a fixing of bodily relations into a form governed by the conventions of the image, a code that must be learned, exactly as the conventions of a diagram must be learned, before the image can be read. The anatomical image passes the test proposed above. A good diagram of gross anatomical structure can be drawn, and could be drawn once the image achieved the necessary fidelity, because the course of a vessel and the attachment of a muscle are nonrecursive facts that hold still under representation, unaltered by having been depicted.

Where the anatomical image falls short of algebra is not, then, in being narrative rather than coordinative, but in the completeness of its demediation. It remains coupled to appearance in a way an algebraic symbol is not, because its coordinative conventions are built on top of a visual resemblance to the structure it depicts, and this partial coupling is a real limitation, though a different one from the limitation that kept alchemy narrative. Anatomy demediates from ordinary language and organises what it demediates coordinatively, and this combination, the joint condition of transformative power argued for above, is exactly what let anatomy revolutionise the description of the body. What anatomy did not achieve, and what the physical sciences went on to achieve, was a notation cut free even from the appearance of its objects, one whose elements mean what they mean by combinatorial relations alone and not by any residual resemblance to how a structure looks. To that further step of demediation the physical sciences owe their peculiar power, a point the next two sections develop.

VII. Newton, or the notation before the physics

The strongest case for the thesis is Newton's, because in his work the priority of the symbolic innovation over the physical discovery is not a matter of interpretation but of chronology and of his own testimony. The physics for which Newton is remembered, the account of motion and gravitation that reorganised the heavens and the earth under a single set of laws, was not the first thing he did. The first thing he did was to build a new mathematical notation, a method for representing and operating on quantities that change continuously and on the rates at which they change, and it was only with this notation in hand that the physics became possible. The notation came first, and Newton knew it came first, and the physics is in a precise sense the application of the notation to the phenomena of motion.

Consider what the physics of continuous motion requires. It requires that one be able to represent a quantity that is changing at every instant, and the rate of that change, which is itself changing at every instant, and to reason rigorously about the relations between the two. This is exactly what the older mathematics could not do. It could represent static quantities and their ratios, and handle certain curves and areas by ingenious specialised methods, but it had no general symbolic technology for the continuously changing quantity and its instantaneous rate of change. Without that technology, the physics of motion could not be formulated, because the laws of motion are relations among quantities the available notation could not represent. No amount of observation of falling bodies could yield the laws, because bodies had been falling, and being watched as they fell, for the whole of human history. What was missing was the notation in which the law of their falling could be written, and until that notation existed the law could not be found, however long the falling was observed.

So Newton built the notation. His method of fluxions, his representation of changing quantities and their rates of change and the operations relating them, is a symbolic technology of exactly the kind this article describes, a system whose terms mean what they mean by their place in the notation and whose operations proceed by rules internal to the system. And with this technology in hand, the physics of motion became not merely possible but almost inevitable, because the phenomena of motion, once represented in the notation, revealed their lawful relations as features of the representation that could be manipulated and derived. The famous derivations of Newton's physics were, in the finding of them, operations within the notation, whatever form he later gave them for publication. The apple, if there was an apple, was inert without the notation. It became evidence for a law only within a symbolic technology that could represent the law, and that technology Newton had to build before the law could be stated. The notation came first, and the physics is what the notation made thinkable.

One precise qualification belongs here, because it sharpens the claim rather than weakening it. When Newton came to publish the Principia in 1687, he recast most of his proofs in the classical geometric idiom of the ancients rather than presenting them in fluxional notation itself, judging that his contemporaries would find geometry more legible than a calculus almost no one else yet knew. The published demonstrations are not, on the page, operations conducted in fluxions. But the qualification changes nothing about priority. Newton needed the fluxional method to find the results before he could translate them into a geometry capable of demonstrating them to others, and a result reached through one notation and re-expressed in another for an audience's sake is still a result that the first notation, and not the second, made discoverable. The geometry of the Principia proves what the calculus first found.

The Newtonian case also refutes a natural objection to the thesis, that notations are developed to express discoveries already made, so the discovery is really prior and the notation merely its convenient dress. This objection has the order of events exactly backward in Newton's case. He did not first discover the laws of motion and then cast about for a notation to express them. He developed the notation as a piece of mathematics, driven by mathematical problems about curves and rates and areas, and the physics came after, as the application of the mathematical technology to the phenomena of motion. The notation was not the dress of a discovery already made. It was the instrument without which the discovery could not have been made, developed before the discovery and for reasons partly independent of it. A science waiting for its revolution is very often a science waiting for a notation that has not yet been built, and when the notation is built, whether within the science or imported from the mathematics, the revolution follows.

Newton's case also passes the test in its purest form, and shows why the test and the priority thesis are not two separate claims but two faces of one. A body's continuously changing position and velocity can be diagrammed, plotted as a curve against time, and the notation of fluxions is, in effect, a general method for drawing and manipulating exactly this kind of diagram in symbolic rather than pictorial form. The motion of a nonrecursive body holds still enough, instant by instant, to be fixed into such a diagram without falsification, and this is precisely the condition that let the notation, once built, achieve the fit that made the physics follow. Had the objects of Newton's physics been recursive, responsive to their own description in the way a mind or a market is, no fluxional notation, however ingeniously constructed, could have achieved what Newton's achieved, and the case would belong with the hard case this article turns to at its end rather than with its triumphs.

VIII. The general pattern in the physical sciences

What holds for Newton holds, with variations, across the history of the physical sciences, and the recurrence of the pattern is itself part of the evidence for the thesis, because a pattern that repeats across many independent cases is harder to dismiss as coincidence than any single case. The physical sciences are, more than any other domain of knowledge, a sequence of notational innovations, each of which opened a space of inference that had not existed before, and each of which was followed by a wave of discovery that the notation had made possible.

The representation of geometric relations in algebraic form, the coordinate method that allowed spatial configurations to be written as equations and equations to be read as spatial configurations, was a notational innovation of the first magnitude, transforming what could be done with space and motion by bringing algebra to bear on geometry and geometry on algebra. This was not a discovery about space. It was a new way of symbolising space, and the discoveries followed from the symbolisation. The later development of notations for fields, for the continuous distribution of quantities through space and their variation, opened the physics of electricity and magnetism and eventually the field theories that dominate modern physics, again with the notation as the condition of the physics rather than its record. The representation of physical processes by particular kinds of diagram, systems of lines and vertices governed by strict rules of construction, gave twentieth-century physics a technology for calculating interactions that would have been intractable in any prior notation, and the diagrams were not illustrations of a physics already worked out but instruments within which the physics was worked out. A notation is built. It opens a space of representation and manipulation that did not exist before. And within that space, discoveries are made that could not have been made without it.

It is worth pausing on what these physical notations have in common, because it identifies the feature that gives the physical sciences their peculiar power and connects back to the anatomical case. Each represents its objects not by their appearance but by relations no perception could deliver. A field is not something that looks like anything; it is a distribution of quantity through space, representable in the notation but not available to the eye as a dissected body is. The notations of physics achieve a demediation more complete than the anatomical image, cutting free not only from ordinary language but from perception itself, answering in the end to measurement and prediction but not to how anything looks or feels. This is the deepest form of demediation the thesis concerns, and it is no accident that the sciences that achieved it have the greatest predictive power. A science that represents its objects by relations cut free of appearance has opened a symbolic domain whose inferential reach is limited only by the notation and the measurements that anchor it. The power of physics is the power of its notations, and the history of physics is the history of their construction.

Each of the phenomena these notations track, quantities changing continuously in space, distributions of field strength, the interactions of particles, is nonrecursive in the sense this article has defined, indifferent to being represented, holding still for a notation built to hold it. This is not incidental to the power of physics. It is the condition of that power, and it is why physics, of all the experimental sciences, could push demediation to its limit, cutting its notations free not only from ordinary language but from perception itself, because its objects never resisted the freezing that a notation performs. A science whose objects resisted that freezing could not have followed the same path however far its notations were pushed, and the sciences that have tried are the subject of the section that follows.

IX. The hard case: neuroscience, psychiatry, and the notation that cannot come

If the thesis is right, it makes a prediction that is uncomfortable and testable, and the honest course is to state it as a prediction and then examine the evidence with the confounds fully in view, rather than presenting a favourable reading as though it settled the matter. The prediction is this. Where a field cannot construct or import a new symbolic technology adequate to its objects, it will not make substantial discoveries of the transformative kind, however much it observes and however much it accumulates, because it will remain trapped in the coupling of its vocabulary to lived experience, unable to achieve the demediation on which transformative discovery depends. A field without a notational revolution should be a field without a scientific revolution. And the comparison that most sharply tests this prediction is the comparison between neuroscience and psychiatry, two fields concerned with overlapping objects, the one having advanced dramatically over the past century and the other having advanced, by the measure of transformative discovery, remarkably little.

Consider first the confounds, because a responsible use of the case requires that they be named before the notational story is told, and because if they alone explain the difference then the thesis gains nothing from the case. The most obvious confound is tractability of the object. The brain as a physical organ is more tractable to investigation than the mind as a lived phenomenon, and neuroscience has had the advantage of studying something that can be measured, stained, lesioned, imaged, and recorded, whereas psychiatry has had to study suffering as it is lived and reported, which resists measurement in ways the tissue does not. It might be that neuroscience advanced and psychiatry did not simply because the brain is easier to study than the mind, and that notation has nothing to do with it. This confound is real and must not be waved away. A second confound is the sheer complexity of the object; the mind may be harder not because it lacks a notation but because it is, as a matter of fact, the most complex phenomenon known, and complexity alone might account for the slow progress. These are serious alternatives, and the notational thesis does not get to ignore them.

But the notational story is not thereby displaced, because it offers an account of the confounds themselves rather than standing merely alongside them. Ask why the brain is tractable and the mind is not, and the answer is not simply that one is physical and the other lived. Neuroscience was able to import and construct notations adequate to its objects, and psychiatry was not. Neuroscience represents its objects in the notations of electrophysiology, of biochemistry, of imaging, of the mathematics of networks and signals, each cut free from the ordinary-language vocabulary of mental life and governed by relations internal to itself. The tractability of the brain is achieved by the notations neuroscience brought to bear, and a phenomenon becomes tractable precisely when a notation adequate to it is found, which is to say, precisely when the phenomenon passes the test proposed earlier. An action potential can be diagrammed. A receptor's binding kinetics can be diagrammed. A tract of white matter can be diagrammed. Each is a nonrecursive fact, or a nonrecursive residue of a larger recursive process, and each admits the coordinative treatment that has made neuroscience transformative. The confound and the thesis are not competitors here. The thesis explains the confound. The brain became tractable because it was brought under notations that fit it; the mind remained intractable because the vocabulary brought to it was never coordinative, and, deeper still, because what it was reaching for does not hold still to be coordinated at all.

For psychiatry, by contrast, has never achieved its demediation, and this is the heart of the case. Psychiatry theorises its objects in a vocabulary that remains, after more than a century of effort, essentially the vocabulary of ordinary language, refined and systematised but not demediated. Its central terms, the names of the disorders and the descriptions of the symptoms, are coupled to lived experience through and through. Depression, anxiety, mania, delusion, compulsion, and the rest are ordinary-language terms, and even where they have been given formal definitions and operationalised into checklists, the definitions are enumerations of ordinary-language descriptions of lived states. The great classificatory systems of psychiatry are not notations in the sense this article requires. They are systematisations of ordinary language, lists of criteria each of which is itself an ordinary-language description of a lived experience, and they therefore inherit exactly the limitation that afflicted alchemy. Their terms mean what they mean by their coupling to lived experience, not by relations internal to a formal system, and so they cannot sustain the kind of internally governed inference on which transformative discovery depends. Psychiatry has refined its ordinary-language vocabulary to a high degree, as alchemy refined its vocabulary of principles and spirits, but refinement of an ordinary-language vocabulary is not demediation, and it does not confer the inferential power that a genuine notation confers.

This diagnosis can now be sharpened past the point where an earlier statement of it had to stop. It is not merely that psychiatry has not yet achieved its demediation, as though the achievement remained available in principle and psychiatry had simply not reached it, a matter of time like chemistry's slow reform of its vocabulary. The general law gives a harder answer. Apply the test proposed earlier directly to the central phenomenon of psychiatric suffering. Can a good diagram of a depression be drawn, one that fixes the phenomenon's elements and their relations in a form that can be read at once and reasoned from, the way a reaction can be diagrammed or a vessel's course or a body's trajectory. A century and a half of the field's own most sophisticated attempts, rating scales that fix a mood to a number, checklists that fix a disorder to a list of present or absent features, circuit diagrams that fix a disturbance to a location in the brain, say no, not through want of trying or of technical sophistication, but because what these attempts diagram, when they succeed at diagramming anything, is never the suffering itself but a nonrecursive residue of it, a symptom count at a moment, a scan taken at a moment, a score on a day. The suffering itself, the self-amplifying loop of a mood that worsens sleep that worsens the mood, the constitution of distress between a person and those who do or do not recognise it, does not hold still long enough to be drawn, because it is selfrecursive and interrecursive in exactly the sense the general law describes, and the failure to diagram it across so many independent and highly motivated attempts is evidence, cumulative and hard to discount, of the kind of phenomenon psychiatry has been trying to notate.

This resolves, more sharply than an earlier statement of this argument could, a question that statement had to leave open. It is not a real possibility, to be weighed alongside others, that a demediated psychiatric notation with mesocosmic fit might yet be built by a science not yet clever enough to have built it. For the recursive core of psychiatric suffering, no such notation is possible, not because psychiatry lacks ingenuity but because a notation of a selfrecursive or interrecursive process is not a hard achievement awaiting a talented builder. It is not an achievement at all, in the way a square circle is not a difficult figure to draw but no figure. What remains open, and what the general law predicts should remain open and productive, is the notation of the nonrecursive fragments folded into the same phenomenon, the pharmacokinetics of a drug clearing the bloodstream, the circadian mechanism a mood disorder sometimes rides on, the metabolic substrate of a postpartum depression tied to a steroid pathway. These fragments can be diagrammed, are being diagrammed, and psychiatry's genuine successes cluster exactly here, at the margins of its domain least like the suffering and most like an ordinary physiological fact. The uneven, partial record of psychiatric discovery, real progress at the edges and none at the centre, is the exact pattern the thesis predicts for a phenomenon of mixed constitution, notatable exactly as far as it is nonrecursive and never as far as it is recursive.

The neuroscience comparison, treated this way, neither proves the thesis nor is idle to it. It cannot prove the thesis, because the confounds of tractability and complexity are real and cannot be fully disentangled from the notational story in a single historical comparison. But it is far from idle, because the notational story, sharpened now by the general law and its diagram test, explains the confounds rather than merely coexisting with them, and because the pattern it reveals is the same pattern found across every case in this article. A field advances transformatively when it achieves the severing of its vocabulary from lived coupling, organises what it has severed coordinatively, and finds that the resulting notation fits a phenomenon that holds still to be fitted, and a field that cannot achieve all three does not advance transformatively however much it observes. Neuroscience achieved them and advanced. Psychiatry has achieved, at most, a systematised ordinary language, and has not advanced in the same way, whatever its real clinical accomplishments, and the deepest reason it could not is not a failing of the field but a fact about its object.

X. Where notations are born: invention, importation, and Newton's exception

The cases assembled so far have shown, again and again, that a notation had to exist before the discoveries that followed it could be made. They have not asked a second and separate question, where the notation itself came from, and this question turns out to have its own answer, one that qualifies the priority thesis in an important way. The common assumption, when a field's transformation is credited to a new notation, is that the field built the notation for itself, out of its own materials, to meet its own need. The cases considered here suggest the opposite is closer to the rule. A field far more often imports an already-working notation from elsewhere than invents one from nothing, and the field that first develops a notation is, more often than not, a different field from the one in which the notation eventually does its transformative work.

Consider where the very idea of classifying diseases came from, since the question bears directly on the hardest case this article has examined. The idea is not native to medicine, and it did not begin with the International Classification of Diseases, which is a comparatively late administrative instrument, first assembled by the statistician Jacques Bertillon in 1893 as a classification of causes of death for the purposes of vital statistics, and adopted by the World Health Organization only in 1948. The systematic classification of disease as such begins earlier and elsewhere, in the eighteenth century, when the physician Francois Boissier de Sauvages, and separately though in close correspondence with him the naturalist Carl Linnaeus, took the method Linnaeus had built for classifying plants, the hierarchy of class, order, genus, and species set out in his Systema Naturae of 1735, and applied that same method, term for term, to disease. Sauvages's Nosologia methodica and Linnaeus's own Genera Morborum sorted thousands of named conditions into classes and orders exactly as Systema Naturae had sorted the plant kingdom. Medicine did not invent disease classification. It borrowed the classifying apparatus of botany, and everything medicine has done with classification since, including the Bertillon list and its descendant the ICD, is built on that eighteenth-century import. Psychiatry's own diagnostic manual sits at the end of this same borrowed lineage, and this matters for the diagnosis offered earlier in this article, because it means psychiatry did not merely fail to demediate a native vocabulary. It inherited, from botany by way of general medicine, the very idea that a domain of phenomena should be sorted into a tree of classes and species in the first place, an idea built for organisms that hold still to be classified, and never designed with a recursive phenomenon in mind.

Psychiatry's case is in fact the clearest illustration of the general pattern, because psychiatry appears never to have originated a notation of its own at any point in its history. Every symbolic technology it has used was brought in from elsewhere. The classificatory apparatus came from botany by way of general nosology, as just described. The statistics in which prevalence and trial outcomes are expressed came from mathematics and from the actuarial and demographic traditions that produced modern probability theory. The rating scale, the device that turns a mood into a number, is a direct import from experimental psychology and psychometrics, fields that built the technique of scaling for their own purposes, chiefly the measurement of sensation, aptitude, and attitude, well before psychiatry adopted it for mood. And the biological and genetic notations on which the more recent research frameworks depend were built by neuroscience and molecular biology, for objects those fields could already hold still, and were only afterward brought across into psychiatry in the hope that they would hold its object still as well. A field that imports every notation it uses, and that has never had to solve the problem of inventing one, has also never had the occasion to discover, from the inside, whether a notation built for its own object was even possible. It has instead spent its history discovering, import by import, that other fields' notations, however successful where they were built, do not fit an object those fields were never trying to hold.

One body of work might seem an exception, and is worth naming so the claim is not thought to ignore it. Psychoanalysis produced two attempts at a symbolic architecture built from within rather than borrowed from without, Freud's structural division of the psyche into id, ego, and superego, and Lacan's later mathemes, algebraic-looking formulae meant to fix the structure of desire and the unconscious. Neither is a notation in the sense this article has defined. Freud's model remains an illustrative heuristic, a way of narrating conflict among agencies rather than a system whose terms are fixed by relations internal to a formal calculus, and it licenses no operation that could not equally be stated in the ordinary language it dresses up. Lacan's mathemes borrow the look of algebra without its discipline, since they support no derivation that a reader could check, extend, or find mistaken independently of the analyst deploying them, which is precisely what a coordinative notation must support to count as one. The near-exception confirms the rule rather than breaking it. It shows what an attempt at a native notation for this domain looks like when the domain will not permit one, and shows that even the most determined such attempt could not get past narration wearing the costume of a symbol.

The pattern is not confined to imports between one science and another, and the case of anatomy, examined earlier in this article as an instance of pictorial rather than algebraic demediation, turns out on closer inspection to be an import as well, and not from within medicine. The coordinative anatomical image, capable of fixing a vessel's course or a muscle's attachment as a structural claim rather than a likeness, was pioneered within the workshop practice of Renaissance art, where painters and sculptors dissected bodies not to advance medicine but to render the living figure correctly, and developed, for that purpose, techniques of cross-section, multiple viewpoint, and layered exposure that no anatomist had used before. Leonardo da Vinci's own anatomical studies, made between roughly 1489 and 1513, are the most striking early instance of this technique, decades ahead of anything medicine itself had produced, though his notebooks were not published in the relevant period and his direct influence on what followed remains debated among historians. What is not in doubt is that when Andreas Vesalius assembled the illustrations for his own Fabrica in 1543, he turned to a trained artist, most probably from Titian's workshop in Venice, rather than to any tradition internal to anatomy, because the coordinative image was already a craft possessed by painters and not yet a craft possessed by physicians. Medicine imported its decisive visual notation from art in the same way it had imported its classificatory notation from botany.

Nor is the pattern confined to importation between fields that would recognise each other as forms of inquiry at all. Photography's dominant technical lineage did not begin as an instrument of science. Louis Daguerre, whose process defeated its rivals and gave photography its name, spent the seventeen years before his invention as a theatrical scene painter and the proprietor of the Diorama, an immersive spectacle of painted, lit tableaux that drew paying Parisian crowds, and it was in trying to fix and improve the very images that his theatre already depended on that he arrived at the process that bears his name. Photography, in its most consequential origin, was an import from spectacle into science and not the other way around, even as it was also, from its first public unveiling, promoted as an instrument for both. Motion pictures make the same case still more concretely. The frame rate above which a sequence of still images stops looking like a flicker and starts looking like continuous motion is a genuine fact about human visual perception, and the specific rates that solved the problem in practice, on the order of sixteen frames a second multiplied by a shuttered flash to a much higher effective rate, were converged on empirically by the commercial cinema industry, under the pressure of exhibitors who needed to satisfy a paying audience without wasting expensive film stock. The numbers the industry needed and found converge closely with what vision science would separately describe as the flicker fusion frequency and the phi phenomenon, a quantified fact about perception pinned down as much by a spectacle-driven trade as by a science of perception.

These cases support two hypotheses that the priority thesis of this article, on its own, does not decide. The first is that the origin of a notation is very often genuinely difficult to locate, because notations migrate through craft traditions, correspondence between practitioners in different fields, and informal borrowing long before any single act of invention can be dated and credited, so that asking where the classifying of disease or the coordinative anatomical image truly began is asking a question that may not have a clean answer even in principle. The second is that the field in which a notation becomes canonical and transformatively successful is, for exactly this reason, rarely the field in which the notation was first built, since a notation is built to serve the purposes of wherever it happens to be built, a botanist's need to sort a kingdom, a painter's need to render a figure, a showman's need to keep an audience watching, and it is only afterward, and often by other hands, that the notation is carried to a domain whose fit with it was no part of its original design. Newton is the striking exception that proves how rare the coincidence is. He built his notation and found its transformative application within a single sustained undertaking, driven by mathematical problems that were already, in effect, physical problems in disguise, so that the inventing and the succeeding happened in the same hands and very nearly the same act. Most of the history assembled in this article did not happen that way, and psychiatry's history least of all, since a field that has only ever imported never had the chance to discover, in the act of building, whether its object would permit a notation built expressly for it, and imported every fit failure along with every import.

XI. What follows, and where the thesis stops

If the thesis holds, several things follow for how the history of science should be read and how the prospects of a stalled science should be assessed, and it is worth setting them out, together with a careful statement of where the thesis stops, because a thesis of this ambition is most credible when it is most explicit about its own limits.

The first thing that follows is that the history of science should be read as a history of symbolic technologies at least as much as a history of discoveries and the experiments that produced them. The conventional histories, organised around findings and the great experiments and observations that yielded them, tell a true story badly framed, because they credit the encounter with nature for transformations that the encounter with nature could not by itself have produced. Behind each of the great findings stands a symbolic innovation that made the finding thinkable, and a history that foregrounded the notations rather than the findings would be closer to the causal truth. It would be a history of the construction of symbolic domains, of the successive achievements by which one field after another cut its vocabulary free of lived coupling, organised what remained into a structure of internal relations, and found, or failed to find, that the structure fit a phenomenon capable of being held still by it. Such a history would recognise the notation, and not the experiment, as the characteristic act of scientific revolution, and it would recognise the builders of notations, who are often not the same people as the makers of the famous discoveries, as the decisive figures they were.

The second thing that follows concerns the prospects of a field that has stalled. On the conventional picture, a stalled field needs better data, more observation, a decisive experiment. On the present thesis this is very often a misdiagnosis, and a field that has accumulated data for decades without transformative result very often has all the data it needs and lacks the notation that would let the data add up to a science. This reframing matters practically, because a field that misdiagnoses its own condition will pour its resources into the encounter with nature when what it needs is a revolution in its symbols. The argument of this article adds a cheaper diagnostic step before that revolution is attempted. A field can ask, of its central phenomenon, whether a good diagram of it can be drawn, and can treat sustained, skilled, repeated failure to draw one as evidence, worth taking seriously before further investment, that it faces a phenomenon admitting no notation at all, in which case its resources are better spent on the narrative disciplines than on the notational ambition that cannot hold it. The falling bodies and the careful dissections and the weighed reactions were available long before the sciences that made sense of them, and what was missing in each case was the notation, and further, the notation could succeed because the phenomenon was of a kind a notation could hold. A stalled science should ask itself four questions in order: whether it is waiting for a fact or a symbol, whether the symbol it lacks would be coordinative or merely a more disciplined narration, whether its central phenomenon is of a kind that a coordinative symbol could ever fit, and, before attempting to build anything, whether a notation adequate to it has already been built somewhere else and need only be imported.

Now the limits, stated plainly, because the thesis overreaches the moment it forgets them. The claim of this article is that symbolic innovation is a necessary enabling condition of transformative scientific discovery, the condition without which the discovery could not be made, and that it is very often prior to the discovery and developed for reasons partly independent of it. The claim is not that symbolic innovation is a sufficient condition, that a new notation guarantees a discovery, which is plainly false, since notations can be built that lead nowhere and many are. Nor is the claim that the encounter with nature is dispensable, that discovery is a matter of symbols alone with no answering to the world. The notations of a science answer, in the end, to measurement and prediction, to an encounter with nature that confirms or refutes what the notation permits to be inferred, and a notation that answered to nothing would be an empty game and not a science. A discovery requires both the notation that makes it thinkable and the world that confirms it, and the thesis asserts only that the first of these has been systematically underrated and the second systematically overrated, while granting fully that the notation without the answering world would be nothing. Demediation opens the space of possible inference. The world decides which of the possible inferences are true. Both are needed, and the thesis corrects only the traditional neglect of the first, without denying the second.

There is a further limit, sharper now than when this argument could appeal only to demediation, and it is the one that keeps the thesis from becoming the skeleton key it was at pains at the outset to avoid becoming. Not every symbolic innovation is a demediation, and a field can refine and systematise its ordinary-language vocabulary, as psychiatry has done, without cutting that vocabulary free of lived coupling. Not every demediated vocabulary is coordinative, and a narrative can be freed of folk association and conducted with great technical discipline while remaining narrative in form, unfolding in time and referring outward rather than fixing its elements into a system of internal relations. And not every coordinative notation, however demediated and however fully organised into internal relations, achieves fit, because fit further requires that the phenomenon notated be nonrecursive, and a coordinative notation built for a recursive phenomenon is coherent without being fitted, a game whose moves are legal and whose conclusions are empty. The thesis of this article is therefore the conjunctive claim that transformative discovery requires a notation that is demediated, coordinative, and fitted to a nonrecursive phenomenon, that this triple achievement is typically prior to the discoveries it enables, and that a field which cannot accomplish all three, for whatever reason, including the reason that its central phenomenon will not permit the third, will not have its revolution however much it observes.

One further observation belongs here, because the argument of this article concerns itself and not only its cases. An account of why recursive phenomena resist notation is itself an account of a domain, the space of possible symbolisations and their fit, that includes its own activity of arguing within that space, and a theory that reflects on its own conditions of possibility is engaging something that responds to how it is engaged, which is one of the marks of the recursive as this article has defined it. It should not be surprising, then, that this article is a narration and not a notation, that its claims could not be fixed into a diagram without falsifying them, since a diagram of the relation between narration and notation would have to hold still a distinction whose very statement is an act of the narrative kind it is trying to characterise. This is not a failure of the argument to practise what it preaches. It is a small and, I think, genuine confirmation of the argument, offered in the very form the argument takes, that some domains are adequately known only by being followed in the telling and not by being fixed in the diagram, and that the theory of notation, reflexively applied to itself, turns out to belong to the domain it says notation cannot reach.

The notation comes first, not always and not sufficiently, but far more often and far more decisively than the history of science, told as a history of findings, has been willing to see.