Tags
A mathematics professor named Sam stands at a blackboard with an equation that will not come out. It has been not coming out for a long time. At the end of his patience he says what people say, that he would sell his soul to get the thing right, and a demon in sunglasses appears in the classroom to collect.
This is “I of Newton,” a short piece from the 1980s revival of The Twilight Zone, made from a story by Joe Haldeman, with Sherman Hemsley as the professor and Ron Glass as the demon.
Sam protests that he was only venting. Too late: what he said, it turns out, had exactly the right sound for an invocation. But a demon enjoys a contest, so terms are offered. Sam may ask three questions about the demon’s powers, and then he must set a question the demon cannot answer or a task the demon cannot perform. Sam says, “Really?” The demon informs him that he has just used his first question.
The second establishes that the demon can break the laws of physics and move through time as he pleases. The third: is there anywhere, in space or time or any other universe, from which the demon could not find his way back? Nowhere. The demon, enjoying himself now, tells Sam to go ahead. Sam says it is not a question but a command, and after a long pause gives it.
Get lost.
The demon vanishes with a howl. Sam drops the sunglasses in the bin and turns back to the blackboard, grumbling that the fellow wasn’t any help at all.
Before going further I want to separate two things that were present in that classroom, because everything after this turns on the difference.
There was a man looking for an answer.
And there was a being who had all of them.
Look at what happened to the first question. “Really?” is not a request for information. It is the sound a search makes while it is still searching — open, not yet aimed at anything, a man finding out where he is. The demon took it as a question spent and entered it in the account. That is the whole of what follows, told as a joke: the being who knows everything cannot hear a question except as a withdrawal.
Before the answer
To sense is to search. Looking for the slip in the third line. Waiting for the next step to show itself. Listening to find out whether that was the sound. The searching is not something done to a sensation that has already turned up; there is no arrival standing apart from the looking, waiting to be looked at. A measurement does not happen without the question that motivates it, and the question is not asked of the measurement afterwards — it is what put a man at the blackboard.
That is what I mean by questioning. Open, expanding, prior to any answer. Not a stance taken toward sensing but the shape sensing has.
Doubt is not the opposite of that. Doubt is its perversion. War is the perversion of peace, hate the perversion of love, despair the perversion of sadness — in none of these is the second the absence of the first. The second is made out of the first, with the same energy and the same motion, and what has been removed is the return. Sadness moves and passes; despair is sadness that will not move. Love reaches; hate is love that has fixed. Questioning opens; doubt is questioning that has closed and goes on asking anyway, which is why it cannot arrive and cannot stop.
The skeptics were named for the searching and not for the doubting. Zetesis: to go on looking. Sextus divides it three ways in his opening pages — the dogmatist says he has found it, the Academic says it cannot be found, and the skeptic is still searching. Only the middle one is doubt. It is the one that stopped and kept the motion anyway.
The classroom has all three positions in it, with one actor doubling. The demon has found everything, and on that strength he is certain that what Sam is looking for cannot be found. Dogmatist and Academic in one pair of sunglasses. Sam is the one still searching.
So doubt needs a claim to aim at, and the claim comes later. To doubt a sensation you must first have finished with it. You must have taken a number off it.
None of this makes the appearance certain, or sharp, or a floor to build on. Nothing is being received here that could be certain. The searching has no edges, because edges are what a formula brings, and it brings them afterwards.
The white in the middle
Vagueness is not a defect in the searching. It is the condition of anything appearing at all.
Shine red, green and blue light on a rough mirror. Then on a finer one. Then finer again. The finer the roughness, the more the three colours mix, and at the end of that progression you have a white surface. Every white thing in the world is a very finely rough mirror. Which means that if every difference had to stay a different appearance, no white thing could appear, and in the end no thing could. The white is not a coarse report of an underlying array of exact wavelengths. The white is what there is to see, and it is there because the differences are not being held apart.
Magnification does not undo this. Roughness is ordinarily what a microscope is for. But build the mirrors as a progression and the indistinctness persists at every magnification, so the move that is supposed to end vagueness runs forever without ending it.
This is the first thing the demon cannot see. Ask him the colour of the finest mirror and he will give you every wavelength leaving it, correctly, with the angle of each. He knows every difference and holds every one apart; that is what it is to know everything. So there is no white for him. White is only available to someone who does not know too much.
I built the box for a classroom and it turned out to be an argument. The children did not receive a verdict from it. They looked longer, and then looked again, and the look times ran long at exactly the mirrors where the theory would not apply. One of them said the colours were beautiful and she did not want to write them down. Another said he could not remember what he saw. What the roughness does is not deliver a false report. It keeps you looking.
Vagus: to wander. Vagueness is wandering, and the wandering is the searching that has not been told where to stop.
Get lost is vagus in the imperative.
No second kind
The felt body, emotion, the texture and arrival and duration of a thought. These are searched for the same way and found the same way. Looking for whether that was fear is the same act as looking for whether that was the sound. There is no inner theatre with a smaller observer seated in it, sunglasses or no, and no second kind of finding that would need a different licence.
The standing objection is that introspection is a bad instrument. People confabulate. People report reasons for their behaviour that demonstrably were not the reasons. People miss large changes and then deny that they missed them.
I concede all of it without reservation, because every one of those findings is about a number taken. Confabulation is a false report of a cause. Change blindness is a failed claim about persistence across a gap. Memory revision is a claim about a past made out of a present. Each is an error in the place where the searching was declared over. Not one is a case of a search going wrong while it was still going.
So the psychology of introspective failure is not an argument against the immediate. It is a careful map of where the stopping happens, and I would rather have it than not.
Notice that I have not called the mind an instrument for receiving inner objects. That picture is what is being argued against here, and it does not improve by being extended inward.
Two motions
Aristotle allowed two natural motions. Gravity, which goes down in a straight line. Levity, which goes up and eventually turns and goes round.
Newton opens his first law with corpus omne — every body — and the distinction is finished. One kind of body. One natural motion, uniform and straight, laid out between a positive and a negative. Heidegger’s remark is that the Aristotelian division between earthly and celestial has by then become obsolete, and obsolete is the right word. Levity was not refuted. It was dropped, because the new motion had no place to put it.
Haldeman’s title is a witch’s ingredient with two letters added: eye of newt, I of Newton. I take it more seriously than it was meant. There is an I in Newton, and it is exactly what corpus omne has no place to put.
I want this vocabulary because it does not have a subject or an object anywhere in it. It is not about minds facing things. It is about which motions are admitted.
Questioning is levity. It rises, and then it turns, and the turning is the return. Doubt is levity with the turn forbidden — rising and rising and never coming round, which is exactly what it means to say that doubt cannot arrive and cannot stop.
And that motion has a mathematical statement. A bounded monotone increasing sequence. Monotone: it never turns back. Bounded: it cannot get away. Forbid the turning and forbid the escaping, and the axiom of completeness will hand you a single point at the top and tell you the point is real.
The demon’s answer to the third question is the second clause spoken aloud as a boast. There is nowhere he cannot come back from. He cannot get away. Omnipotence, in the one respect that turns out to matter, is a boundary. He is the point at the top of the sequence, real because the axiom says so, and wearing sunglasses.
Monotone is vagus struck out by stipulation. The sequence does not wander, not because anyone found out that it doesn’t, but because wandering was excluded in the setup. Nothing was discovered. A condition was imposed, and uniqueness came out the other side.
The induction of inductions
Measure a plank. Whole metres first, then decimetres, then centimetres, and at every stage there is a little more plank left over than your last mark. The process ends when the tape runs out, or the eyes fail, or the measurer loses interest. The axiom then assures you that a real number exists for the actual length, and that there is an actual length.
Now make the same move with people instead of decimetres. Science does not rest on one reading. It compares records made on different nights by different hands, judges which differences are systematic and which are noise, and concludes that what has held will go on holding.
Hume named the difficulty and I will not spend long on it. The inference requires that unobserved cases resemble observed ones, and that principle cannot be established deductively without assuming itself. He concluded that induction is not the work of reason but of the imagination.
The general form is what I am after. All replicated measurements, never finished, may be treated as finished.
That is the induction of inductions, and it is the axiom of completeness made outside the language of numbers. The structure matches at every point. A sequence of replications, each narrowing the estimate. Bounded, since the value is not infinite. Monotone in precision, since the intervals are supposed to nest and shrink. And never finished, because the experiment can always be run once more, and there is always a further place of precision the method has not reached. Then the move: assert the limit, and call the limit the true value.
Consensus is the least upper bound, and it is the least one that matters — a least upper bound is unique. That uniqueness is the whole trick. Where there were many observers with many offsets there is now one number, and the many-to-one has become one-to-one, which is Russell’s definition of the non-vague achieved by stipulation rather than by looking. The observers are supposed to drop out at the limit. Their offsets cancel and what remains is the object itself, unmediated. That is the escape from the first person. It is asserted, not found.
What remains when all the observers have dropped out has a name, and it is not a joke that the name is a demon. In 1814 Laplace imagined an intellect that knew, at a single instant, every force and every position in nature, for which nothing would be uncertain and the future would stand as plainly present as the past. His readers took to calling it Laplace’s demon. It is the least upper bound of all observers, imagined as somebody. The equation summons it, said the right way.
The demon in the classroom has a place in mind for Sam’s soul: a universe run by binary digits, where Sam could start out as a decimal and gradually work his way up to a fraction. Consider what that promotion is. A decimal that goes on without ever repeating is the least governable thing in arithmetic. A fraction is a decimal that has agreed to repeat. Damnation, in this cosmology, is a career in which you advance by becoming more determinate. The profession runs the same ladder and calls it rigour.
The error bar is the “…”. It is notation standing in for the rest of an investigation nobody carried out. A meta-analysis is a sequence of studies with a limit taken.
And the cost is the cost of the plank. Is that precise enough recurs at every level of magnification, and it is answered by declaring the unfinished sequence finished. Which shuts out more than it means to. The motive for taking the limit is not curiosity; it is the craving for certainty, and the fear of doubt, and those two names describe one condition. That is the curse word woven into the invocation. A mathematician cannot rest in an unfinished sequence. A profession cannot rest in a result that might not replicate. So the sequence is closed by decree, and because doubt and questioning are the same motion with and without the turn, forbidding the one forbids the other. The monotone condition does not distinguish between them and cannot. It only says: do not come back.
In this perfect wakefulness the light does not radiate, does not twinkle, no-one blinks.
Except that the equation is on a blackboard, and it is looked at by a man who blinks.
What does not stop
Something survives all this, and it is not evidence.
Some searching will not finish where the formula said it would finish. The bridge is not behaving, and the engineer goes back and looks again. The trial fails and someone runs it again. The result will not replicate no matter who runs it, so it goes on being run.
I used to describe this as an appearance overruling an inference. That was wrong in a way worth marking, because it smuggles back the picture this essay is against: two items, each with content, and a contest between them settled by contradiction. There is no such contest.
Here is what happens instead, and the shape of it is already in the mathematics. A positive definition states what a thing is. A dialectical one states the limits of a proposed definition — it does not aim at the essence, it aims at where the proposal gives out. Analysis students think continuity through dialectically, epsilon answering delta, and then remove the dialectical shape and put a proof in its place. An open set could be defined as not its boundary, and then some boundaries would have to be found by ongoing inquiry rather than declared.
The bridge falling is a boundary that ongoing inquiry has to find. The engineer had a positive definition, complete, with a stopping-place written into it. The stopping-place did not arrive. He is returned to the dialectical case, where the limits of the proposal are what is at issue and somebody has to go and look for them.
That is why it costs. Nobody has been refuted. Everybody is still looking.
Which is the same event as the axiom, seen from the other side. The axiom declares an unfinished search finished. This is a search that will not be declared finished. One is the apparatus closing the sequence; the other is the sequence refusing to close. The cost is the same cost either way, and the only question is whether it is paid by the decree or by the looking.
And it falls in either domain. The engineer in front of a structure that is not behaving. The practitioner who goes on seeing, plainly and unwillingly, what cuts against his teacher, his doctrine and his hopes for himself. The professor at the blackboard after the demon has gone. None receives a refutation. None can stop looking. Same event. Same currency.
Comparing observers requires an observer
Here is where the standard picture cannot be repaired.
Astronomy found this out in the most literal way. In 1796 the Astronomer Royal dismissed his assistant for timing star transits later than he did. Twenty-six years later Bessel compared his own timings with those of colleagues no one would have dismissed, and found steady personal offsets larger than the one that had cost the assistant his post. The correction built from them was called the personal equation.
But Bessel did not step outside himself to discover that he and his colleagues differed. He looked at the records. He judged the difference systematic and not noise. He decided how many nights were enough. Every one of those acts is the kind of act the personal equation exists to correct for, and every one was performed in the first person, unassisted, by a man with an offset of his own.
The only thing that could stop this regress is an observer who needs no correction: one who compares all the observers without being one of them, who has no offset because he has no position, who can come back from anywhere because he was never anywhere in particular. That is the demon’s job description exactly. And the demon is a character in a television programme.
There is no regress-stopper. There never has been. Each correction of first-person judgment is designed, applied and evaluated by first-person judgment, and if you go looking for the level at which this stops being true you will not find it. You will only find people, looking.
Which means the apparatus of objectivity is not a way of getting out. It is a way of getting a better result from inside, and the difference matters enormously in practice and not at all in principle.
What the agreement costs
The serious defence of third-person method is not that any individual scientist errs less than any contemplative. It is that errors are uncorrelated. My misreading and yours are independent, so when we pool, noise cancels and signal stands.
So look at what the profession does to produce its agreements. Shared training. Common instruments from common manufacturers. A standardized vocabulary in which the phenomenon may be described. Peer review by people trained in that vocabulary. Replication by protocols fixed in advance. Textbooks that settle what a competent observation is before anyone observes anything.
Every one of these is a correlating operation, built deliberately and well to make two observers more alike. The agreement they produce cannot then be cited as independent convergence upon the object, because independence was the thing being removed. Consensus is both the output of the correction machinery and an input to it, and the relation runs the wrong way for comfort: the tighter the correction, the less the eventual agreement tells you.
Bugaev, arguing in 1897 against the mathematical fixation on completeness, called incompleteness a manifestation of independent individuality and autonomy. He was talking about the number line and he was talking about this. A unique least upper bound is individuality removed — one value standing where many observers stood. The guild’s shared training and the axiom’s uniqueness clause are one operation performed at two scales, and that is why the sociological complaint about guilds and the epistemological defence of method are not two conversations. The institution erodes its own justification in the act of performing it.
Follow the road to its end and you arrive at the demon: consensus with nobody left in it. One knower, perfectly agreed with himself, and no independence remaining to spend.
I am not saying the corrections are bad. They are why the bridges stand. I am saying they buy reliability by spending independence, and you cannot spend it and still have it.
The inversion
The standard picture has trained attention waiting outside, hoping to be licensed by objective method — hoping that one day the disciplines of the inner will be validated by instruments and thereby allowed to count.
The dependency runs the other way.
Discriminating what is there from what you expected to be there. Noticing that you are tired, that you are invested, that you have already decided. Holding attention steady enough to register a small change nothing prompted you to look for. Every gauge-reading in the history of the world has run on these.
And when you examine what the apparatus consists of, you find prosthetics for known defects in exactly these capacities. Blinding manages expectation. Pre-registration manages the drift of a hope into a hypothesis. Instrumentation replaces a judgment that fatigues with a trace that does not. The personal equation corrects a constant in a man’s attention that the man could not feel.
You cannot build a prosthetic without knowing what is failing. Each device encodes something known about attention, and then conceals where it came from by working so well that nobody has to think about the defect again.
Imagine the prosthetics finished — every correction applied, nothing left over to correct — and you have the demon again. A superb artificial leg with no one walking on it.
Attention is not the worse instrument. It is the instrument the others are corrections to.
The line and the return
There are two shapes a correction can take, and the difference is the difference between an equation and a return.
Identity is a line. Even written as A = A it needs two positions and a sign between them holding the gap closed; you cannot write “A =” and have said anything. Selfhood on that model is an assertion of sameness across a distance, and it needs the gap covered, exactly as the axiom covers the gaps in the line.
The demon’s terms are lines of this kind. A question and its answer. A task and its completion. Two terms and a sign between them, the gap to be closed by his powers, and his powers close every such gap there is. Sam’s move is to decline the shape. Not a question: a command. And get lost has no completion the demon could hand back. Its fulfilment is a going with no guaranteed return, which is the one shape his powers were not built in.
He can come back from anywhere. That sounds like the return and is its opposite, because he has never been away. A return needs a going it does not control. What the demon has is a line folded double.
The other shape needs no equals sign. One object, and coming back to it. Aristotle’s point that circular motion cannot be defined by an opposition is the relevant one: the present you, reading, and returning to the present you, reading, is enough for motion, and there is no second term. Nothing is claimed to be the same. Heraclitus has it already — we step and do not step in the same rivers.
Sitting still and watching is the second shape. It is a correction too, but of a different kind from the astronomers’. A linear correction has an output: a number you subtract, after which the matter need not be attended to again. The return produces no number. Nothing is stored, nothing accrues, and tomorrow you do it again with no credit from today. The output of the practice is the practice.
That is why the objectivity apparatus can forget its origin and the practice cannot.
And the two treat wandering in opposite ways. Blinding, protocol, pre-registration, fixed vocabulary — every one is engineered to keep the mind from going where it will. The return does not prevent the going. It consents to it and adds the coming back. One approach removes the wandering. The other keeps the wandering and adds the turn.
Which gives the perversion its shape. Doubt is wandering with the turn taken out, and an apparatus built of protocols against a wandering never permitted to come round does not contain doubt. It institutionalizes it, and then covers the hole with a limit so that the asking stops by decree rather than by arrival.
Constants
The impermanence of constants is not a mystical objection and I want to state it carefully, because the careless version is easy to dismiss.
Physics does check this. Variation in the fine-structure constant has been probed against the natural fission reactor at Oklo, two billion years old, against absorption lines in quasar spectra, against the abundances of the light elements formed in the first minutes. The bounds are tight, and they reach back very far. Anyone who says nobody has looked deserves to be stopped at the first paragraph.
But every one of those bounds is derived inside a framework that assumes the stability it is testing. The Oklo constraint is read out of nuclear physics taken to have been the same nuclear physics. This is not a scandal and there is no way around it. It is what testing a constant necessarily looks like from inside, and the error bar on the result is calculated using the assumption in question.
The constant feels constant. That feeling is a fervour about something only time can report on, and the report is being written in the language whose stability is at issue.
The demon, asked his second question, said he could break the laws of physics and step through time whenever he liked. It is the one claim in the episode that could only be checked by the party making it. Nobody in the classroom thought to ask how he knew.
Two things I am not saying
That a profession’s first job is self-preservation, I hold to. Its first job, not its only one. People inside these institutions do careful, honest and expensive work, and much of it survives the pressure it is under. If “first” slides into “only,” the argument becomes a complaint, and a complaint is easy to put down.
And contemplative institutions gatekeep at least as hard as universities. Lineage, authorization, the approved description of what you saw, the quiet disappearance of the student whose experience did not fit the map. I concede this early because it costs the argument nothing. My claim concerns the structure of investigation, not which guild has nicer people. Every guild correlates its members. That is what a guild is for.
Where this sits
Husserl described objective science as a construction out of the lifeworld that forgot it was a construction. Merleau-Ponty treated the objective attitude as a derived posture concealing its origin. James took introspection as a primary method and gave it the seriousness of any laboratory procedure. Fleck showed that a thought-collective determines what can be seen at all. Gieryn named the boundary work by which a profession draws the line around itself. Daston and Galison traced mechanical objectivity as a discipline of self-restraint — the observer suppressing his own judgment, an inner exercise throughout. Varela built a research programme on the dependency instead of arguing about it. Schaffer read the personal equation as observatory discipline, a chronometric regime for the surveillance of subordinate observers.
I want the demon for something else: for the licensing relation, for the fact that the tools of objectivity are made out of what attention has learned about itself and cannot be made any other way, and for the one task that the finished observer cannot perform.
Back to the blackboard
There is a detail I have saved.
When the demon is gone, the equation is still on the board, and still wrong. The episode never solves it. Sam goes back to it, complaining that the demon wasn’t any help at all.
I want to mark what follows as mine rather than Haldeman’s. He wrote a joke, and a very good one, and this is laid over it rather than found in it. But the complaint is exact. The demon could have answered anything Sam asked and could not help, because what Sam wanted was never the answer. He wanted to get it right, and getting it right is done at a blackboard by a man who blinks, going back to the same line and finding it not quite the same line. An answer handed down by a being who has never been lost is a number with nobody’s looking in it. Omniscience was no help with the equation for the same reason it could not see white.
Science has this ending too, and tells it less well. Having established the personal equation, Bessel found that it drifts — with the year, with the instrument, with the man. So the correction had to be remade, and remaking it meant going back to the same place: a person attending to a star and to himself at once. Astronomy got the motion right and the description wrong, and it was the description that was institutionalized.
We have been going back ever since. We only stopped saying so.
And there is a reason to care beyond the accounting. If the searching is not in doubt, then the endless refinement toward a precision that never arrives is the intellect sharpening a vagueness it did not find sharp, and the appearances we were told to look past were never deficient. What follows from that is happiness, which is a longer argument and one I have made elsewhere. Here I only want the ground it stands on: this, now, still being looked for.
