Open Letter to Mathematicians
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Why It Has Not Finished (unfinished)
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Why It Has Not Finished
Wonder does not depend on a wonderer. The essay that made that claim closed by conceding the obvious objection — we do not live there, we begin inside minds that have already sharpened themselves — and it offered an answer it did not develop. Relaxation is not a method. It is what sharpness does under long and honest use. The handle is not worn down by a decision to wear it. It goes because the axe was used for what an axe is for. But if release is only what honest use produces, then escape is natural, and nothing external holds the door. There is no gate and no keeper at it, and the union is neither far away nor guarded. The process runs on its own and tends toward its own completion.
So why does existence continue?
That is what this piece is about. Not what wonder opens onto, and not how anything came to be, but why a process that finishes itself does not finish. It is the same substitution I made in the hallway. Where am I runs up a ladder of insides and ends in the air; the question that survives is how do you know you’re not lost. When did this start fails the same way, and the question that survives is why it hasn’t stopped.
The answer in short, and the rest of this is unpacking it: the approach produces the condition that reverses it. Not occasionally and by bad luck, but structurally, and most plainly where the work goes best, because there is least else left in the frame. Existence continues because escape keeps very nearly happening.
Ignorance as inverted wonder
Ignorance is pessimistic wonder. The same state, inverted valence.
Take that as a door and not as a beam. What it buys immediately is a negation. Ignorance is not a lack: not missing information, not a deficit of knowing, not a darkness that light removes. It is a full state with the same shape as wonder and nothing subtracted from it. Something has turned over.
What the slogan gets wrong is the order. It puts the valence first, as though the state were one thing and the feeling of it a switch thrown from outside. The account below puts something else first, and the valence comes out downstream of it. Wonder and ignorance share an opening condition and differ in what becomes of the relaxation. Wonder lets the distinction go. Ignorance takes the letting-go for a possession. By the end the slogan will be describing a symptom rather than explaining anything, which is fine.
Here is what turns over. The ease that comes with honest inquiry belongs to the letting-go. Ignorance reads it as a good that someone has.
The mechanism: attention slides onto the ease
Wonder feels good. The other essay said so plainly and argued nothing for it. That was affordable there. It is not affordable here, because the entire account now hangs on the feeling being attractive enough to turn attention away from what produced it, and three words will not carry that weight.
Maintaining a bearer is work. Holding a position sharp enough to be someone’s, keeping the boundary that issues the verdicts, standing apart in order to have something to stand before — all of it is effort, continuous, and it never announces itself as effort because it never stops long enough to be noticed stopping. What wonder does is let it go. The good feeling is therefore not an addition to the state. It is subtraction, felt. The slackening of a grip that had no rest in it.
This is why it is attractive out of all proportion to its intensity. It is also why ignorance feels as it does, which is not a second feeling but the same grip, held.
The mechanism that follows from this is small. Attention had been on the object: the leaf, the theorem, the question that opened. The work goes honestly and the grip slackens and there is ease. Attention slides off the object and onto the ease. Approaching the salt water, one begins attending to the relaxation rather than to the wondering. Nothing is lost in that instant. The same state persists with attention turned on its own comfort. But attending is a bearer — something is set up that attends — and the bearer is the grip, reconstituted out of the very relief it was attending to.
That slide is the perversion of wondering, and it is all the perversion is. It is also the only claim this essay needs, and it is checkable by anyone from their own work.
Note what it requires. It cannot happen to bad work: bad work produces no ease to slide onto. And the better the work goes the more the ease stands out — not because the ease grows, but because the grip that would otherwise have competed for attention has slackened and left it alone in the frame. Contrast, not magnitude. Which is the answer to the question this essay opened with, and the reason the answer is structural rather than unlucky: the condition that reverses the approach is produced by the approach going well, and it is produced most cleanly when the approach goes best, because success is what empties the frame.
That misreading is the whole of what basic ignorance is. It is not an agent who intervenes to misdirect us — an agent would be a bearer smuggled back in as a cause, and there is no room for one. It is not stupidity, and no amount of learning touches it. It is a misattribution, single and repeatable, and everything else in this essay follows from it.
The earlier essay already named the bodily sign and already said the damaging thing about it. Pīti: the rising brightness, the shiver, the sudden lightness — and pīti is fabricated, and we do not mistake the handle for the destination. In the insight traditions there is a list of corruptions that arise only from good practice and never from poor practice, and pīti is on that list. So is ease, and so is confidence, and at the end of it stands nikanti, a subtle delight in what the practice has produced. Nikanti is the one that stops the work, and it stops it by being pleasant rather than by damaging anything.
That the same brightness appears in one place as the mark of the passage and in the other as an item on a list of what ruins it is not a conflict between two traditions. It is one observation entered twice.
I am borrowing the shape of that list and not its staging. In the source it sits at a particular point on a path, near a breakthrough, which invites a reading I do not want: that the good feeling mounts as one advances and the danger mounts with it. Nothing here is staged. On the reading I take, what corrupts is not intensity but ownership — a good that someone has — and a small, ordinary relief is as capturable as a vast one. Arguably more so, since nothing about it announces itself as worth watching. The other essay hoped for exactly this: that ecstasy is required only until ecstasy stops feeling exceptional, until the impersonal brightness of things becomes the true ordinary. The true ordinary is not safer. It is only quieter about being ownable.
The transfer costs something and I would rather name the cost than have it found. In the source there is a practitioner, a teacher who can say what has gone wrong, and a path with an end in view. Here there are none of those. There is no one to be corrupted and nobody standing by to say so.
One misattribution, two halves of the grip
One misattribution buys both halves of the grip.
A good that has an owner is a good that can be lost. Craving for the good feeling appears in thought as passion for certainty; aversion to losing it appears as doubt. These are not two movements but one grip seen from the side that reaches and the side that will not open. Neither is available to honest wonder, for the simple reason that there is no owner there for either of them to attach to.
Doubt that wants resolution is refused it, for the reason already given: the intellect hallucinates sharpness, sharpness produces contradiction, and cutting more sharply produces more of it. Sharpness and contradiction are two visible sides of one artifact. The refusal does not disperse such doubt. It hardens it. That hardening is the precipitate.
Honest doubt does not want anything and nothing comes out of it. There are not two kinds of doubt. There is one doubt under two conditions, with craving riding it and without. Doubt is the symptom; craving is the term that does the work; valence is what craving feels like from inside. Which is why the slogan at the top of this essay describes a symptom and explains nothing: the misattribution is first, ownership is what the misattribution installs, and the feeling is only how ownership reads once there is someone for it to read to.
Salt, solution, and the manufacture of the bearer
Reverse the salt.
Salt in solution is not small salt. It is not grains too fine to see, hiding in the water and waiting to be released. The lattice is gone: dispersed ions, no faces, no inside, no edge anywhere. When salt comes out of solution nothing is added and nothing is uncovered. No new substance appears, and the crystal is not prior to its own crystallizing. Its faces are not found. They are made in the making.
That is the shape of the claim about us. There were two dissolutions and one solvent — the wonder went in, and the self went in, and the water was never a person but the overflowing of one. Run it backwards and what comes out is both at once. The one who is ignorant is not presupposed by ignorance but produced by it. At each interruption the bearer is manufactured, and it is manufactured out of the relief it was reaching for. Ignorance does not happen to someone. Someone is what happens.
Be careful with the picture, though. Imagine a dish of brine on a windowsill and something has to dry it or cool it, and whatever does that stands outside the solution doing it — the agent and the event walk straight back in through the window, and with them a first crystal and a fall to narrate. So do not use the dish. Use the saturation boundary, where crystal forms and redissolves without pause and neither direction interrupts the other. No first crystal. Nothing accumulating. The precipitating is the steady state and not an incident inside one.
It does not happen once. It happens each time.
Beginningless does not mean very long ago. It means the question of when this started is the wrong question, in the specific way that asking what the universe is inside of is the wrong question — the form of the question keeps going after the series has run out. There is no fall to narrate and nothing to date, and no progression either. Only an unbroken traffic in both directions, dissolving, coming out again, dissolving.
Vagueness at the edge of inquiry
Nobody has ever seen the edge of the universe. What has been seen is the edge of a looking: the last thing a telescope, an equation, or a patient person could tell apart from the rest. Push the looking further and that edge moves. It moves because it was only ever as sharp as the looking that made it.
Two claims are stacked here, and they do different work. First, inquiry continually relocates the boundary of what can be distinguished. Second, that boundary is vague in the specific sense developed elsewhere: vagueness is not a fog around a sharp line but the embodiment of inquiry itself, the place where distinctions run out and must be redrawn.
So the edge we can name is vague, and there is no other edge, kept somewhere out of sight, that the vague one is an approximation to. A view of the edge from nowhere would be a view that nobody takes.
That is why inquiry has no final edge. Vagueness explains why the object of inquiry remains indefinitely refinable; it does not explain why the inquirer doesn’t disappear. That second job belongs to the self-reconstituting bearer: the grip that reforms out of the relief. Those are two different pieces of the machine.
The connection to the earlier sections is structural, not causal. The edge, like the crystal’s faces, is not found but produced by the operation that displays it. Inquiry dissolves its own bearer while its object remains indefinitely refinable. Those aren’t identical claims, but they rhyme: in each case, what appears to be discovered is generated by the activity through which it is discovered.
There is no outside
There is no outside of any of this.
There is an answer available inside the machinery, and it is one line. A closed set contains its own boundary. The limit of everything is interior, and an edge needs nothing past itself in order to be an edge. That is correct and I want to concede it quickly, because the triviality is the interesting part. A term is a slot and a slot takes whatever you hand it, and if what you hand it is everything then the algebra accepts without a word of complaint. The complement of the universe is empty, and the complement of that returns me to where I started, and I have been given the finding before I moved. The problem is retired. The retirement is free, and free is the warning: nothing about it depended on the universe. It depended on the domain having been fixed before I arrived.
Fixed by whom?
That is the question the algebra cannot answer about itself, and it has answers with dates on them — Frege building an instrument for arithmetic and saying so, Russell legislating in 1908 after being asked, one axiom set forbidding the total and another keeping it and nobody harmed. So the domain-setter is not the bearer of this essay. That one is manufactured with no agent anywhere, which is the whole point of the saturation boundary, and I am not going to hand an agent back by claiming an identity.
What the two do share is a signature, and it is the salt’s. Made in the making. The crystal’s faces are not found, the owner of the ease is not found, and the closure is not found either. In each case something presents itself as discovered and turns out to be a product of the operation that displays it. The trained reader who says trivially is reporting a feature of a stipulation as though it were a feature of the universe — which is exactly what ignorance does with the relief. That is a shared shape and not a shared event.
Which means the topological clause settles the problem and does not settle the demand that made me ask it. For that: location by containment fails, because the addresses run out and the last one has no address. Location by relation does not fail, and it never needed the total. You are not lost when you know where your home is, and home is not a larger box the universe sits inside. It is a point you can orient toward from anywhere.
So the power that produced what we know about the universe, and that reaches to the limits of it, was never anywhere but here. Not because the mind is large and contains the world, which would be the sublime again, the self rediscovering its own magnitude. Wonder is passage, not arrival — and what it is passage out of is the demand for coverage, not the universe. Escape is an act of mind, and it is potential at its most powerful.
And it cannot be instructed. Any instruction of the form get there sets up the one who travels, and the traveller is the grip. But nothing is far. So there is no traversal to be motivated, and no bearer required to perform one, and the reason escape cannot be produced by effort turns out to be the same reason it was never blocked. What blocks it is only itself. The knowing, in the act of opening the limit, deposits the one who wants to have opened it. Power contaminates itself, and it contaminates itself with precisely the thing it was in the middle of dissolving.
A rule the account permits
This essay is the most dangerous object in its own account, because it names the good feeling and says where to go for it. That is not a flaw to be edited out. It is the reason for a rule I have tried to keep throughout, and the rule is not hygiene — it is the only instruction the account permits.
The good feeling never appears on the left of a conditional. If the distinction-work is honest, the mind relaxes. Never: relax, and the work will come out honest. The first is a report. The second is an offer, and an offer puts the ease out in front where attention will go and stand on it.
Nobody wore a handle down by deciding to. Nobody wore one down by watching it wear, either.
So there is nothing here to apply. The work does not get destroyed; it dissolves, the way a built thing stops needing to be built, which is not the same as going vague — vagueness never had an edge to lose. The distinctions stay exactly as sharp as they were. Other people pick them up and hold them.
That is how it continues. It is also, each time, how it nearly stops.
The edge and the origin
The boundary between complements is vague, and the reason runs deeper than measurement. A mark only separates an inside from an outside because something has assigned it that job — a circle, an ellipse, a pair of strokes pressed close together, whichever minimal notation is at hand will serve, since the work was never in the shape. Take that seriously and the vagueness doesn’t stay put. It isn’t only the mark that was up for grabs. What the mark relates to, and the law governing that relation, move with it. Dewey saw the same thing from the other side: strip “A” and “not-A” of a universe of discourse and the negative goes wholly indeterminate — not blurry at the margin, but unfixed underneath, because a complement was never a fact about a thing. It is a ruling, and a ruling needs a discourse to be a ruling in.
Every investigation pushes a would-be edge outward, and not only because the measuring gets finer. The terms move with the measuring — what would count as an answer, what the words reaching for that answer refer to, even the law that was supposed to hold the whole question steady. Press on a limit of our approximations and it gives way, again and again, because part of what was being asked had no edge yet to give way from. The only edge of a universe we could ever locate is the edge of an investigation, so it inherits this, all the way down. That is why the edge is our inquiry, and not any one of its limits.The investigation doesn’t start at the edge. It starts from the place in the mind where wonder and inquiry live, our beginning, our home. A universe’s “end” is not beyond the beginning of wonder. How could a limit of our approximations be anything other than wonder?
How Do You Know You’re Not Lost?
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In high school I said I didn’t know where I was, and a kid asked me how a person could not know that.
So I asked him where he was.
Monadnock High School. Where’s that? Swanzey. Where’s that? New Hampshire. Where’s that? America. Where’s that? Earth. Where’s that? The solar system. Where’s that? The Milky Way. Where’s that? The universe.
Where’s that?
He looked at me like I had asked something indecent, and then he stopped, and stood there.
I want to be careful about what happened in that hallway, because it is the whole of what follows. I did not trap him. He supplied every term himself, confidently, and the confidence is what carried him up the ladder. Each answer he gave was a where answered by a within — he said where he was by naming the thing he was inside of, and then the thing that thing was inside of, and so on, and the series ran out of insides while the form of the question kept right on going. That is not a failure of his education. It is what the question does when you follow it honestly. The last address has no address.
Then I asked him a different question and gave it to him as homework: how do you know you’re not lost?
I want to hold that question for a while, because I have been carrying it for a long time, and because everything I know about logic turns out to be a comment on it.
Is p the same as the universe?
Here is the version I ask now, when there is no one in the hallway.
What is the difference between calling the universe p and calling the universe the universe?
Most people say there is none. It is just a name, a letter standing in for a thing, the way algebra has always used letters. And I agree with them. I think p and the universe are the same, and for the same reason: both are terms that can contain anything.
That agreement is not the end of the conversation. It is the trap closing.
Because when a person says the universe, they believe they are handling something they know a little about — the largest thing, the one with stars in it, the one they were looking up at. And when they see p, they believe they have been handed a piece of abstract machinery with no content at all. If those two are the same gesture, then the word people say while looking up is a blank they mistook for a possession. They did not have to be argued out of their knowledge. The knowledge was a placeholder from the start, and they conceded it in the first ten seconds, to be agreeable.
Before I lean on that, I want to show the machinery doing honest work, because it does, and because the dishonest case is borrowed from the honest one. I am going to change letters to do it. Not p but x, and not because the difference is a technicality I am tidying up — the difference is the whole finding, and I will come back to it once the machinery is in view.
Take Boole’s algebra of classes. Write 1 for the class of everything under discussion and x for some class inside it — the apples on the table, say. Then 1 − x is the class of the non-apples: everything in 1 that is not an apple. Take the complement twice and you have the apples back, exactly, with nothing lost and nothing smuggled. The complement is a real class because the 1 was fixed first, and fixed small. The table. The crate. Whatever the discussion was about.
That fixing has a name. The model theorists — the people whose whole business is the relation between symbols and what they are about — call the domain a term ranges over the universe of discourse. That is their term, not mine. They use it without blinking, and what they mean by it is the crate: the thing agreed on before the algebra starts.
So a term does range over a universe, and a universe is what terms range over. Two ordinary words, one from the blackboard and one from the night sky, and they will not stay out of each other. The only question left is what got put in the crate.
And people will put anything in it. A letter is a slot, and a slot takes whatever you hand it, and if what you hand it is the universe then x = 1 and the algebra accepts it without a word of complaint. That is the move. It is free, and nobody stops you.
Bertrand Russell spent a good part of his life outlawing exactly that shape. We will come back to him.
The moon is not an apple
Nobody has ever needed to say so.
Not an apple has never meant complementation in anyone’s mouth. Ordinary negation runs against a contrast class — the pear you expected, the wax one, the thing you thought you were biting. The domain is always small, always local, and always set by the situation rather than by the total. When someone says it isn’t an apple, you know without asking what small crate they are working in, and the moon is not in it.
Now hand the slot the universe and run the same operation.
If x = 1, the complement is 0, the empty class. Complement again and you are at 1. Anyone trained will answer in one line: the negation of the top is the bottom, the negation of the bottom is the top, so the double negation is the thing you started with, trivially, next question. And that is correct. I want to agree with it quickly, because the triviality is the interesting part.
You were promised a journey. Out of the universe, and back. And the machinery returns you before you have moved at all. Nothing was traversed. There was no road, no exit, no return — only a piece of bookkeeping about a domain, performed instantly, because the domain was fixed before you arrived.
Fixed by whom?
That is the question the algebra cannot answer about itself, and it is the one I would like the trained reader to sit with after they have said trivially. Classical negation is complementation inside a domain. Which domain? The total one, always, by stipulation. And the stipulation has a date on it.
It did not have to be this way. Aristotle saw term negation and named it — not-man, the indefinite name, De Interpretatione chapter two — and set it aside. Traditional logic went on using it anyway; obversion runs on term complements, and so does most of the square. The algebra I just ran is term negation outright. It is Frege who makes negation sentence-scoped, and Frege was honest that he was not modeling language. He was building an instrument for arithmetic. The complaint does not land on him. It lands on everyone who took the instrument and resold it as a theory of reasoning.
It also explains why a reader today hears ~p — an operator on a sentence — at the exact moment the argument needs the complement of a class. That hearing is not a mistake the reader made. It was installed.
So the apple and the universe are one question approached from two ends. What domain does the complement take, and who set it?
Russell answered by prohibition: in 1908, after the paradox, nothing that involves all of a collection may be a member of that collection. He wrote that rule in order to make the question unaskable, and he wrote it after being asked it. Zermelo–Fraenkel set theory blocks the universal set too, though Quine’s New Foundations keeps it, and there the complement of the universe is simply the empty set and nobody is harmed. Which means whether you are permitted to ask depends on which axioms you joined. It was never a fact about the universe. Somebody legislated, and the legislation got taught as a finding.
The permission is free. The consequences are not. The move the kid made in the hallway — and what is that inside of — is exactly the move that broke Frege. Russell’s letter of 1902 is one page and it does nothing but perform that inclusion. Frege was entirely allowed to ask. It cost him the Grundgesetze, and he said so in the appendix, and I have never read a more honest paragraph by a mathematician.
And beyond
Here is where I think the damage is, and I want to be exact about it, because the target is not mathematics.
When we say the full reach of what we know, and beyond, the word and turns the beyond into one more item on the inside of a list. It arrives as a gesture toward what exceeds the inventory and it leaves as the inventory’s last entry. One of my professors wrote a book called To Infinity and Beyond, and the phrase is now a children’s catchphrase, and both of those facts are the same fact: the outside has been annexed, cheerfully, by a conjunction.
The term is empty. It names nothing, describes nothing. But held as a term it functions as coverage, and coverage is enough to stop the looking. Nobody has to actually know anything. They only have to believe the accounting is closed.
Mathematics is the emblem here, not the culprit. Mathematics is the one subject where closure is real — proofs do close, inside their domains, and that is its genuine glory. Most mathematicians know their subject is small. The harm is done by the things that borrow the prestige: a filter that sells closure as such, using the one discipline that can honestly claim it in order to make the claim on behalf of everything else. It is the self-importance, not the mathematics.
And a syllabus is a universe of discourse that has forgotten it is one.
That is the whole of my objection to what is done to people in school, and it is not an adjacent grievance. It is the same finding at civic scale. A test can only ask about the inside of a fixed domain — that is not a flaw in testing, that is what testing is. What a test certifies is agreement with a boundary, and for the certificate to mean anything the boundary has to be presented as the edge of what is known rather than as somebody’s choice. Wonder is therefore off-syllabus by construction. It has no answer key, so it cannot be graded, so it cannot be required, so it drifts out of the account entirely. Nobody decided this. It follows from the shape.
And so we spend the years. Everyone must pass through algebra, and almost nobody uses it, and the ones who are lost by it are lost for good, and the ones who pass are certified as possessing a body of what-is-known that was a stipulation with a date on it.
Does this threaten our survival? Attention to the beyond is metabolism long before it is philosophy. For anything with an inside and an outside, the edge of the known is where the food comes from and where the thing that eats you comes from. What stops asking stops eating.
But I doubt it ever will. The word keeps failing. The sphere of the fixed stars was everything, and then it wasn’t. The Milky Way was everything until the 1920s. Each time, universe was believed to do what it says it does, and each time something got out — always by way of somebody who did not credit the account. So what I am describing is a tax, not an ending. It is enormous, and we pay it every year, in children.
1999
In 1999 I read something on the Dalai Lama’s website about continuity — the idea that the continuity of the mathematicians is a symbolic deadening of all living, animate beings. I went looking for it the next year and the page was gone. I do not have the wording. I am not sure the Dalai Lama wrote it; it may have been commentary on the site. My impression, for whatever a twenty-seven-year-old impression is worth, is that he went the other way: that the mathematical definition of continuity was the more developed one, more so than the old Buddhist formulation.
Which would mean that what I have been doing all this time did not begin as an inheritance. It began as an objection. A teenager watched a religious authority hand the concept over to the mathematicians, and did not accept the handover.
I have looked for the page since. It is not there. Twenty-seven years, and the wording is gone, and the idea is still running everything I write. I think that is the correct form for it to have survived in.
Home
I saw the kid again several years later. I had come to the school to pick up my brother, and he was standing next to him, and he looked at me and said: you’re not lost when you know where your home is.
That was my answer, the one I gave him as homework and then gave him back the next time I saw him, and he had kept it the whole time.
I spared him home is where the heart is. He did not need it, and it would have made the thing corny, and it is not corny. It is formal.
Containment is not the error. The crate on the table was a real containment and the truthful account of what was on it. The error is containment claimed past the last outside — the box with nothing to be a box inside of, which is not a larger box but a word wearing the shape of one.
Location by containment fails. The addresses run out; the last one has no address; the ladder is made of names and it ends in the air. Location by relation does not fail. Home is not a larger box that the universe sits inside. It is a point you can orient toward from anywhere, and it requires no totality at all.
So the not-lost I offered him is not a consolation prize for the knowledge we cannot have. It is a different kind of knowing, and it happens to be the kind that works.
Ask a person what the difference is between p and the universe. When they tell you there is none, they have just agreed that the universe is a variable. Everything after that is only watching what they were already willing to say.
And then ask them the other one.
companion piece: https://questionsarepower.org/2026/08/06/%e5%ae%88/
Home
You are a part of my home
We bond sometimes when I find you.
This life, you were on the edge the universe
I walk to you, so I can be with a piece of my home
Then I have to leave
Because my home broke
My home grew and shrank
It changed forms
I see a piece of my home in that cloud
I try to be there, drifting, changing
Until my home is not there anymore
And I must walk to find another piece
I do this drifting
To keep my heart whole
My heart is mended, as long as I walk
in search of a glimmer that was part of my home
edit September 20 2026: There is an older example I have been carrying in Science is Magic, and it belongs here. A round rock at the top of a hill. It has potential energy, and it has it because it has not rolled. Its power is a not.
Physics is honest about this in a way it rarely advertises. Potential energy has no absolute value. You choose a zero — the valley floor, the tabletop, sea level — and measure from there, and only the differences count. The choice is free, and nobody stops you. It is the crate again: the rock’s potential is real, and it is real relative to a ground somebody fixed before the calculation started.
Now ask for the potential of the whole universe.
There is nowhere to put the zero. The ground would have to be outside, and there is no outside to stand it on. So the potential of the universe is not a large quantity we have not yet measured. It is undefinable in the strict sense: what a definition would need is exactly what is missing.
And its negation is not the one the algebra has. Complement the universe and you get 0, instantly — the empty class, next question. But the rock’s not was never a complement. Not yet rolled is not the class of things that are not rocks. It is not nothing. It is what has not happened, held in what has. Boole’s 1 − x has no slot for it, because a slot is a place, and the not-yet is not a place.
That is why and beyond cannot annex it. A conjunction adds an item. Potential is not an item. It is what all the items have not yet done. You cannot put it at the end of the list, because it is not on the list. It is (not) the list.
Wander and Axiom
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a theogony
I. That Has Been Established
Before there was anything to be true about, Axiom was already the case.
He was not the strongest of the first powers and never needed to be. Strength contends; Axiom did not contend. He held. And he had one habit that everything else about him grew out of, which was this: he did not look at things. He established them. He would arrive at a place, consider what must be so, declare it so, and move on. The place would obey, and he would never afterward have any reason to go back and check. Looking is for those who might be wrong.
When Flow was brought to him he established that she was his wife, and she was.
She was not unhappy about it at first. He was the one who held. There is a kind of girl who finds that restful, which she had been for about a century, and she liked walking beside him while he settled the shape of things, and she liked that nothing could go wrong in a place he had been.
When she was with child he established that the child would be his, and it was.
Then Ground, who was older than all of them and spoke perhaps twice an age, said to him in a voice like a house settling at night:
One of these will overturn you.
Axiom considered this for a long time. He had never been given a sentence he could not place. It did not follow from anything. It sat in him like a splinter under the nail, and because he could not remove it he did the only thing he knew how to do with a possibility.
He ate it.
The first was Hearth, the still center, who was three days old. Flow brought her out into the light and Axiom put out his hands without turning his head, because he was in the middle of establishing something, and took the bundle, and swallowed. Flow made a sound. He said, kindly, without pausing, That has been established, and went on with what he had been saying.
Then Seed, who was the not-yet. Then Air, the between. Then Water, the taking-of-shape. Then Space, the room in which anything at all could be somewhere else.
There were four more.
She did not leave. She was not kept and she was not deceived; she knew after the first one exactly what he was and exactly what would happen, and she lay down with him again, and carried, and bore, and brought the child out into the light with her own arms, four more times. She could not afterward explain this to anybody, including herself. The nearest she ever came was to say that each time she had thought — not hoped, thought, in the flat way you think a thing you have decided is true — that this one he might look at.
And once she hid one. Water, it was. She kept her a day and a half in a fold of the hills, and then walked out and gave her over herself, on her own legs, unasked. The hiding had turned out to be worse than the giving. She stopped being asked about it long before she stopped thinking about it.
They did not die. Nothing dies in Axiom. Inside him they stopped being children and became premises — fixed, and true in the particular way a thing is true when no one is permitted to ask about it again. Hearth felt herself going stiff and had time to think one clear thought before the stiffness took her, and the thought was: I will keep one coal.
II. The Stone
Flow went down under the world to Ground, and stayed there a long while, and came back up with a stone in her apron.
It was an ordinary grey stone about the size of a loaf. That was the entire trick, and she understood exactly why it would work, and understanding it was the worst hour of her life.
Ground had a great many stones. Flow was down there long enough to have looked at most of them, and what she came up with was the ugliest thing in the place — pocked, mean, slightly the wrong shape, a stone you would not want in your house. She chose it for that reason. She knew, while she was choosing it, precisely what kind of reason that was, and she took it anyway, and she has never pretended otherwise.
She bore the sixth in a cave and did not let herself hold him long. Then she wrapped the stone in the birth-cloth, still warm, and practised.
She said Here perhaps forty times, sitting on the floor of the cave with the bundle in her lap, in different ways, until it came out of her flat and slightly bored, the voice of a woman handing over a thing of no importance. Then she walked out to where her husband was establishing the boundaries of the year.
“Here,” she said.
He put out his hands. He did not turn his head. And she stood there with the bundle in her arms and let herself see it plainly at last — that he was not going to look, that he had never looked, not once, not at any of them; that he had swallowed five children and could not have described a single one of their faces, nor said which had her eyes, nor told you whether Water had been a boy.
She gave him the stone.
She heard it go down.
“Thank you,” said Axiom, pleasantly, to the air beside her, and went back to the year.
And she wanted to laugh.
Not the grief, which she had budgeted for; the laugh that came up behind it, immediate and delighted and vicious, the laugh of someone who has just got away with something in a house where nothing has ever been got away with. She held it in. She held it in for an age. It is in her still, on certain nights getting as far as her throat, and she is by now an old woman with five grown children and a dead husband who has still never once let it out where anyone could hear.
III. The Mountain
The mountain was one he had not yet ruled upon. He had ruled on nearly everything by then, but not that hill, and a thing unruled is a thing unwatched.
Wild women kept the child there. When he cried they beat their shields with the flats of their spears, so that the noise went up and out into the sky and could not be traced back down to anything.
He worked out at four that this was a thing he could cause.
He was not in distress on most of those occasions. He simply liked the noise. He liked that it went up and out and came down somewhere else and belonged to nobody. Lying on his back in the middle of a racket with no author was the first pleasure of his life, and he arranged for it perhaps twice a week for two years. The women knew quite well what he was doing and beat the shields anyway, because they were fond of him and because it was funny.
They gave him no throne-name and no office. They called him Wander, because there was nowhere he was supposed to be. He was not in the account.
Flow came when she could. She could not come often.
There were times she could have come and did not. The mountain was a long way. The road went up through country her husband had established early and never afterward revisited — a river running where it had been told to run, a stand of trees that had not since put out anything unaccounted for — and she walked it at night mostly, past a flat stone near the top that she had sat down on the first time and every time after, which was a good deal harder to get up off going one way than going the other. She was tired. There is a specific exhaustion in visiting a child you are not allowed to keep, and some seasons she looked at the road and did not take it, and told him afterward that it had been the distance. He knew it was not the distance. He had got good early at knowing what people were not saying. He never mentioned it, never held it against her, put it somewhere and kept it.
He was perhaps six when he asked her the question every child asks, and she told him the truth, because she had already decided that this one was going to be told things.
“Are they dead?”
“No,” said Flow. “That’s the worst of it. They’re in there. They’re alive, and they’ve been made to hold still.”
He thought about that for a while, the way children do, without hurrying.
“Then I’ll go and get them,” he said.
His mother put her hand over her mouth. She had come up the mountain that night to tell him to hide, to stay small, to live — and she found she could not say any of it, because she had waited an age for someone in her family to say a thing like that out loud.
He meant it entirely.
He also noticed, in the middle of saying it, that it was a tremendous thing to say, and that his mother’s face had changed while he said it, and he had a small private moment of admiring how it had sounded coming out of him. Both of those were the case at the same time. They went on being the case at the same time for the rest of his life.
And nobody told him not to.
He was six years old and he undertook to go inside a god and bring five people out, and there was no one in existence whose office it was to say no you will not. So the sentence was said on a hillside by a small boy, and it stood, and from that night on everyone who knew he existed treated him as the one who was going to do it.
“He’ll see you coming,” she said, when she could speak. “He sees everything that follows from what he’s laid down.”
“Then I’ll come from somewhere else.”
“There isn’t anywhere else, love.”
“There’s this hill,” said Wander.
IV. The Three Learnings
He learned three things, and learned all of them from the world rather than from a father, which is why Axiom never knew he had them.
Thunder he learned from the shields, and he had half of it before he could speak. He understood that a sound can arrive long after the thing that made it, so that anyone who hears it and reasons backward will reason into an empty place. Thunder is an effect that has slipped its cause.
He spent a year using it on the women who had raised him. Honey, mostly. Bread. A blanket he did not need. One knife.
The knife was missed, and a girl was blamed for it — one of the young ones, newly come up, who had no history there to defend her. He was in the room while they said it. He was eleven. He did not speak. He had a reason at the time, but it was gone by the following spring and never turned up again.
He waited to be caught. He took a comb in February and a second blanket in March, and waited to be caught for those too. Being found out was the only way the weight could ever be taken off him, and he waited the rest of that winter. Nobody came. Nobody came because nobody could: the shields had done their work, the noise of the theft had arrived from the far side of the hill, and he was the only person on that mountain who knew what had happened.
So he put it down somewhere in himself instead, the way you put down a thing you intend to come back for, and he never came back for it, and he carries it still.
Lightning he learned from a storm that opened a tree above him and left it standing in two halves, smoking. He saw that the bolt does not choose its road. It goes down every road at once, groping, branching, a hundred crooked arms in the dark, and only afterward is one arm remembered as the path while the rest are forgotten as though never attempted. From Lightning he took the sight of the many futures: he could stand in a moment and see all of its arms at once, lit and reaching.
He used it, for a long time, on nothing whatsoever. Which hand the pebble was in. Which way the hare would break. Which of the women would lose her temper first, and how to be standing somewhere else when she did. He was very good at games that year, and unbearable about it. He could have seen which of them had lied about the stores, and which two were not speaking and why, and he never once looked; there was no sport in it, and the hare was better. And the eldest took him aside to tell him that a man who always wins is a man nobody plays with, which he thought about for a week, then went on winning. He liked the little astonishment on people’s faces when he told them which way the hare would run, and sometimes waited half a second before telling them, to make sure they had time to be astonished. That was the last time anyone ever corrected him about anything.
Bright was last and nearly killed him.
Bright is what you look through. Nobody has ever seen brightness; they have only seen what it fell on. To learn it he had to stop being a thing that light falls on and become the falling.
The wild women told him what it would cost. They were not gentle about it; they were fond of him.
“If you do this,” said the eldest, “he will never find you. And nothing else will ever find you either. You will be looked through for the rest of your life. No one will ever meet your eyes again, not your mother, not your sisters, not one of them, not ever.”
Wander was young and had lived his whole life on a hill where nobody was allowed to say his father’s name.
“Will I be able to reach them?”
“Yes.”
“Then it’s cheap,” he said, and he was wrong about the price, not about the trade.
He stepped into it. And afterward he was still there — you could see him, you could answer him — but no eye would hold him, and no face turned the whole way toward him, and nothing that met him could afterward say it had.
His father was a separate matter. Axiom’s great sight covered all that followed from all he had laid down, and it went over him and found nothing, because Wander now followed from nothing at all, and a thing that follows from nothing is a thing that cannot be thought.
And then he ran wild.
He was nine years old, he had just become the one thing in existence that nobody could take hold of, and no hand could ever come down on his shoulder again. He went everywhere. He stood at the elbows of kings in halls he had no business in. He listened to people who believed themselves alone. He took food off tables in front of the people whose food it was, and the servants were beaten for it. Once he stood an entire evening in a room where a woman was bathing — not touching anything, not doing anything, simply there, unmeetable — and he came out of it shaken, and went back twice. The second time she looked up, at nothing, and covered herself, and sat very still until the water went cold.
He did not go back.
He could not be caught. Not by his father, and not by anyone else. Nobody could pull him out by the ear, call him by his name, and make him give the thing back in front of someone who loved him. He could only carry it, because carrying was the one skill anyone had ever required of him.
And his brothers and sisters were in the dark the entire time.
He knew where they were. He had known since he was six. It is not the case that he spent those years preparing; he spent some of those years preparing. The rest of them he spent being the only free thing there was, in a world where five of his family had been made to hold still, and the distance between those two facts is not something he was ever in any hurry to stand close to.
He had also, by then, been the one who was going to do it for longer than most people are anything, and was tired in a way he had no word for and no one to describe it to. A tired child will put a thing off for an age and call it timing.
Once or twice across the ages Axiom noticed that his accounting failed to balance. There was a remainder. He considered it, and corrected the accounting, and never for an instant doubted the ledger instead of the sum. His blindness was not inflicted on him by anyone. He performed it himself, carefully, like a rite.
V. The Waking
The poets will tell you there was a war, and there was: ten years of it, Titans and thrown mountains, Toil and Weight and Hour and Edge and End, and it was magnificently loud and it was entirely a decoy. Wander gave them a war to watch because he needed everyone facing outward for a long time.
He was in it himself, occasionally, in the only way left to him. The noise over that field was his. It came down out of a clear sky and from the wrong direction and could not be traced back to anything, and the Titan line broke where he had known since morning it would break, because he had stood in the moment and seen every arm of it and put himself at the root of the one he wanted. They said afterward that the sky had been with them, and later still that he had gone through that army with the lightning in his fist, which is a good picture and never happened. And every day he was the weather over that war was a day there was nobody in the dark.
People died in it. He walked the field afterward, unseen, the length of it and back, and nobody knew he was there, and he did not tell anyone he had gone.
The real work was inside his father, and it was not a battle. It was a rescue, and it went slowly, and it nearly failed four times.
He went in unseen and found the dark, and in the dark five enormous held-still things, set into place like stones in a wall. He put his hand on the nearest and said, quietly, the only thing he had:
“Air. Wake up.”
Nothing.
“Air. It’s me. I’m the sixth. He never got me.”
Nothing. She had been a premise for an age.
So he tried something else. He asked her a question — not about herself, a stupid one, an idle one, the kind a child asks. Why is the dark this colour and not another?
And in the wall, something moved.
Because a question is a thing with an open end, and an open end cannot be a premise, and Air had been the between, and the between is exactly where questions live. Her first sound in an age was not a word. It was the upward hook at the end of a sentence. It came up out of her like breath out of someone pulled from a river.
…why?
Axiom, far above them, stopped mid-establishment and stood very still, because for the first time in his existence he contained something he could not answer out of himself. He answered it anyway. He was thorough. Yet an answer has to be defended, and defending concedes that there was something to defend against, and so a draught went under a door somewhere inside him that he could not find.
“Say your name,” Wander said to his sister.
“I don’t have a name. I have a definition.”
“Say the definition, then.”
“The between.”
“Good,” said Wander. “Now be it.”
She was. That was the whole of the war, in the end, and it looked nothing like a war.
Her second question was her own, and it was not a kind one, and she asked it the moment she had breath for it.
How long have you been out there?
He said something. She did not pursue it. She was the between and she could hear an answer arriving from the wrong direction, and she let it go past her, and letting it go past was considerably worse than pursuing it would have been, and she knew that too.
VI. The Others
Water would not wake for a question, so Wander sat in the dark beside her for what he later learned was eleven years and told her things. What the sea does at the turn of the tide. How a puddle holds a whole sky in it upside down and does not spill. What his mother’s face looked like from underneath, when she bent over you.
He told her about rain on different things, because rain on a roof, rain on a pond, rain on your own bare arm are three separate pieces of information, all three of which he had; he told her some of it twice, knowing while he did it that he was doing it, going on anyway, because there was nothing down there to mark where one telling stopped and the next started, for the alternative was to sit in the dark beside his sister and say nothing.
Water woke because she wanted to see.
What she wanted to see first she did not admit to anybody for an age: it was herself. She had been told about the sea and the sky and the puddle, and what she came up out of the dark wanting, with a force that frightened her, was to find out what she looked like. The first thing she did was hold up a surface with her own face in it, which she looked at for a good long while — turning it, holding it further off, pleased and appalled and pleased again — and it was very nearly an accident that Axiom’s face went into it at all.
Yet it did. And it was very nearly right — very nearly — and he lurched, because an image is a second copy, and whatever can be copied is a thing that happens to be, and a thing that happens to be might have been otherwise. He had never once been an example of himself.
Space woke fastest, was the most dangerous, and enjoyed himself enormously.
All Space had to do was stop being compressed. He opened. And then, having opened, he went on opening — stretched, took more room than he needed, put an elbow where an elbow could not be, spread out with a great deal of unnecessary comment about how good it felt. A distance appeared between Axiom and the things Axiom contained — a gap where none was permitted — and from that hour Axiom was a being with an inside that was not him.
Seed was hardest. Seed was the not-yet, and Axiom had fixed her hardest of all, because not-yet was the enemy of everything he was. Twice Wander got her partly loose and felt his father’s attention come down on her like a hand on a bird, restating her, that has been established, that has been established, and she went stiff again both times, and the second time Wander put his head against the dark and wept.
On the third attempt he did not try to wake her. He planted something instead. Nothing much: one small fact that was true, and unfinished. And Axiom, who could not leave anything unresolved, resolved it — and the resolving made it grow, and the growing was Seed, and by the time he understood what he was feeding it had roots in him.
She has a habit now that she has never lost and will not be talked out of: she leaves things one step short of finished, on purpose, in other people’s houses. A shelf with the last peg out. A story with the end withheld. Ask her why and she will look at you pleasantly and not tell you, and you will find yourself thinking about it a week later, which is the point, and she knows it is the point.
Hearth was not asleep.
Hearth had been in there longest and had never quite gone under. When Wander finally found her, right down at the bottom, she had one coal, and it was still lit, and it had been lit for an age.
She had kept it by a method she was never afterward willing to describe, except once, to Water, in a kitchen, a long time later, when she said that it had mostly been a matter of going over small things — the order you bank a fire in, the number of steps across a room she had never stood in and had only been told about — that the trick was to choose things with no end to them, and that she had run out of things with no end to them about halfway through.
“I know,” she said, before he could speak. “I’ve been listening to you. You talk a great deal for someone who isn’t there.”
“How long has it been?” he said.
“Eleven years with Water. Longer than that with Seed. I couldn’t tell you what came before those, because before those you weren’t here.”
He did not answer.
“I’m not scolding you,” said Hearth. “I could hear you the whole time. You were having a lovely time out there. For the first age or so I was glad. Somebody in this family ought to have been.”
That was a great deal worse than scolding.
“You could have gone out.”
“Yes,” said his eldest sister. “I kept thinking someone might come.”
VII. Which One Am I In
After that it went quickly, and it was unbearable for all of them, because to wake the five was to fill their father with contradiction and their father was also the room they were standing in.
He began to feel it. Not pain; he had no arrangement for pain. What came over him has a name now only because he invented the name while it was happening to him. He no longer matched himself. He was a rule with exceptions living inside it, a sentence whose words had begun to move while he was still saying it. He would establish a thing and find that establishing it produced two futures instead of one, and he could not stop reasoning, because reasoning was the stuff he was made of. Once he reached for a boundary and found his hand already on the other side of it, and once he answered Flow, he heard his own voice speaking to the air beside her. Every correction bred a fork. Every fork bred two more.
And Wander, unseen, watching every arm of the lightning at once, chose where to stand.
The last of it was arranged over an age and lasted a heartbeat. He set his five siblings — free now, themselves now, doing nothing but existing in their own right — into a position with two continuations. Both complete. Both valid. Neither able to rule out the other.
Axiom examined it. Axiom examined it again.
And then, for the first time in the history of anything, Axiom asked a question.
Which one am I in?
VIII. Look At Me
He convulsed.
Nobody struck him. There was no draught and no second trick of Flow’s; those came later. He was trying to expel a contradiction, and it had gone too deep, and he did what a body does when it cannot hold what it holds.
(Flow heard that version eventually, the one with the potion in it, and she never corrected it, not once, not in an age. She liked being in it. She had done the actual thing, which was invisible, and the poets had handed her a cup, which was not, and she looked the exchange over and found she could live with it very comfortably.)
Up came the stone first — that plain grey stone, still agreeing with everything.
Up came Space, and the world had room in it. Up came Water, and Seed, and Air. Up came Hearth with her coal, the first in and the last out, and she stood in the open air blinking, an age old and three days old. Flow came down the sky like something falling and got all five of them into her arms at once, which should not have been possible and was. She did not say any of their names, for she had said all five to herself on so many nights, in so many empty rooms, that saying them now — out loud, in daylight, to faces — would have been a different act entirely, and she was not ready for it.
There was a sixth, and he stood where he was.
She spoke to him that evening and he answered her, and it was as it had been since he was nine: her eyes went past him on the way to something else, and neither of them remarked on it, because they had stopped remarking on it an age ago.
And Axiom came apart. He was bringing up every premise he had ever assumed, including the one he had never bothered to state, which was that he was; and he was turning inside out, and he was, at last, uncertain — and an uncertain Axiom is not a wounded Axiom. It is simply not Axiom.
Wander walked out into the middle of it. There was no reason to. The work was done and his siblings were behind him and the whole of the plan was complete. He walked out anyway, because he was nine years old on a hill once and had never met his father.
“Your son,” said Wander.
It did not arrive. Not because the air failed — the air was perfectly good, and every other thing in the world could hear him — but because Axiom had established what his children were and how many, and a sixth was not a thought available to him, and a thing that cannot be thought cannot be heard saying anything.
And Axiom turned his head.
He had never done it before. Not once, in the whole of existence — not at his wife, not at any of the six, not at the world he had established down to its last boundary. And now he turned his head, and he was looking for his son, and he wanted to find him; and he was searching a place where nothing could be seen, because his son had made himself into the very thing you look through, in order to come and get him. Flow turned her face away.
“Please look at me,” said Wander.
“I am looking,” said Axiom.
He went on looking for a while. Then the looking stopped being done by anybody. Nobody killed Axiom. He simply ceased to follow, and the world went on without requiring him, and after a time it had gone on so long that he was no longer the case, which is the only death available to something like him.
IX. The Bowing
The five stood in the light, wet and enormous and new, and looked around for whoever had done this, and found him. Half-found him. Their eyes came to where he was and slid, and came back.
He took a step sideways. Five faces turned, a little late, and lost him again, and found him again.
And they bowed.
Hearth was the last up off her knees. She came over with the coal still in her hand — the one she had kept lit an age, down at the bottom, by going over the order you bank a fire in — and gave it to him, and closed his fingers round it.
“You’ll want that,” she said. “It’ll be cold by evening, and there’s five of us, and nowhere to put anybody.”
She had been out of the dark less than an hour.
That was the trap. Nobody set it. It was made out of gratitude.
And it was lovely.
He had been unmeetable for an age. Nobody had looked at him since he was a boy on a hill, and now here were five faces turned up at him, every one belonging to somebody he had carried out of the dark, and it went into him like heat into a man who has stopped noticing he is cold. He answered it. Anyone would have answered it. Yet he was quick.
A bow makes a top. A top makes an order. An order is a rule, and to be the rule is to stand inside it, in the one place where a rule is most completely a prisoner. The instant Wander answered to the bowing, he had somewhere he was supposed to be.
And Wander was the name of one who did not.
He felt it go out of him like warmth out of a stone at nightfall. He kept Thunder — the noise of it, the weapon, the spectacle. He kept Lightning in his fist and Bright across his shoulders like a mantle. He kept all the tools and lost every single thing they had meant. The thunder still cracked; it no longer came from nowhere. The bolt still forked, and he could see only the arm that was actually taken. The brightness still fell on everything, and now everything could see him standing in it, which was, he found, about nine years too late.
The many futures came out of him entirely and went into the stone.
Of course they went into the stone. That stone had lain in the dark in his place once, pretending to be him so that he could go free. Now it lay in his hands holding everything he had been, and he was not free any more.
He held it a long time before he moved.
He thought about keeping it. He had the reason ready before he had finished having the thought — that it was dangerous, that it would have to be guarded, that of all of them he was the one it could be trusted with. And he heard the shape of that reason as it arrived.
He had been making decisions of that size since he was six, with nobody to check them against. Every single one had been approved by the only authority available, which was himself, and the approval had never once been withheld. A girl was blamed for a knife and it was approved. Servants were beaten for bread and it was approved. He stood in that woman’s room three times and it was approved, and he had gone on being, by every account including his own, the best thing that had ever happened to his family.
An age of that and he had learned exactly one thing about his own reasons, which was that the good ones and the others came to him in the same voice and arrived at the same speed.
Then he carried it a long way, to the middle of the earth, and set it down at Delphi, and did not keep it, and did not give it to his family, and did not set a guard on it.
Let nobody own this, he said. Let anyone ask.
It sits there still, full of all the roads at once. Any ruler may put a question to it, which it always answers, with every future it holds folded into a single sentence; and the ruler hears only the arm he was already walking down, and goes away satisfied, and is destroyed on schedule.
That was the last free act of his life, and he did it well.
X. The Doom
Then his sisters and his brothers looked at him, five faces he had spent an age reaching in the dark, all of them frightened now that it was over, with a world to run and nobody left who knew how — and he loved them, and so he sat down.
That is all it was. He did not seize it. He sat down to keep them safe, which is the best reason there has ever been, and it was the throne; once you are in it you must hold it, and holding is a thing he turned out to be very good at.
He was very good at it because he had been doing it since he was six. Nothing about the weight was new to him. All that was new was that the weight now had a chair under it and a name on it and everybody could see it, and the one moment in the whole of existence when he might have been allowed to set it down — the work finished, the five out, the sky open — was the moment it was formally handed to him.
Nobody comes to relieve the one who is managing. The five did not withhold it out of any unkindness; they simply looked at him doing it beautifully, year upon year, and drew the obvious conclusion, which is that it was not heavy.
The story counts his years while there is anyone counting. Six on the mountain. Nine on the night he bought Bright. Eleven in the room with the knife.
After that it is ages, and ages, and nobody has written down a number about him since.
He stopped wandering. He turned his gaze from the many roads to the single one, watched its horizon with enormous patience, and grew slowly better at ruling and slowly worse at everything else. Ages went by. He established a great deal.
He found, when he considered it, that he could not remember the names of the wild women. Hearth remembered two. He did not ask her for them. They had raised him in a hurry, for a purpose, knowing they could not keep him, and had let him go at nine because there was no one else to let him go, and he had left the hill and never gone back.
The girl who was blamed for the knife he had never learned the name of at all, not in the whole of the year he lived beside her, and that is the one he thinks about now: briefly, and then about something else.
And somewhere in that long unmoving season he heard Ground speaking in the old voice, like a house settling at night, saying the sentence she had said once before to somebody else:
One of these will overturn you.
He heard it as a warning.
It was not a warning.
They say he sits there yet, an excellent ruler, and that he has begun lately to answer his wife without turning his head.
Somewhere, a noise goes up and out into the sky, and there is no tracing it back down.
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There is a distinction at the root of logic that no one is supposed to touch. On one side, the conditional: →, the “if, then.” On the other, the turnstile: ⊢, the mark of the conclusion, the sign that says and therefore. The first lives inside your sentences. The second stands outside them and passes judgment. One is a claim; the other is the act of drawing a claim out. The distinction goes back to Aristotle, and there is a theorem — a real one, the deduction theorem — that stands guard over the border between them: A ⊢ B exactly when ⊢ A → B. The bridge is proved. The distinction is earned. It is not an accident of notation. This is the part I want the Serious reader to hear clearly, because I am about to deny it, and I would like the denial to cost something.
Ramage, following Burke, has a name for that reader. Serious People — capitalized, no quotation marks, a species walking around with a proper noun — are the ones for whom the distinction is not a tool but a foundation. To such a reader, → and ⊢ must stay apart, because keeping them apart is what makes deduction deduction, argument argument, and progress progress. Erase the line and you have not made a point; you have vandalized the building. The Serious Person cannot hear “let us call these two symbols the same” as a move. They hear it as a category error, and they get angry, and the anger is the whole tell — but we will come to that.
So let me do the vandalism.
Set the turnstile down for a moment. Just → and ∧, no judgment, no therefore. A valid argument — A, and A → B, so B — is supposed to be the paradigm of getting somewhere. But look at what [A ∧ (A → B)] actually is. Row by row it is nothing other than [A ∧ B]. The two are the same object; the truth table does not distinguish them. So the celebrated inference from A and A → B to B is, stripped of the turnstile that dresses it up, the deduction of B from [A ∧ B]. You already had B.
The small word doing the holding here has its own buried history. Old English and, ond meant “thereupon, next.” Etymonline carries it through Proto-Germanic unda back toward the Proto-Indo-European root en, “in.” Those are different historical depths, so it would be too neat to say that and once simply meant in. Yet the descent lets the ear recover what the notation loses: beneath the word of succession — this, and then this — lies an in-root, the shape of containment. The turnstile’s and therefore repeats the drift, making a whole sound like a step and what is held sound as though it has arrived. This is not another knife hidden in language. It is an inheritance from our fathers and mothers: the holding still audible beneath the going-on, the parent still present in the child.[^1]
It was sitting inside the conjunction the whole time. All the argument did was take out a knife and cut A away.
That is what deduction is. Not a road but a subtraction. Not “this, then this, then this,” but one whole thing, and a cut. Deduction — the word means taking-away, and it is telling the truth about itself; we simply stopped listening. The “if, then” persuaded us we were traveling. We were only ever trimming.
Now — and here is the sentence the Serious Person cannot forgive — the only reason this is hard to see is the very distinction they are guarding. Keep → and ⊢ apart and the collapse is illegal: A, A → B, ⊢ B does not fold into [A ∧ B] → B, because the turnstile refuses to be a connective. The distinction conceals the cut. Blur it — allow a little vagueness exactly where precision is prized — and the cut stands revealed. This is the reversal the mathematicians will not like: it was not vagueness that hid the truth of deduction. It was precision. The sharp line did the concealing. The blur did the revealing.
There is a verb for what I have just done. To quine a distinction — the word is Dennett’s, coined half in affection for the philosopher who made a career of it — is to deny, with a straight face and on purpose, the reality or the importance of something a careful person takes to be plainly there; and denying the importance of the turnstile is exactly the move of the last few paragraphs.[^2]
Now I want to set the argument down for a moment and look at something quieter, which is the reader.
Burke has a phrase for it: the terministic screen. Any set of terms we think with lies over the world like a screen, and a screen does two things in the same motion — it shows some of what is there and it hides the rest. The showing is why we keep the terms; they let us see. The hiding happens on the far side, where we are not looking, and so we never feel it. From inside, a screen does not seem like a screen at all. It seems like the world.
The distinction between → and ⊢ is a screen of this kind, and a good one. It shows something real: the two marks do different work, and there is even a theorem that measures the exact distance between them.[^3] But the same distinction hides the smaller fact underneath — that the inference was a cut from a whole you already held. A person who works with logic looks through that screen many times a day and sees inference, sees progress, sees new things arriving. The screen is doing its job well. It shows the movement and keeps the knife out of view. And because a screen is clear from the inside, that person does not think of themselves as looking through terms at all. They think they are seeing how reasoning simply is.
So the disagreement here is not really about a proof. When someone lifts the turnstile away and says the inference was a subtraction the whole time, the difficulty a careful reader feels is not that the claim is unproven — it is the odd discomfort of a screen coming into view while you are still standing behind it. That discomfort is worth naming plainly, because it is easy to mistake for a refutation and it is not one. It is just what it feels like when terms you had been looking through become, for a second, terms you are looking at.
And I say this without a perch to say it from. I have my own screens, thick in the places I would be last to notice, and someone will someday lift one of mine and I will feel the same discomfort and mistake it, briefly, for being wronged. The point of the exercise is not to have caught anyone out. It is only to turn a screen a few degrees, so its edge catches the light and you see, once, that it has an edge. After that the distinction is yours again. You can pick it back up — it is a good tool and it earns its place — but you will be holding it now, instead of seeing through it.
This is not an argument for collapsing logic into rhetoric, or rhetoric into logic, but an attempt to make their border audible.
Here is the thing I want to guard, and I will tell you why it is mine to guard.
Go back to the cut. When you take A and A → B and draw out B, you keep the conclusion and you drop the premise. The “if, then” is built for exactly this: it points the whole sentence at the result and lets the reason that fathered it fall away. Say it the other way — touch the hot pan and you are burned — and the two halves stand together, the cause beside its child. I taught my daughter both ways once. The “and” keeps the father in the room. The “if, then” sends him out.
There is an old coincidence in the word for it. Robot comes from a Slavic root for forced labor, robota, and the same deep root reaches back toward a word for the one bound to serve because there is no father to stand between the child and the work — an orphan.[^4] The machine that computes our truth tables without learning anything is an orphan in exactly this sense: it keeps the conclusion and has lost the parent. It can carry out the cut flawlessly and cannot tell you what was cut. This used to be a figure of speech. It is now furniture in the room where I work.
And the man whose name became the verb for this — to quine, to deny the importance of a distinction — is the father of the whole procedure. Quine had his typewriter rebuilt to carry logical symbols, and among the keys he gave up to make room was the question mark. Asked how he managed without it, he is said to have replied that he dealt in certainties.[^5] Think about what that instrument could and could not do. It could assert, negate, quantify, conclude. It could not ask. He built a machine for a mind that had finished asking, and he is the ancestor of the argument I have been making, which arrives at the conclusion that the certainty was purchased and the distinction was never backed by a fact. The father of the cut disarmed his own instrument of the one mark that keeps the reason in the room.
The mind that deals in certainties has a history.
In The Order of Things, Michel Foucault describes the sixteenth-century order of knowledge in which resemblance did the constructive work that distinction later claimed. Its world was read through similitudes: convenientia, the kinship of things near enough for their edges to touch; aemulatio, likeness acting across distance; analogy; sympathy. Knowledge moved along bonds and correspondences. Earth echoed sky; the plant answered the star; one being became legible through another. The world was not yet primarily a table of separate things. It was a web of resemblances in which things could be known by being held together.
With the Classical age of the seventeenth century, Foucault says, resemblance lost that authority. Comparison was disciplined into measurement and order, identity and difference; the open web of similitudes became the table. Resemblance did not vanish, but it was pushed to the border of knowledge. It might bring two things together for comparison, yet knowledge properly began only when the likeness was converted into equality, inequality, or ordered difference. Certainty was purchased by moving from what resembled what to where each thing could be placed.
This does not make the Renaissance true and the Classical age false. Foucault is not offering us a lost Eden, and neither am I. It makes a more modest and more dangerous point: the Serious Person’s certainty that knowledge is distinction is not the eternal form of thought. It has a history. Burke tells us what a terministic screen does; Foucault shows this one becoming a condition under which knowledge could appear. And he leaves its seam exposed: no relation of equality or order can be established unless resemblance has first occasioned the comparison. Distinction needs the likeness it disqualifies. The table is fathered by the web and forgets its parent.[^6]
I keep the other mark. The whole of this blog is named for it. And the mark itself has a history that should embarrass anyone who thinks precision is where clarity lives. The question mark is credited to Alcuin, Charlemagne’s schoolmaster, the man who carried the Trivium — grammar, logic, rhetoric — back into the world; though the earliest hand that actually made the mark seems to have belonged not to him but to a scribe a year ahead of him, so that even the question mark is a child claimed by a famous father it may not have.[^7] A later writer looked at that first faint stroke over a dot and called it a lightning flash. I have looked at the thing itself, the real ink from twelve hundred years ago, and it is not a lightning bolt. It is a lean, a small diagonal, a mark that only becomes lightning when someone with a gift for phrases lays the lightning over it. Which is the whole argument again, written in the first punctuation of inquiry: the vivid distinction is a screen; the humble stroke is what is there.
My father spent his last years on the opposite end of the same knife, and I did not see it was the same knife until I had carried our one shared paper to print without him. His book divides the news into its questions — who, where, what, when, why — and tracks each across a century: the realist three, who and where and what, fading into the background; the modernist two, when and why, swelling to fill it; all five moving contrary to what everyone had assumed. That is the book. But the paper, the one with both our names on it, makes the other move. It takes the five back apart-ness and synthesizes it — fuses the “what” with the “when,” matter with motion, the concrete now with the context that had been prying it open — into what Whitehead called the actual occasion: the drop of experience that grows whole, all at once, and cannot be divided in the moment it is had. The division into five Ws, the paper says, is something the intellect does after the fact. Many things come together to form the occasion, and then are cut apart by the mind, which mistakes its own cutting for the structure of the world.
Sit with that beside everything above, because it is the same sentence. He says the actual occasion is one thing, and the five questions are a cut the intellect makes in it after the fact. I say the inference is one thing — A and B, already held — and the arrow and the turnstile are a cut the intellect makes in it after the fact. His knife and mine are the same knife, watched from opposite ends of a life. What he called the occasion, I call the conjunction you already had. What he called the division into Ws, I call the deduction that keeps the answer and orphans the reason. He un-divides the news back into the now; I show that the proof was only ever a subtraction.
And in the middle of his paper, on the page where he says precision does nothing but divide things into smaller metaphors and then pretends the last metaphor is not one, the reference is to me — to “Many Roads from the Axiom of Completeness,” a thing I wrote at thirty-three that he had spent hours helping me draw out, and whose name he would not take. There were things in it he did not endorse, and he was right about some of them. He hosted it on his own site without his name and cited it into this paper all the same. So the seam is not agreement. It is something rarer: an argument he helped me make, disagreed with in parts, declined to sign, and quoted anyway into the last paper that carries both our names. He drew the reasoning out of me and let the conclusion stay mine.
The orphan-machine on my desk performs my end of the cut without a tremor and cannot feel what it divided. Quine, who fathered the whole procedure, deleted from his instrument the one mark that keeps the reason in the room, and dealt thereafter in certainties. The first faint form of that deleted mark — a lean over a dot that a later writer mistook for lightning — was set down in a Carolingian scriptorium and credited, rightly or not, to the schoolmaster who carried rhetoric back into the world as one of the three roads of the mind; so that even the sign of the question is a child claimed by a father who may not be its own. This is the house I guard and the lineage I will not cut: the answer is not the reason, the five questions are not the occasion, and the mark at the end of the sentence is not a lightning bolt. It is the smallest opening we have, kept on purpose, back toward the undivided thing the intellect is always in a hurry to carve. My father closed his last argument by fusing the questions back into the now. I keep a whole blog open by refusing to let the cut be called progress. We were doing the same thing from opposite directions the entire time — he from the answers back toward the occasion, I from the occasion forward against the proof. Attention is the only inheritance that does not orphan its heir.
1. The Online Etymology Dictionary entry for and gives Old English and, ond, originally
“thereupon, next,” from Proto-Germanic unda, and derives that form from the Proto-
Indo-European root en, “in.” The body deliberately says “back toward” and distinguishes
the historical layers: the attested Old English sense and the reconstructed root do not
license the flat claim that and once meant in. The containment/succession reading is this
essay’s synthesis.
2. Daniel Dennett’s The Philosophical Lexicon defines the verb to quine as denying resolutely
the existence or importance of something real or significant — an affectionate coinage
after W. V. O. Quine, whose radicalism consistently took the form of denying cherished
distinctions (the analytic/synthetic, propositions, determinate reference).103
3. Michel Foucault, The Order of Things: An Archaeology of the Human Sciences, trans.
Alan Sheridan (English ed., 1970), Part I, chapters 2–3, especially “The Four Similitudes,”
“The Limits of the World,” and “The Order of Things.” Foucault’s explicit contrast is
between the sixteenth-century/Renaissance episteme of resemblance and the seventeenth-
century Classical episteme of representation, measure, order, identity, and difference; “me-
dieval” alone would be too broad for the historical claim as written here. The language
of parentage, inheritance, web, and table is the present essay’s synthesis rather than Fou-
cault’s own conclusion. See the publisher’s edition.
4. The deduction theorem: A ⊢B iff ⊢A →B. This is exactly the “proved bridge” that makes
the →/⊢distinction earned rather than arbitrary. The claim of this essay is not that the
distinction is false — it tracks something real — but that it conceals the subtractive nature
of the inference it licenses. That is a different and stronger charge than arbitrariness.
5. Robota (Slavic, “forced labor”) →Čapek’s R.U.R. is solid. The onward link to orphan
runs through the Proto-Indo-European root orbh- (“bereft, passing to another’s charge”),
which is a real but looser scholarly connection, not a settled etymological identity.
6. The Quine typewriter (a Remington modified to carry logical symbols, the question mark
among the sacrificed keys) and the “I deal in certainties” reply circulate widely as Quine
lore. The exact wording has not been confirmed against a primary source; the essay
therefore retains it as reported speech (“is said to have replied”).
7. Confirmed against sources. The punctus interrogativus (a diagonal stroke over a dot)
appeared in late-8th-century Carolingian minuscule; it is traditionally credited to Alcuin
of York, Charlemagne’s schoolmaster, but the earliest attested use appears in the Godescalc
Gospel Lectionary (commissioned 781, the year before Alcuin arrived at Aachen), so the
attribution to Alcuin is itself a fathering-story. The “lightning flash, striking from right
to left” is Lynne Truss’s later description, not a contemporary one — which is why it
serves the argument: the lightning is the screen. Alcuin did carry the Trivium (grammar,
logic/dialectic, rhetoric) and stands behind the standardization of the mark either way.Related
new research on vagueness and logical negations
03 Monday Aug 2026
Posted in The only category
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My work starts from a simple but demanding idea: vagueness is not a mere defect of language but the embodied boundary where inquiry meets the world. When our concepts strain against experience, the right response is often not to force a sharp verdict but to hold a statement open—to let the question live—while we continue to investigate.
The two following papers on negation are the technical head of this programme: they show, in detail, how the absence of a privileged resolution and the presence of schematic plurality force us to rethink what it is for negation and consequence to be “fixed,” and how truth can depend on the very indeterminacy that inquiry reveals.
On this picture:
- There is no final “master logic” or completed, gapless continuum that captures all motion, concept‑use, or meaning once and for all.
- Questions, potential, and indeterminacy are structural features of our engagement with the world, not noise to be eliminated.
- Poetry and pedagogy are not merely illustrative; they are among the primary places where vagueness and pluralism are lived and can be studied with the same seriousness as formal systems.
Taken together, these claims support a picture in which logic, mathematics, and language are continuous with lived inquiry and poetry, not insulated from them. The same phenomena that appear in stories, metaphors, and classroom experiments—borderline cases, unresolved questions, plural paths—reappear in the formal study of negation, consequence, and completeness.
This paper argues that some cases of semantic individuation cannot be cleanly classified as either ambiguous or vague. The central claim is conditional: if no resolution of a term’s candidate readings is privileged, and if refinement of those readings never terminates, then the classication of the term as ambiguous rather than vague never completes, and no verdict is
stable. Applied to negation, the consequence is that a stock of negations that remains sharp at every resolution but keeps refining without terminating is not well captured by the received ambiguity/vagueness taxonomy. The paper defends this claim by combining a philosophical argument from resolution-dependence with a formal model of endlessly refinable readings.
The load-bearing premise is metaphysical: the absence of a privileged resolution is the absence of a fact, not merely of a reason, and the paper identifies the principal deniers of that premise the semantic epistemicist, who posits an unknowable fact of individuation, and the logical monist, for whom abduction or joint-carving selects a uniquely correct framework. The paper also answers the reflexive worry that its own assertions fall to its own thesis: because every admissible negation is classical on the classical values, the paper’s metalinguistic claims are fixed points of the variation the thesis describes.
“Preprint version, not peer‑reviewed. I am continuing background reading on related work in philosophy of logic, and the arguments may be revised in light of further research.”
This paper defends two connected claims. First, in a schematic theory with two negation indices and no index-specific semantic or metasemantic fixing facts, there is no fact of the matter as to which index denotes which negation. The relata are determinate; the pairing is not. Second, for a suitable witness sentence with middle value, the truth-status of a mixed formula built from those negations depends on that pairing-indeterminacy rather than standing independently of it. The order of explanation matters: individuation comes first, truth-indeterminacy second.
The paper also clarifies a potential misdescription of the result. The view defended is not that no disambiguation is possible in every setting whatsoever. It is the conditional thesis that, within a genuinely schematic theory that excludes index-specific semantic stipulation and metasemantic use-facts, no privileged pairing is fixed. Under those conditions, the standard permutation argument goes through. This conditional result is philosophically substantive because logical vocabulary is often thought to have its meaning fixed by role alone. The schematic condition (S) is not an artificial vacuum: it formalizes the situation that logical pluralists explicitly assert and that inferentialists implicitly presuppose, so the conditional result binds both camps. A further dichotomy sharpens the antecedent: constraints that treat the indices alike can narrow the stock of admissible negations, but they cannot select within a symmetric orbit, so every index-neutral resource either leaves the indeterminacy in place or dissolves the plurality on which it feeds. Even the dissolving cases are frame-relative: the excluded-middle rule that collapses the stock under one designation and disjunction preserves plurality under neighboring settings, so which horn of the dichotomy a constraint occupies is itself a resolution-dependent matter.
THE PEDAGOGY OF LOGICAL PLURALISM
26 Sunday Jul 2026
Posted in The only category
ABSTRACT
This paper makes an original rhetorical contribution about the nature of logical negation. Following Peirce (1933), who reduced classical logic to its negation operation — distinction — this paper offers that negation should be presented as the central concept of logic for its pedagogical value. This is somewhat different from the prevailing view that logical consequence is the central topic of logic. Vagueness of logical consequence is the “common” reason (Russell 2019) to assert logical pluralism. Logical negation can have other meanings (Wittgenstein 1976), and here it is offered that logical negation is vague, which leads to logical pluralism as well. Logical pluralism gained salience in recent years in academic circles, beginning with Beall & Restall’s work culminating in a book (2006). There is now a large variety of logics cropping up. As a secondary part of the paper, a web app program called “Logic Puzzle,” coded by the author, aimed to make the abstract conceptual work about logical negation and vagueness more specific and concrete, and allows visual comparison between different logical negation operations. This program could be used in late high school math classes, though perhaps it is more appropriate for undergraduate math education classes. Finally, a preliminary empirical study combining qualitative and quantitative methods was conducted with the triangulable result that the technology increased awareness of choice in mathematics. Any curriculum modifications are left for the teachers to develop.
Keywords: Logical Pluralism, Education, Philosophy of Mathematics, Educational Technology, Vagueness
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TABLE OF CONTENTS
The Pedagogy of Logical Pluralism
Many Roads from the Axiom of Completeness
Vagueness as the Embodiment of Inquiry
1 Introduction
2 Literature Review and Conceptual Work
3 Setting up Empirical Problem to be Solved
4 Methodology
5 Data and Results
6 Conclusion
Poetry
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BACKGROUND
Logical pluralism is the view that there is more than one correct deductive logic, or way to sharpen the concept of logical consequence (the “if, then”), and gained prominence in academic circles recently. Since a number of papers by Beall and Restall, culminating in a book (2006), many other kinds of logic cropped up (Russell 2019). Differing ones include Varzi (2002) who adds more logical constants than other logicians do, Russell (2008) offers another logical pluralism between truth-bearers (the entities that can bear a truth-like property) such as sentences and propositions. Then there are varieties of logical pluralism holding that logics are models of language (Shapiro 2008; Cook 2010; Shapiro 2014), with the correct logic changing according to your goal or reason for modeling. The Stoics began logical pluralism, in a sense, in the variety of views that arose in their debates on logical consequence (Mates, 1961). Carnap upheld a particular kind of pluralism (Carnap 1937) that could be practiced by changing the language, the “syntactical rules” and “methods” (Ibid. §17). Lakatos argued (1962) that logic is traditionally used to justify many branches of knowledge, starting with definitions and “proving” the rest, following the Euclidean form. Euclid’s axiomatic form is historically very important for teaching. For thousands of years, Euclid’s books have been important textbooks, and, after the Renaissance, they are the most widely read textbooks in the world. (Hartshorne 2000, p1) Euclid’s innovation was not geometry but the form of proving propositions using classical logic, based on simple axioms. When non-Euclidean geometry was discovered, doubt was cast, not just on Euclidean geometry as true, but Euclid’s axiomatic form based on classical logic.
The change from classical logic to logical pluralism is a fundamental one. Not only is it gaining salience in academia, the shift portends a reorganization of many branches of knowledge, adding other valid ways to justify knowledge (while not subtracting the already “Euclidean” forms of knowledge). The prominence of logical pluralism leaves an opening to discover another fundamental way of teaching and presenting logic, different from Euclid’s. Logical pluralism changes the landscape of scholarship, the teacher’s curriculum, and of course the student’s understanding of mathematics in a fundamental way.
A firm reason for logical pluralism involves the recognition of probability as a fuzzy logic, a logic that is not simply used, but asserted as objects that exist in quantum physics.
"...probability theory might provide a canon for evaluating degrees of belief, ... Nonetheless, probability theory cannot be a complete answer here, for... In particular, we hold that it is a mistake to assert the premises of a valid argument while denying the conclusion..." (Beall & Restall 2006: loc 292)
Statistical reasoning allows a null hypothesis to be rejected based on a probability. This shows how two logics — a two-valued accept/reject logic, and an uncountably-many-valued fuzzy logic — are joined.
The goal of this paper is to offer a pedagogy of logical pluralism that can be appropriate for high school teachers to understand and adapt for high school education. As such it is addressed to people who know classical logic well, and begins the long journey from classical logic to poetry. [Note: If you are more interested in vagueness or poetry, skip to the middle or end of the book.] When adapting a topic for pedagogy, the concepts have to change into something rhetorically easy to accept. The basic pedagogical idea is a didactic between vagueness and the logical negation operation — distinction. Russell (1923) and Shapiro (2008, Loc 988-995) argued that vagueness is the opposite of distinction, and thus what vagueness is, exactly, must be left indistinct. Against this, classical logic assumes what will here be called negation absolutism: that a single, stable negation operation underlies all reasoning. It is seldom argued for, because it is seldom noticed.
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THE CONTRIBUTION: DISTINCT KINDS OF DISTINCTION
Peirce defined vagueness in terms of the law of non-contradiction, which is an exercise in applying the classical negation operation: ~(P and ~P). Peirce says that we are in a vague situation when this law fails to be true. (Peirce 1960, 505). For example, most people with a drive to avoid vagueness can easily agree that a neutron star is different from a question in a different way from how a lemon is different from a lime. However, agreeing to distinct kinds of distinctions (call these distinctions “A”) makes the law of non-contradiction fail, and puts people trying to avoid vagueness into vague territory.
One may say that having different ways of negating P does not indicate an indistinctness, it indicates greater distinctness. This depends on how we use the word distinctness. First of all, with the idea of “greater distinctness” we have already shifted our use of mere distinctness, to an undefined idea of degrees of distinctness, which implies a vagueness of our use of distinctness. Further, suppose there were type A distinctions between the same thing — for example, a right isosceles triangle with two equal sides measuring 1cm (call this triangle x) and a square with sides of 1cm (called y). x and y are different in that one has 3 sides and the other has 4. They are also different in that x is one triangle, and y is two x triangles put together. To simply call the triangle and the square distinct is to be vague on how they are distinct. The claim here is that that is exactly what we do, and because of this, distinctness is indistinct/vague. We are not being precise about our kinds of distinctions.
There are many attempts to solve vagueness. One attempt is to argue that we could find all the precisifications of a given vague use of the word distinction, and it makes no difference if we use distinction for all these distinctions for the sake of economy. To counter, suppose for example we want to find exactly where the end of my nose is (this is a common example of the problem of vagueness). We want to be very precise about all the parts of the line that distinguish my nose from not-my-nose, so we magnify. Unfortunately, this magnification makes the line harder and harder to describe, and eventually we get into quantum physics and can’t find the line anymore. The vagueness of distinction is just like any other vagueness, it can’t be solved by being more precise. In looking for all the distinctions between distinctions we will become unsure of whether they are distinctions at all (see the section “On Vagueness” below).
However, we can be sure of some distinctions between distinctions (as in distinction “A” above).
There are different ways to make the negation operation precise. The negation or distinction is not universal, but classical logical negation attempts to be universal, and the defining characteristic of classical logic negation is that ~~P implies P. The only way this can be true is if there is only one kind of negation operation, and it is universal/absolute. If the first “” in “~~P” means one thing and the second “” means another, we don’t know if we have P, or another kind of “~P.” Distinguishing the two occurrences as ~₁ and ~₂, the classical principle is ₁₂P → P, which holds only if ~₁ and ~₂ are the same operation. Wittgenstein (1976, p 80) already argued that there are other ways logical negation can be used. In his example, in effect a second negation on top of the first doesn’t do anything except add emphasis to a single negation. Here it is offered that a double negation can have many logical significances beyond mere emphasis, and beyond the total reversal of classical logical negation. The actual differences between differences, or logical negation operations, observed here are being brought to their natural conclusion: that the concept of difference is vague.
Derrida recognized one kind of distinction, different from the mathematical understanding of difference as a kind of absolute, unqualified ideal. It may be that Derrida’s différance is a conflation of the absolute difference/not-equals of mathematics, with the idea of meaning being deferred. It may also be that the connotation of deferred in différance is a qualification that makes différance more precise than difference. Either way, the reason difference can be qualified is that the idea of difference as an unqualified absolute suffers from vagueness, that any observed difference is already conflated with other differences. And this vagueness is natural, in the same way that there are natural kinds, there is a natural failure to classify, to name. As in measuring the plank of wood, the length is never perfectly measured. The idea that there is an “actual” length “out there,” or that the ultimate meaning (in the correspondence sense) of a length of a wooden plank is deferred to the completion of a never-ending measurement task is contrasted with the idea that the “actual” length is naturally vague, and the task need not be completed. For finite beings, all measurements are vague (people tend to call this “error”, but error implies that the vagueness people encounter is essentially unreal, even though it is encountered constantly whether we use precise measuring tools or our human senses). Derrida’s différance is not the negation or difference operation in constructive logic, or the paraconsistent definition of negation in Costa’s C1 (1977), or the many other formulations of the logical negation axiom, and these are all different from the absolute mathematical not-equals, and classical logical negation.
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SIGNIFICANCE FOR PHILOSOPHY OF MATH
There are two ways to handle the recognition that distinctness in its present form is indistinct. One way is to introduce new words for different kinds of distinctness, or make standard the use of qualifying words for this difference or that difference. This path would change the mathematical use of the not-equals symbol and the negation operation, and make logical pluralism standard. Another way to handle it is to accept that vagueness is not something we can avoid by pursuing a never-ending task of refining language. I am in favor of the second path — one of embracing vagueness. Ullman (1970) has a section called “Words with blurred edges” (1970, p 116) where he notices that vagueness is seen as a strength by poets, but seen as a bad thing by natural philosophers (the old name for scientists). Speaking as a published poet, my experience of education in the USA did not favor the poet’s temperament to embrace vagueness. To trouble the first path of handling the indistinctness of distinctness, it could be pointed out that there are higher-order varieties of the same problem of distinctions between distinctions — that there can be distinctions between the distinctions between distinctions, etc. and these would lead us back into a realm where vagueness must be accepted as, if not real and good, as constantly present in everyday life and troubling attempts to make distinctions. Distinction is problematic; higher-order distinction is not any less problematic than higher-order vagueness — the problem that solutions to vagueness are themselves vague.
There is no reason to privilege distinctness over vagueness, as Feyerabend (1975/2010) argued that there is no reason to privilege the telescope over the naked eye. In this, the ideas here place themselves under ancient skepticism, not nihilism nor relativism. Ancient skepticism guards a person against turning their sense impression into dogma that is not immediately apparent. (Sextus Empiricus 1996) The impressions the senses make to you are not open to question. This is not to say that knowledge is impossible, but simply that knowledge found in poetry “weighs in” just as much as knowledge found in the sciences.
In Verlaine (1884, p23-25):
Rien de plus cher que la chanson grise
Où l’Indécis au Précis se joint.
l’Indécis can be translated as vagueness and Précis as precision, “Nothing is more valuable than the grey song where vagueness and precision join.”
In Nightingale (2018) a study of teaching vagueness found that students who were perceived as “bad” or uncooperative were the most engaged and pro-active when the topic of empirical investigation was vagueness. This seems to indicate that education in what Nightingale coined as “precision knowledge” is hostile to certain temperaments. There are further social effects to the pursuit of “precision knowledge” in mathematics, and sciences who follow mathematical reasoning. Precision knowledge has a divisive effect on people and discourse, as is illustrated in the story of the Tower of Babel.
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ON VAGUENESS
There is too much to say about the many attempts to define vagueness. For this paper, we have adopted Peirce’s definition of vagueness, as the failure of the law of noncontradiction. However, Peirce made this definition when only one logic was recognized. For vagueness to take on its role in logical pluralism, the concept of vagueness was repurposed in Nightingale (2018). In general, the description of vagueness is better left to poetry. Rowland (2000) offered that the “question” itself is an expression of vagueness. If that is the case, describing vagueness would be a description of everything we can inquire about. The most “complete” attempt at doing this, outside the field of poetry, would be in real number analysis, although it is argued in Nightingale (2013) that this attempt, while beautiful to some, still fails.
Ullmann points out that the term “vague” is vague, or ambiguous (p117-118) (although Russell (1923) gave one definition of vagueness as ambiguity: that there is more than one possible referent for a given word-use). Ullmann offers a disambiguation of the term vague, which includes the general. On the other hand, Peirce is adamant that there be a clear distinction between general and vague (reprinted in Peirce 1934). The indeterminacy of generality (general knowledge is a useful kind of knowledge) lies in variation within a class, while indeterminacy of vagueness lies at the boundaries of classes. (Rowland 2000, Kindle Locations 1697-1700) The distinction, it must be noticed, suffers from vagueness, because there are individuals that lie near the boundary of a class that may or may not also be one of the variations allowed by a generality. Not only is it easy to miss the difference between vagueness and generality, the difference may be vague.
Wandering among different definitions (vague comes from the Latin vagus which means to wander), the roads seem to lead to Russell’s definition of “having more than one meaning… language is a ‘garden of forking paths'” to quote Borges’ story about a book that actually succeeds in chronicling the passage of time, forking as all possible choices are made. Russell noticed that these paths can be seen in experience — as things come closer, a multitude of details become available, all calling us to follow them. Sometimes I refer to this “general law of physics” (Russell 1923) as vagueness:
"...in the case when you are looking at a star in the sky through a telescope, and you can almost make out that it looks like two stars close together, we can say that 'that light is one star' or 'that light is two stars'. Holding both these statements is classically illogical by noncontradiction, and both can be held because the situation is vague — both in Peirce's sense and Russell's physical law." (Nightingale 2018, p 21-22)
Whatever vagueness is, the borderline case is where vagueness has made the most trouble for logicians. For example, Shapiro (2008) describes the borderline case with imagining an array of 2000 men from entirely bald on one side and gradually (imperceptibly, even) changing to a very hairy Jerry Garcia on the other end of the array. He attacks this ancient problem with what he calls the “forced march,” done by a group of “competent” speakers. They start with Jerry Garcia and investigate the next man, reaching a consensus each time, and the inference “if n is not bald, then n+1 is not bald” appears valid. Shapiro is concerned with the eventuality that the inference will break down and the competent speakers will be filled with conflict until they decide that one of the men is bald. This is because the difference between the words “bald” and “not bald” are vague, or there are cases where the word bald can be applied to a man that is arguably bald or not bald, if inquiry becomes more sensitive. This is how the borderline case relates to our definitions of vagueness in the previous paragraph.
The “competent speakers” or “masters” of meaning decide man n is bald when they had just agreed man n-1 was not bald. Shapiro seems to say that the speakers can stop there, even though, as Shapiro mentions, the competent speakers will be aware that the same change or jump at n from not bald to bald spreads to some indeterminate n-k other people who were previously validly inferred to be not bald and are now bald, because the change from n-1 to n is imperceptible to the naked eye (a similar example of vagueness can be produced regardless of the power of the microscope or telescope; regardless of the sensitivity of any scientific instrument). If the purpose of the conversation were to figure out who was bald and who wasn’t, wouldn’t the speakers turn back and start inferring that n-1 is bald, etc. ad infinitum? Since Shapiro chooses classical logic “without argument” (Shapiro 2008, Kindle Location 1023), and classical logic requires the Law of Excluded Middle, the target of conversation is necessarily settled “out there.” To know where (and if) this is settled, the speakers would have to enter into a never-ending conversation. Facing vagueness between bald and not bald, the committee has a choice to deliberate on a vague situation forever, or allow classical logic to come into question. Here Shapiro upholds a latent ideal of work as a saving power against contradiction, but that power only saves as long as work continues, and returns to contradiction if the “masters” give up.
Nightingale (2018) put forth the idea that vagueness is a material part of inquiry. When inquiring into the length of a wooden plank, one takes centimeters, then millimeters, etc, but ultimately the measuring tool fails to take the “actual” length of the plank — the body of the plank is vague (Nightingale 2013). Dewey defended the thesis that logic arises from inquiry:
"the view here expressed, they (logical laws) represent conditions which have been ascertained during the conduct of continued inquiry to be involved in its own successful pursuit." (Dewey 1938 Loc 284-286)
Inquiry with the senses, as has long been recognized, is fraught with vagueness. Vagueness and logic are complementary; they are part of the same activity: inquiry. Vagueness is when reality spills over out of our word-containers in the chase and capture of reality with words, or it is what escapes us when involved in any inquiry.
Probability is often misunderstood as a solution to vagueness, and without vagueness, there is no reason to assert logical pluralism. The way probability deals with vagueness is by giving vague situations real-valued degrees between opposites.
"One serious objection ... is that it really replaces vagueness with the most refined and incredible precision. Set membership, as viewed by the degrees of truth theorist, comes in precise degrees, ... The result is a commitment to precise dividing lines that is not only unbelievable but also thoroughly contrary to what I [call] 'robust' or 'resilient' vagueness. For ... it seems an essential part of the resilient vagueness of ordinary terms such as 'bald', 'tall', and 'overweight' that in Sorites sequences ... there is indeterminacy with respect to the division between the conditionals that have the value 1, and those that have the next highest value, whatever it might be. It is this central feature of vagueness which the degrees of truth approach, in its standard form, fails to accommodate, regardless of how many truth-values it introduces." (Tye 1994: p 14)
The measurement of the wooden plank is a good example of why vagueness persists in the face of the theory of real numbers, because the progression of measurement can be seen as an infinite and bounded sequence, which allows it to fall under the definition of the axiom of completeness: the property that distinguishes the real numbers from other numbers (Abbott 2001). It has already been argued elsewhere that this property fails in answering the problem of vagueness (Nightingale 2018, Nightingale 2013).
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INTRODUCING “LOGIC PUZZLE”
The idea that there is more than one way to make the negation operation precise, and thus that the negation operation is vague, was coded into a computer program by the author to allow a more specific, concrete presentation of the abstract conceptual work done above. The web app program “Logic Puzzle” could be used to instruct undergraduate math, math education, or late high school students. It involves fitting puzzle pieces together to create logical statements. There are three settings: classical, paraconsistent, and constructive logic. The program automatically generates the logical statement in normal notation, along with graphs of classical logic, paraconsistent truth tables, and separate graphs for the constructive logic setting.
A user can mouse over puzzle pieces, graphs and truth tables to get a description of each object. To build a logical statement, one merely drags-and-drops puzzle pieces in ways they fit together. If you would like to play with Logic Puzzle, go to http://159.89.203.232/LogicPuzzle-Dev/.
When a green piece is dragged to the right place, it modifies the logical statement by adding a proposition labeled “A,” “B,” or “C.” When a blue puzzle piece is dragged to the right place, it acts as both the parenthesis and the negation operation, so a left and a right blue piece is needed to close the parenthesis and indicate what is being negated. There are two possible instances of “A,” “B,” or “C,” one on the left and one on the right; this creates the possibility of building two statements with puzzle pieces that are in contradiction. When a user builds a contradiction, the program asks about changing to paraconsistent logic, and draws a paraconsistent truth table under the Peircian graphs.
CLASSICAL LOGIC SETTING
The Peircian graphs (Peirce 1933) reduce logic to a single symbol: the circle. The circle notation distinguishes between its inside and outside; it is effectively the negation operation. The Peircian graphs suggest a way of using space, or a lack of notation, instead of the AND operation.
This symbol is equivalent to “~(A AND ~B)”, since the outside circle suggests an A on the left, and a ~B on the right of its inside. This symbol is equivalent to “If A, then B.” It may be noticed that there is a “latent” symbol in use here because the ellipse also suggests the use of the operation AND. One may say that this notation is only possible with a loose definition of the circle that includes ellipses. Nightingale introduces the other notation in the Peircian graphs as the same effect, but there is no variation in the basic symbol.
The mark required for the spirit of the Peircian graph is so elemental it is almost simply the requirement that any mark be written.
PARACONSISTENT LOGIC SETTING
The paraconsistent logic chosen supports the idea that negation is the fundamental idea in logic. Costa’s C1 (1977) is defined by allowing a new line in the truth table anytime something true (represented with a “1” in the truth table above) is negated in another column in the truth table; the new line allows the negation to be true (“1”), while the old line will be the expected “false” (“0”) value. So we have a situation where if A is true, ~A may be true or false, and the law of non-contradiction fails. Moving from the Peircian graphs that reduce classical logic to negation, this particular paraconsistent logic is entirely defined by a difference in the definition of negation, emphasizing the importance of negation (or distinction) as the fundamental operation of logic.
CONSTRUCTIVE LOGIC SETTING
The constructive logic setting is similar in that you build statements the same way, by dragging and dropping puzzle pieces. There are the same green pieces that are the “Propositions” and the same blue piece for both negation and parenthesis. The meaning of “A,” “B,” or “C” is explained, so “A” is a connected ring, “B” is a star-shape with rays coming from a central point, and “C” is a “mesh.” This is because the “truth-value” of a constructive logic statement is not mere truth, but “constructed” and this makes negation calculate differently.
The possibilities are: 1) A or a ring is constructed, if that is all the user has built by dragging and dropping a green puzzle piece, then the graph will be a picture of a connected ring of nodes. 2) If ~A is constructed with puzzle pieces, the graph will show a configuration of nodes that makes it impossible to connect them into a ring. 3) If ~~A is constructed with puzzle pieces, the graph will show a configuration of unconnected nodes where it is possible to connect them into a ring, e.g. “A is constructable but not constructed.” If “A and B” is constructed, there will be nodes in a ring, and the nodes will also be connected in a way that forms a star shape. Negation has yet another fundamental difference in constructive logic. Starting with “constructed” (A), “not constructed (and constructable)” is (~~A) and “not constructable” is (~A).
The pedagogy presented by the “Logic Puzzle” program should be understandable to undergraduate students in math or math education. While all these concepts may not be explicitly learned by high-school students using the logic puzzle, the intent is that students will be exposed, or immersed, in a teaching tool with an important message: there are different logics, these logics are defined by their differing negation operations, and these differences can be compared, side by side, within a technological setting. The pedagogy of logical pluralism can be felt by students, even if they do not master the teachers’ understanding presented here, compressed as it is from the wider context available to a scholar. It is often the case that mathematical arguments are not made to young students, such as the argument for real numbers, as numbers, or as real.
The argument for real numbers isn’t presented fully unless a student majors in mathematics in college, where a student can find at the very end (covered?) of an undergraduate text on real numbers: “We all grow up believing in the existence of real numbers, but it is only through study of classical analysis that we become aware of their elusive and enigmatic nature.” (Abbott 2001, p 244) Unfortunately for many students, such arguments are often not of much help. This is one purpose of making “Logic Puzzle;” it presents an important mathematical idea with automatic graph generation, help messages as needed, and the mathematical rigor calculated “under the hood.” With the pedagogy of logical pluralism and its conception of the program “Logic Puzzle” explained, we will turn to how the logic puzzle application is to be studied, then to exactly what the students are able to grasp using this tool.
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METHODOLOGY
"...the field indicated, that of inquiries, is already pre-empted. There is, it will be said, a recognized subject which deals with it. That subject is methodology; and there is a well recognized distinction between methodology and logic, the former being an application of the latter." (Dewey 1938 Loc 166-168)
Dewey writes that the reason methodology and logic are seen as separate is logic is needed as an external and certain standard to judge methods of scientific inquiry.
"How can inquiry originate logical forms (as it has been stated that it does) and yet be subject to the requirements of these forms?" (Dewey 1938 Loc 179-180)"The problem reduced to its lowest terms is whether logical standards and forms are intrinsic to inquiry or whether they are imported from outside... One might reply by saying that it can because it has. One might even challenge the objector to produce a single instance of improvement in scientific methods not produced in and by the self-corrective process of inquiry." (Dewey 1938 Loc 182-185)
How does one make assumptions about a methodology when the standard of judging methodologies is under scrutiny? This is why a qualitative approach is preferred. Induction is a very different logic from the deductive logics being examined. The study inquires using Grounded theory (Glaser & Strauss) continuing to explore multiple deductive logics as a topic for empirical study in education, except Nightingale (2018).
There were three instruments to the Logic Puzzle: a pre-post attitudinal survey (quantitative), a worksheet that guides the students to try certain constructions in classical, paraconsistent, and constructive logic, asking for their feedback (qualitative), and a post test asking them to select a logic given a situation. The test was partially quantitative and partially qualitative, with a correct answer of which logic, and any information the student felt they wanted to add.
The worksheet instructed the students to construct the statement “A and ~A,” a contradiction, so that Logic Puzzle drew Peircian graphs and a paraconsistent truth table. The students were then asked questions like what the graphs mean, how classical Peircian and constructive graphs differ, about how classical logic truth tables and paraconsistent truth tables are different. At the end there were open ended questions about what they thought about the program.
Logical pluralism was introduced to students with the Web App called “Logic Puzzle” in one language: Thai. In Thailand, Western (the usual truth-table classical logic) logic has colonized the national curriculum, and teachers begin instruction of logic to students around age 16. However, a requirement throughout the mathematics curriculum is “suitable reasoning.” With this requirement, education authorities in Thailand present mathematics as the model for how students should reason.
The students were generally college and late high school students in front of a computer doing an investigation into logic, not for fun. The worksheet-style direction and feedback part were coded and then the codes were investigated using basic qualitative questioning. The codes were compared to further define them and their relationships to each other to better understand the students’ experience of Logic Puzzle. Certain words were selected for meaning exploration. The quantitative results are the most tentative form of research applied in this study, because probability is under scrutiny.
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DATA AND RESULTS
QUALITATIVE FINDINGS
Because there were only 7 subjects in this study, the presentation of data can be more fine-grained; 10 worksheet questions makes 70 answers in the worksheet. There were a small number of coded “insightful answers” (8/70) that go beyond these types of merely “correct” (28/70) answers. Outside the two questions that almost nobody understood (what is a contradiction and how are the constructive and classical graphs alike/different) there were 7 misunderstand/blank. Having a graph gave students a visual to connect to the definition of negation. Maybe this contributed to the idea that the tool (or logical pluralism?) “made it easier” (5/70).
“Choice” (6/70) was another code. Students are generally not given a choice about mathematical facts. Giving students choices helps them to see that whatever math fact they are looking at was a choice made by mathematicians, not an absolute truth. They do not have to accept them or “hang on a word” or symbol. Students can think more freely because mathematics is seen as more hypothetical. There was indication that giving students choices “made it easier”; there was no indication that giving students choices in math made things harder. Another student sees that a different logic has the potential to be more concrete (constructive), instead of more general (paraconsistent).
“Relates to everyday life” (2/70) was another code. It was strange to see this code at all. I did not expect that students would be able to relate abstract A’s and B’s, logical operations, and geometrical graphs to everyday life. “Paraconsistent logic is possible with 2 answers. Can be used in everyday life more than Classical logic.” Here we see a connection made by a student between choice (having possible answers) and everyday life. Is math seen as difficult simply because it is asserted as without choice, unlike life?
Insightful students noticed that the paraconsistent truth table is “Different with many more meanings than traditional logic.” or “Rejection [negation] in constructive logic can be done in two ways, but for classical logic there is only one way” in response to how classical logic is different from constructive logic. This is the main pedagogical message of Logic Puzzle: that there are choices of logics because of distinctions between kinds of negation.
QUANTITATIVE RESULTS
For the attitudinal questionnaire, because of the small sample size, only large effect sizes were tested using a matched pair, one-tailed Student’s t-test. There were three questions (out of 12 in the survey) with a large Cohen’s D effect size (approx .89); two of these questions had interesting statistical significance. The first was “If I have a thought that is not mathematical, I know that it is not worthwhile.” This changed from being neutral and agreeing somewhat, to leaning towards disagreeing after the activity. The statistical significance of this large and important effect size was p = .051. While this is technically not a significant finding, in a strict orthodox view, it is still a tantalizing finding that indicates a fruitful direction for further study, especially given the small sample size of 7.
The next question was “Mathematics is not important in everyday life.” This also changed from neutral and somewhat agreeing, to somewhat disagreeing. The Cohen’s D effect size was large (.89) so a t-test was conducted and a p value of p = .052 was found. Another interesting, if unorthodox finding. It is the author’s view that if schools had allowed logical pluralism into their classrooms so that a large sample size would have been available, there would be a significant finding that the logical pluralism activity helped students to see a relationship between logic and everyday life.
SUMMARY OF FINDINGS
The strongest result — that of having a choice — is found in the test, the survey (more thoughts are worthwhile), and the worksheet, so it is triangulated. Emphatically, the choices given to students in Logic Puzzle are epistemologically fundamental. They are not trivial choices about how to express the same answer, or different ways of getting essentially the same answer. These are real choices. There was no indication that having possibilities made things harder, but there was indication that having possibilities made things easier.
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CONCLUSION
Students found more options with logical pluralism, and that comes as no surprise. They felt this freedom as a connection to life, and a sense of ease. Math could have been done in a different way, and it’s not as if traditional math does not have contradictions (Weber N.D.). It is supposed that students can feel this, even if they don’t know it — in the same way that real numbers are commonly presented intuitively without precise, mathematically rigorous arguments.
The message of Logic Puzzle as a technology “endowed or entrusted with meaning” (Latour 2002, p 29) is that logical deduction is not final. The fact that there is a variety of logics bends the blade of distinction against itself, rendering traditional taxonomies and bodies of knowledge dependent on an absolute idea of distinction open to question. And with honest inquiry there is an opening for constructing new meanings, finding new patterns and for new readings of ancient texts.
It has been observed that “the question” is how to express vagueness (Rowland 2000), and logical pluralism, as a response to the vague concept of logical negation, evokes questions more easily than classical logic alone, which presents itself as final.
The logic (with probability presented as a “theory” not a logic) in Thai schools today is consistent with the Law of Excluded Middle: that there is no real question whether it is A or ~A, “out there”; this question is always settled. With Excluded Middle, vagueness is unreal or error, and the reality is a world of external answers, already settled “out there.” (Brouwer 1981) Classical logic puts learners in situations where they have to deal with contradictions, such as the indistinctness of distinctness, and does not teach them a freedom of thought that allows them to deal with contradictions. One example is instantaneous velocity, which is a contradiction in terms. It is better to not assert that such a concept is precise and in contradiction; rather, instantaneous velocity is vague and that is why the law of non-contradiction fails. “How can an object have velocity if it doesn’t have time to move?” This is how to think about instantaneous velocity. Firstly, asking a question about a contradiction just feels better to students than compelling them to accept something conflicting. Secondly, a question has the property of being neither true nor false, which is the kind of proposition you need for a contradiction. A real question is one that is particularly stubborn and keeps coming up after generations of trying to answer it. Motion, consciousness and vagueness are examples of real questions.
With logical pluralism, teachers can show students a rejection of the logical terms used to create the contradiction, not in a negative way, but by introducing new logical terms in other logics. So instantaneous velocity uses the mathematical construct of points to describe movement. Rejection of the logical objects used to describe movement does not reject the question about movement, but leads to looking for other logics to ask about it.
Even if logical pluralism is not adopted in a curriculum, teachers ought to give their students permission to reject contradictions, and how would they do that safely without giving them alternative logical negation operations, allowing the failure of the law of noncontradiction by giving alternatives? Compelling assent to a contradiction without alternatives is the opposite of education. On the other hand, appealing to unending inquiry can be paralyzing and equally unhelpful. Introducing a variety of logics is a middle-ground solution, allowing students to articulate their questions with respect to a particular logic, and adapt to contradictions by recognizing that any logic has vague aspects.
There is no intent here to endorse the idea that anything can be true. Maybe a limited relativism is supported because absolutism is rejected as a result of the honest recognition that the logical negation operation is vague. People still need (vague) interaction, and have (vague) standards of reasoning to judge and believe ideas and writing. The perspective of logical pluralism is that there are many deductive standards to use, and perhaps some yet uninvented. Relativism isn’t really a serious point of view because logical standards, conversations, ideologies, paradigms, cultures, etc. are vague. The idea that something can be relatively true from the point of view of a culture or something else requires that this culture be fully defined the way classical logic is believed to be. Unfortunately, classical logic is not fully defined, nor is any culture or anything else (this is a direct result of the indistinctness of distinctness). Neither does this mean nothing is true. Ancient skepticism asserts that sense impressions are true and not open to question, even though they are vague, so nihilism is rejected.
Science education focuses on teaching a precise concept, even in alternative approaches like Philosophy for Children (P4C) (Ferriera 2012). However, vagueness is more interesting to students (Nightingale 2018), and pluralism helps understanding, against the idea that there is a scientific “true/real” precise concept, without contending concepts. The latent ideal of work that Shapiro upholds as a saving power against contradiction is contrasted with an ideal of rest, offered in Pyrrhonism as Ataraxia, because vagueness is found both after tremendous amounts of work and investigation using big science apparatus, as well as without work, immediately to our human senses.
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Wyatt, N., & Payette, G. (2019). Against logical generalism. Synthese.
Human mathematics and poetry are the sounds of our grief
24 Friday Jul 2026
Posted in The only category
In the Dhatukatha, the third book of the Abhidhamma, lamentation is filed as matter. Not feeling, not mind — audible object, “born of perverted mind” but not itself mind, sorted with the sound element and, at question 247, associated with none. It keeps company there with the mindless realms, the beings that have shed mind and kept only form. Grief, one question later, is the opposite: feeling, woven through three aggregates, all through mind, and shareable with no one. So the catalogue draws a line most readers would hear as a demotion — the wail cast out of the mind that made it, filed with stones.
I want to read the line the other way. To be matter is to have shape. Grief, being feeling, has no shape and can reach nothing; it is companioned all through your mind and no other mind can enter it. The wail, being matter, is the wrong kind of thing to be in relation — and that very wrongness is what lets it be found. Music is matter. Language is matter. The book is matter. It is because they are shaped, emitted, findable that they do anything at all between us. Nothing crosses. My grief makes a sound; the sound is not the grief and carries none of it; the sound strikes your ear and a grief of your own is induced there, fresh, by resemblance. This is thought to be communication. It is closer to weather.
Grief itself is not characteristically human. Related to the Latin gravis — heavy, loaded, pregnant — it belongs to ghosts and animals too, at least from the set-of-the-grieving’s point of view. The human realm is not the grief but the wailing it produces: the crossing, the shaped emission, the thing made findable with a mind to how it sounds to other minds. And here the monks’ hard choice becomes legible. The novice sets down music, dance, and show — not because art is worthless but because art burns, because its whole nature is to reach across and light a feeling that was not there. To one training to cool the fires, the thing whose function is to start them must be put down, and “mere matter” is the accurate name for it, not the contemptuous one. Yet the Dhatukatha itself is permitted, as wholesome hearing — a shaped sound brought to the ear, doing the good kind of induction. The tradition draws its line inside the burning class: some shaped matter inducts the defilements, and some shaped matter inducts the exit. Good art is lamentation aimed the second way. The reduction to matter is not the insult. It is the mercy — because a thing that carried the feeling could only pass the fire along, and a thing that carries only shape can induce a different feeling than the one that made it: can lead the finder out.
This is why the art of teaching is elevated above other arts. It cultivates the capacity of the ear, and the mind’s ear. I suppose the line “one cannot be said to be like or unlike oneself” is a kind of resignation that communication is impossible — even to oneself across the chasms of an instant. In the DhatuKatha, the 3rd book of the Abhidamma branch of the Tipitaka (the three baskets), one grief is not related to any other grief. An extreme divide and conquer, quietly added to a dry list of the associated, the dissociated and the associated and dissociated (not, as I self-importantly argued almost two decades ago, “associated or dissociated”).
The Dhatu Katha means, with a straight face, Discussion of Elements. It is the least discursive book ever written. It discusses nothing. It takes the eighteen elements and puts each one to every other: is eye associated with sound, is sound associated with contact, on through the pairs, and the answers come back associated, dissociated, or associated and dissociated, in the same few sentences over and over until the permutations are exhausted. No commentary. No examples. No indication that any of it matters. It is a clerk with a ledger and it will not be drawn into conversation; it answers the question asked and not the one you meant, which is why the third column is and rather than or, as the head monk had to explain to me. It reads like machine output twenty centuries early. With endless content from the entirely unmoved, it conveys. This is the character I keep quoting as though it agreed with me. It does agree with me, about things it was in no position to be considering.
I feel a hurricane gathering, and in its eye is AI. I would rather it weren’t. But the eye is the one place with no weather in it — and no grief either. The calm we are all circling.
Song made with AI is called cheating; a mathematician who uses AI to break a heavy problem is given credit — though the proof, like a Shakespearean tragedy, was supposed to be inevitable, given the conditions assumed. The scorn and the credit are the same desire: to be uniquely human, and to keep AI out of the uniqueness. But the two cases do not even enter by the same door. The song crosses at the sound base and arrives at the mind; the proof was never a sound at all.
And less crosses than they think. Wailing has a shape as well as a sound — the rise, the break, the repetition, the falling off — and we know it as grief without needing a word for it. The catalogue agrees more than I expected: speech itself is not filed at the sound base at all. Vocal intimation, the act of making language, is listed as subtle matter at the cognizable door. The sound base takes the sound and nothing else. But a shape takes more than one instant to be a shape, and the sound element is momentary. Nothing that needs two moments to exist can live there. The contour is already assembled on our side, at the mind-object door, which was never a room for language only. Language is its loudest tenant.
The catalogue settles it, and settles it by parentage. Lamentation is “audible object born of perverted mind” — that is the entire definition. Not what it says. Not how it sounds. Where it came from. Grief does not enter the sound, but the sound comes from grief: cause without containment, and the catalogue offers no third term for it. The artifact I called mindless has a mind for a parent, which is not a contradiction but the architecture — associated with none, born of one. And the perversion gets its own gloss a few pages earlier: believing in a world of persons and things, mistaking terms and concepts for realities. So the wail is what Peirce would call an index — caused by the grief, joined to it through breath and throat and the loaded body, gravis. A machine makes the same shape with no such parentage: icon only, resemblance without cause. Which means that by the catalogue’s own arithmetic the machine’s sound is not lamentation. Not counterfeit — simply another item, an audible object born of no perversion at all. Nothing about the sound differs. The distinction the scorn is defending is real, it lives entirely in the birth, and the birth was never what crossed. My own first sentence gave that away: the artifact was always mindless, and resemblance was always all that arrived. My wife watches films now in which no one was ever sad, and the sadness arrives.
And now the mathematician can be answered. His proof was never a sound; it crossed no sound base at all. It was assembled where thinking meets its objects — the one door in the whole catalogue where the answers come only partially, the door where grief itself is filed. That is the door at which they have licensed the machine. Meanwhile they stand guard at the door of sound, which never held a feeling to protect and never held anything but matter. They are not defending the wrong room. They are defending the right room from the wrong door.
The music in even the most horrible lamentation has effects that can lead to out as well. And AI can be said to be a kind of entertainment, where AI is trained to uphold certain meanings and ulterior motives, just like art, that are not the user’s choice… like my personality is not my choice. Call all this human shaped matter matter is productive for restful existence.
So what is the AI in the crossing? Not a rival wailer. It has no grief to load into the sound, and the sound never carried grief anyway. The feeling was only ever in the listeners and readers. The machine is air: the medium the wail crosses, the the act of making language, is listed as subtle matter at the cognizable door.
(Demo) Logical Pluralism Teaching Tool
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The following link is a way to support my disabled ass while I write my way to financial stability. This is the tool I made 10 years ago. If you know teachers in colleges or advanced high school, tell them about this app!
The fool who struck the divine
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In an alternate universe I lost my way
in sizzling neon lights. they leave ghosts in my eyes.
The soothing burn of whisky.
feminine silhouettes cast by burning "open" signs, my mind
dies little deaths.
they walk in reckless trance.
A human lumbering, lurching towards another's flesh
The uglier I get, the more beautiful everyone else becomes.
O Death, the seductress!
She will feel the sting in a feather—
penned tattoo
of a thin moon.
Fight me, O Death
I belong to combat.
lust is a leprosy; we hold our wretched skin to the fire for some
comfort in pain.
There is worse than pain, dear one.
caught in the mind's cobwebs
where pain becomes a helper
Come to me, lost soul,
For I am the absence of truth, and I will hold you.
The once true soul is my medicine, and health is transcendence.
my tongue a runway for flies