Open Letter to Mathematicians

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.

Get Lost

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A mathematics professor named Sam stands at a blackboard with an equation that will not come out. It has been not coming out for a long time. At the end of his patience he says what people say, that he would sell his soul to get the thing right, and a demon in sunglasses appears in the classroom to collect.

This is “I of Newton,” a short piece from the 1980s revival of The Twilight Zone, made from a story by Joe Haldeman, with Sherman Hemsley as the professor and Ron Glass as the demon.

Sam protests that he was only venting. Too late: what he said, it turns out, had exactly the right sound for an invocation. But a demon enjoys a contest, so terms are offered. Sam may ask three questions about the demon’s powers, and then he must set a question the demon cannot answer or a task the demon cannot perform. Sam says, “Really?” The demon informs him that he has just used his first question.

The second establishes that the demon can break the laws of physics and move through time as he pleases. The third: is there anywhere, in space or time or any other universe, from which the demon could not find his way back? Nowhere. The demon, enjoying himself now, tells Sam to go ahead. Sam says it is not a question but a command, and after a long pause gives it.

Get lost.

The demon vanishes with a howl. Sam drops the sunglasses in the bin and turns back to the blackboard, grumbling that the fellow wasn’t any help at all.

Before going further I want to separate two things that were present in that classroom, because everything after this turns on the difference.

There was a man looking for an answer.

And there was a being who had all of them.

Look at what happened to the first question. “Really?” is not a request for information. It is the sound a search makes while it is still searching — open, not yet aimed at anything, a man finding out where he is. The demon took it as a question spent and entered it in the account. That is the whole of what follows, told as a joke: the being who knows everything cannot hear a question except as a withdrawal.

Before the answer

To sense is to search. Looking for the slip in the third line. Waiting for the next step to show itself. Listening to find out whether that was the sound. The searching is not something done to a sensation that has already turned up; there is no arrival standing apart from the looking, waiting to be looked at. A measurement does not happen without the question that motivates it, and the question is not asked of the measurement afterwards — it is what put a man at the blackboard.

That is what I mean by questioning. Open, expanding, prior to any answer. Not a stance taken toward sensing but the shape sensing has.

Doubt is not the opposite of that. Doubt is its perversion. War is the perversion of peace, hate the perversion of love, despair the perversion of sadness — in none of these is the second the absence of the first. The second is made out of the first, with the same energy and the same motion, and what has been removed is the return. Sadness moves and passes; despair is sadness that will not move. Love reaches; hate is love that has fixed. Questioning opens; doubt is questioning that has closed and goes on asking anyway, which is why it cannot arrive and cannot stop.

The skeptics were named for the searching and not for the doubting. Zetesis: to go on looking. Sextus divides it three ways in his opening pages — the dogmatist says he has found it, the Academic says it cannot be found, and the skeptic is still searching. Only the middle one is doubt. It is the one that stopped and kept the motion anyway.

The classroom has all three positions in it, with one actor doubling. The demon has found everything, and on that strength he is certain that what Sam is looking for cannot be found. Dogmatist and Academic in one pair of sunglasses. Sam is the one still searching.

So doubt needs a claim to aim at, and the claim comes later. To doubt a sensation you must first have finished with it. You must have taken a number off it.

None of this makes the appearance certain, or sharp, or a floor to build on. Nothing is being received here that could be certain. The searching has no edges, because edges are what a formula brings, and it brings them afterwards.

The white in the middle

Vagueness is not a defect in the searching. It is the condition of anything appearing at all.

Shine red, green and blue light on a rough mirror. Then on a finer one. Then finer again. The finer the roughness, the more the three colours mix, and at the end of that progression you have a white surface. Every white thing in the world is a very finely rough mirror. Which means that if every difference had to stay a different appearance, no white thing could appear, and in the end no thing could. The white is not a coarse report of an underlying array of exact wavelengths. The white is what there is to see, and it is there because the differences are not being held apart.

Magnification does not undo this. Roughness is ordinarily what a microscope is for. But build the mirrors as a progression and the indistinctness persists at every magnification, so the move that is supposed to end vagueness runs forever without ending it.

This is the first thing the demon cannot see. Ask him the colour of the finest mirror and he will give you every wavelength leaving it, correctly, with the angle of each. He knows every difference and holds every one apart; that is what it is to know everything. So there is no white for him. White is only available to someone who does not know too much.

I built the box for a classroom and it turned out to be an argument. The children did not receive a verdict from it. They looked longer, and then looked again, and the look times ran long at exactly the mirrors where the theory would not apply. One of them said the colours were beautiful and she did not want to write them down. Another said he could not remember what he saw. What the roughness does is not deliver a false report. It keeps you looking.

Vagus: to wander. Vagueness is wandering, and the wandering is the searching that has not been told where to stop.

Get lost is vagus in the imperative.

No second kind

The felt body, emotion, the texture and arrival and duration of a thought. These are searched for the same way and found the same way. Looking for whether that was fear is the same act as looking for whether that was the sound. There is no inner theatre with a smaller observer seated in it, sunglasses or no, and no second kind of finding that would need a different licence.

The standing objection is that introspection is a bad instrument. People confabulate. People report reasons for their behaviour that demonstrably were not the reasons. People miss large changes and then deny that they missed them.

I concede all of it without reservation, because every one of those findings is about a number taken. Confabulation is a false report of a cause. Change blindness is a failed claim about persistence across a gap. Memory revision is a claim about a past made out of a present. Each is an error in the place where the searching was declared over. Not one is a case of a search going wrong while it was still going.

So the psychology of introspective failure is not an argument against the immediate. It is a careful map of where the stopping happens, and I would rather have it than not.

Notice that I have not called the mind an instrument for receiving inner objects. That picture is what is being argued against here, and it does not improve by being extended inward.

Two motions

Aristotle allowed two natural motions. Gravity, which goes down in a straight line. Levity, which goes up and eventually turns and goes round.

Newton opens his first law with corpus omne — every body — and the distinction is finished. One kind of body. One natural motion, uniform and straight, laid out between a positive and a negative. Heidegger’s remark is that the Aristotelian division between earthly and celestial has by then become obsolete, and obsolete is the right word. Levity was not refuted. It was dropped, because the new motion had no place to put it.

Haldeman’s title is a witch’s ingredient with two letters added: eye of newt, I of Newton. I take it more seriously than it was meant. There is an I in Newton, and it is exactly what corpus omne has no place to put.

I want this vocabulary because it does not have a subject or an object anywhere in it. It is not about minds facing things. It is about which motions are admitted.

Questioning is levity. It rises, and then it turns, and the turning is the return. Doubt is levity with the turn forbidden — rising and rising and never coming round, which is exactly what it means to say that doubt cannot arrive and cannot stop.

And that motion has a mathematical statement. A bounded monotone increasing sequence. Monotone: it never turns back. Bounded: it cannot get away. Forbid the turning and forbid the escaping, and the axiom of completeness will hand you a single point at the top and tell you the point is real.

The demon’s answer to the third question is the second clause spoken aloud as a boast. There is nowhere he cannot come back from. He cannot get away. Omnipotence, in the one respect that turns out to matter, is a boundary. He is the point at the top of the sequence, real because the axiom says so, and wearing sunglasses.

Monotone is vagus struck out by stipulation. The sequence does not wander, not because anyone found out that it doesn’t, but because wandering was excluded in the setup. Nothing was discovered. A condition was imposed, and uniqueness came out the other side.

The induction of inductions

Measure a plank. Whole metres first, then decimetres, then centimetres, and at every stage there is a little more plank left over than your last mark. The process ends when the tape runs out, or the eyes fail, or the measurer loses interest. The axiom then assures you that a real number exists for the actual length, and that there is an actual length.

Now make the same move with people instead of decimetres. Science does not rest on one reading. It compares records made on different nights by different hands, judges which differences are systematic and which are noise, and concludes that what has held will go on holding.

Hume named the difficulty and I will not spend long on it. The inference requires that unobserved cases resemble observed ones, and that principle cannot be established deductively without assuming itself. He concluded that induction is not the work of reason but of the imagination.

The general form is what I am after. All replicated measurements, never finished, may be treated as finished.

That is the induction of inductions, and it is the axiom of completeness made outside the language of numbers. The structure matches at every point. A sequence of replications, each narrowing the estimate. Bounded, since the value is not infinite. Monotone in precision, since the intervals are supposed to nest and shrink. And never finished, because the experiment can always be run once more, and there is always a further place of precision the method has not reached. Then the move: assert the limit, and call the limit the true value.

Consensus is the least upper bound, and it is the least one that matters — a least upper bound is unique. That uniqueness is the whole trick. Where there were many observers with many offsets there is now one number, and the many-to-one has become one-to-one, which is Russell’s definition of the non-vague achieved by stipulation rather than by looking. The observers are supposed to drop out at the limit. Their offsets cancel and what remains is the object itself, unmediated. That is the escape from the first person. It is asserted, not found.

What remains when all the observers have dropped out has a name, and it is not a joke that the name is a demon. In 1814 Laplace imagined an intellect that knew, at a single instant, every force and every position in nature, for which nothing would be uncertain and the future would stand as plainly present as the past. His readers took to calling it Laplace’s demon. It is the least upper bound of all observers, imagined as somebody. The equation summons it, said the right way.

The demon in the classroom has a place in mind for Sam’s soul: a universe run by binary digits, where Sam could start out as a decimal and gradually work his way up to a fraction. Consider what that promotion is. A decimal that goes on without ever repeating is the least governable thing in arithmetic. A fraction is a decimal that has agreed to repeat. Damnation, in this cosmology, is a career in which you advance by becoming more determinate. The profession runs the same ladder and calls it rigour.

The error bar is the “…”. It is notation standing in for the rest of an investigation nobody carried out. A meta-analysis is a sequence of studies with a limit taken.

And the cost is the cost of the plank. Is that precise enough recurs at every level of magnification, and it is answered by declaring the unfinished sequence finished. Which shuts out more than it means to. The motive for taking the limit is not curiosity; it is the craving for certainty, and the fear of doubt, and those two names describe one condition. That is the curse word woven into the invocation. A mathematician cannot rest in an unfinished sequence. A profession cannot rest in a result that might not replicate. So the sequence is closed by decree, and because doubt and questioning are the same motion with and without the turn, forbidding the one forbids the other. The monotone condition does not distinguish between them and cannot. It only says: do not come back.

In this perfect wakefulness the light does not radiate, does not twinkle, no-one blinks.

Except that the equation is on a blackboard, and it is looked at by a man who blinks.

What does not stop

Something survives all this, and it is not evidence.

Some searching will not finish where the formula said it would finish. The bridge is not behaving, and the engineer goes back and looks again. The trial fails and someone runs it again. The result will not replicate no matter who runs it, so it goes on being run.

I used to describe this as an appearance overruling an inference. That was wrong in a way worth marking, because it smuggles back the picture this essay is against: two items, each with content, and a contest between them settled by contradiction. There is no such contest.

Here is what happens instead, and the shape of it is already in the mathematics. A positive definition states what a thing is. A dialectical one states the limits of a proposed definition — it does not aim at the essence, it aims at where the proposal gives out. Analysis students think continuity through dialectically, epsilon answering delta, and then remove the dialectical shape and put a proof in its place. An open set could be defined as not its boundary, and then some boundaries would have to be found by ongoing inquiry rather than declared.

The bridge falling is a boundary that ongoing inquiry has to find. The engineer had a positive definition, complete, with a stopping-place written into it. The stopping-place did not arrive. He is returned to the dialectical case, where the limits of the proposal are what is at issue and somebody has to go and look for them.

That is why it costs. Nobody has been refuted. Everybody is still looking.

Which is the same event as the axiom, seen from the other side. The axiom declares an unfinished search finished. This is a search that will not be declared finished. One is the apparatus closing the sequence; the other is the sequence refusing to close. The cost is the same cost either way, and the only question is whether it is paid by the decree or by the looking.

And it falls in either domain. The engineer in front of a structure that is not behaving. The practitioner who goes on seeing, plainly and unwillingly, what cuts against his teacher, his doctrine and his hopes for himself. The professor at the blackboard after the demon has gone. None receives a refutation. None can stop looking. Same event. Same currency.

Comparing observers requires an observer

Here is where the standard picture cannot be repaired.

Astronomy found this out in the most literal way. In 1796 the Astronomer Royal dismissed his assistant for timing star transits later than he did. Twenty-six years later Bessel compared his own timings with those of colleagues no one would have dismissed, and found steady personal offsets larger than the one that had cost the assistant his post. The correction built from them was called the personal equation.

But Bessel did not step outside himself to discover that he and his colleagues differed. He looked at the records. He judged the difference systematic and not noise. He decided how many nights were enough. Every one of those acts is the kind of act the personal equation exists to correct for, and every one was performed in the first person, unassisted, by a man with an offset of his own.

The only thing that could stop this regress is an observer who needs no correction: one who compares all the observers without being one of them, who has no offset because he has no position, who can come back from anywhere because he was never anywhere in particular. That is the demon’s job description exactly. And the demon is a character in a television programme.

There is no regress-stopper. There never has been. Each correction of first-person judgment is designed, applied and evaluated by first-person judgment, and if you go looking for the level at which this stops being true you will not find it. You will only find people, looking.

Which means the apparatus of objectivity is not a way of getting out. It is a way of getting a better result from inside, and the difference matters enormously in practice and not at all in principle.

What the agreement costs

The serious defence of third-person method is not that any individual scientist errs less than any contemplative. It is that errors are uncorrelated. My misreading and yours are independent, so when we pool, noise cancels and signal stands.

So look at what the profession does to produce its agreements. Shared training. Common instruments from common manufacturers. A standardized vocabulary in which the phenomenon may be described. Peer review by people trained in that vocabulary. Replication by protocols fixed in advance. Textbooks that settle what a competent observation is before anyone observes anything.

Every one of these is a correlating operation, built deliberately and well to make two observers more alike. The agreement they produce cannot then be cited as independent convergence upon the object, because independence was the thing being removed. Consensus is both the output of the correction machinery and an input to it, and the relation runs the wrong way for comfort: the tighter the correction, the less the eventual agreement tells you.

Bugaev, arguing in 1897 against the mathematical fixation on completeness, called incompleteness a manifestation of independent individuality and autonomy. He was talking about the number line and he was talking about this. A unique least upper bound is individuality removed — one value standing where many observers stood. The guild’s shared training and the axiom’s uniqueness clause are one operation performed at two scales, and that is why the sociological complaint about guilds and the epistemological defence of method are not two conversations. The institution erodes its own justification in the act of performing it.

Follow the road to its end and you arrive at the demon: consensus with nobody left in it. One knower, perfectly agreed with himself, and no independence remaining to spend.

I am not saying the corrections are bad. They are why the bridges stand. I am saying they buy reliability by spending independence, and you cannot spend it and still have it.

The inversion

The standard picture has trained attention waiting outside, hoping to be licensed by objective method — hoping that one day the disciplines of the inner will be validated by instruments and thereby allowed to count.

The dependency runs the other way.

Discriminating what is there from what you expected to be there. Noticing that you are tired, that you are invested, that you have already decided. Holding attention steady enough to register a small change nothing prompted you to look for. Every gauge-reading in the history of the world has run on these.

And when you examine what the apparatus consists of, you find prosthetics for known defects in exactly these capacities. Blinding manages expectation. Pre-registration manages the drift of a hope into a hypothesis. Instrumentation replaces a judgment that fatigues with a trace that does not. The personal equation corrects a constant in a man’s attention that the man could not feel.

You cannot build a prosthetic without knowing what is failing. Each device encodes something known about attention, and then conceals where it came from by working so well that nobody has to think about the defect again.

Imagine the prosthetics finished — every correction applied, nothing left over to correct — and you have the demon again. A superb artificial leg with no one walking on it.

Attention is not the worse instrument. It is the instrument the others are corrections to.

The line and the return

There are two shapes a correction can take, and the difference is the difference between an equation and a return.

Identity is a line. Even written as A = A it needs two positions and a sign between them holding the gap closed; you cannot write “A =” and have said anything. Selfhood on that model is an assertion of sameness across a distance, and it needs the gap covered, exactly as the axiom covers the gaps in the line.

The demon’s terms are lines of this kind. A question and its answer. A task and its completion. Two terms and a sign between them, the gap to be closed by his powers, and his powers close every such gap there is. Sam’s move is to decline the shape. Not a question: a command. And get lost has no completion the demon could hand back. Its fulfilment is a going with no guaranteed return, which is the one shape his powers were not built in.

He can come back from anywhere. That sounds like the return and is its opposite, because he has never been away. A return needs a going it does not control. What the demon has is a line folded double.

The other shape needs no equals sign. One object, and coming back to it. Aristotle’s point that circular motion cannot be defined by an opposition is the relevant one: the present you, reading, and returning to the present you, reading, is enough for motion, and there is no second term. Nothing is claimed to be the same. Heraclitus has it already — we step and do not step in the same rivers.

Sitting still and watching is the second shape. It is a correction too, but of a different kind from the astronomers’. A linear correction has an output: a number you subtract, after which the matter need not be attended to again. The return produces no number. Nothing is stored, nothing accrues, and tomorrow you do it again with no credit from today. The output of the practice is the practice.

That is why the objectivity apparatus can forget its origin and the practice cannot.

And the two treat wandering in opposite ways. Blinding, protocol, pre-registration, fixed vocabulary — every one is engineered to keep the mind from going where it will. The return does not prevent the going. It consents to it and adds the coming back. One approach removes the wandering. The other keeps the wandering and adds the turn.

Which gives the perversion its shape. Doubt is wandering with the turn taken out, and an apparatus built of protocols against a wandering never permitted to come round does not contain doubt. It institutionalizes it, and then covers the hole with a limit so that the asking stops by decree rather than by arrival.

Constants

The impermanence of constants is not a mystical objection and I want to state it carefully, because the careless version is easy to dismiss.

Physics does check this. Variation in the fine-structure constant has been probed against the natural fission reactor at Oklo, two billion years old, against absorption lines in quasar spectra, against the abundances of the light elements formed in the first minutes. The bounds are tight, and they reach back very far. Anyone who says nobody has looked deserves to be stopped at the first paragraph.

But every one of those bounds is derived inside a framework that assumes the stability it is testing. The Oklo constraint is read out of nuclear physics taken to have been the same nuclear physics. This is not a scandal and there is no way around it. It is what testing a constant necessarily looks like from inside, and the error bar on the result is calculated using the assumption in question.

The constant feels constant. That feeling is a fervour about something only time can report on, and the report is being written in the language whose stability is at issue.

The demon, asked his second question, said he could break the laws of physics and step through time whenever he liked. It is the one claim in the episode that could only be checked by the party making it. Nobody in the classroom thought to ask how he knew.

Two things I am not saying

That a profession’s first job is self-preservation, I hold to. Its first job, not its only one. People inside these institutions do careful, honest and expensive work, and much of it survives the pressure it is under. If “first” slides into “only,” the argument becomes a complaint, and a complaint is easy to put down.

And contemplative institutions gatekeep at least as hard as universities. Lineage, authorization, the approved description of what you saw, the quiet disappearance of the student whose experience did not fit the map. I concede this early because it costs the argument nothing. My claim concerns the structure of investigation, not which guild has nicer people. Every guild correlates its members. That is what a guild is for.

Where this sits

Husserl described objective science as a construction out of the lifeworld that forgot it was a construction. Merleau-Ponty treated the objective attitude as a derived posture concealing its origin. James took introspection as a primary method and gave it the seriousness of any laboratory procedure. Fleck showed that a thought-collective determines what can be seen at all. Gieryn named the boundary work by which a profession draws the line around itself. Daston and Galison traced mechanical objectivity as a discipline of self-restraint — the observer suppressing his own judgment, an inner exercise throughout. Varela built a research programme on the dependency instead of arguing about it. Schaffer read the personal equation as observatory discipline, a chronometric regime for the surveillance of subordinate observers.

I want the demon for something else: for the licensing relation, for the fact that the tools of objectivity are made out of what attention has learned about itself and cannot be made any other way, and for the one task that the finished observer cannot perform.

Back to the blackboard

There is a detail I have saved.

When the demon is gone, the equation is still on the board, and still wrong. The episode never solves it. Sam goes back to it, complaining that the demon wasn’t any help at all.

I want to mark what follows as mine rather than Haldeman’s. He wrote a joke, and a very good one, and this is laid over it rather than found in it. But the complaint is exact. The demon could have answered anything Sam asked and could not help, because what Sam wanted was never the answer. He wanted to get it right, and getting it right is done at a blackboard by a man who blinks, going back to the same line and finding it not quite the same line. An answer handed down by a being who has never been lost is a number with nobody’s looking in it. Omniscience was no help with the equation for the same reason it could not see white.

Science has this ending too, and tells it less well. Having established the personal equation, Bessel found that it drifts — with the year, with the instrument, with the man. So the correction had to be remade, and remaking it meant going back to the same place: a person attending to a star and to himself at once. Astronomy got the motion right and the description wrong, and it was the description that was institutionalized.

We have been going back ever since. We only stopped saying so.

And there is a reason to care beyond the accounting. If the searching is not in doubt, then the endless refinement toward a precision that never arrives is the intellect sharpening a vagueness it did not find sharp, and the appearances we were told to look past were never deficient. What follows from that is happiness, which is a longer argument and one I have made elsewhere. Here I only want the ground it stands on: this, now, still being looked for.

new research on vagueness and logical negations

03 Monday Aug 2026

Posted by Andrew Nightingale in The only category

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logic, mathematics, philosophy

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.

https://www.researchgate.net/publication/411137932_No_Privileged_Resolution_Ambiguity_vagueness_and_the_individuation_of_negation

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.”

https://www.researchgate.net/publication/411137931_Pairing_Indeterminacy_in_a_Schematic_Theory_of_Negation

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 by Andrew Nightingale in The only category

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logic, logical pluralism, mathematics, pedagogy, philosophy, science, writing

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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REFERENCES

Abbott, S. (2001). Understanding analysis. New York: Springer.

Beall, J., & Restall, G. (2006). Logical pluralism. Oxford: Clarendon Press.

Borges, J. L. (2002). Other inquisitions: 1937-1952. Austin: University of Texas Press.

Brouwer, L. E., & Dalen, D. V. (1981). Brouwer’s Cambridge lectures on intuitionism. Cambridge: Cambridge University Press.

Brouwer, L. E. J. (1907). On the Foundations of Mathematics. Thesis, Amsterdam; English translation in Heyting (ed.) 1975: 11-101.

Brouwer, L. E. J. (1908). The Unreliability of the Logical Principles. English translation in Heyting (ed.) 1975: 107-111.

Carnap, R. (1937). The Logical Syntax of Language. London: Kegan Paul.

Cook, R., 2010, “Let a thousand flowers bloom: a tour of logical pluralism,” Philosophy Compass, 5(6): 492-504.

Corbin, J. M., & Strauss, A. L. (2015). Basics of qualitative research: Techniques and procedures for developing grounded theory. Los Angeles: SAGE.

Costa, N. C., & Alves, E. H. (1977). A semantical analysis of the calculi C_n. Notre Dame Journal of Formal Logic, 18(4), 621-630.

Dewey, J. (1938). Logic, the theory of inquiry. New York: H. Holt and Company.

Drago, A. (2018). Suggestion for Teaching Science as a Pluralist Enterprise. Transversal: International Journal for the Historiography of Science, (5), 66.

Sextus Empiricus. (1996). The skeptic way: Sextus Empiricus’ “Outlines of Pyrrhonism.” Translated by B. Mates. New York: Oxford University Press.

Ferreira, L. B. (2012). Philosophy for Children in Science Class. Thinking: The Journal of Philosophy for Children, 20(1), 73-81.

Feyerabend, P. (2010). Against method. London: Verso.

Foucault, M. (1973). The order of things: An archaeology of the human sciences. New York: Vintage Books.

Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research.

Hartshorne, R. (2000). Geometry: Euclid and beyond. New York: Springer.

Lakatos, I. (1962). “Infinite Regress and Foundations of Mathematics,” Aristotelian Society Supplementary Volume, 36: 155-94.

Latour, B. (2002). We have never been modern. Cambridge, MA: Harvard University Press.

Mates, B. (1961). Stoic logic. Berkeley: University of California Press.

Nightingale, A. (2013). Many roads from the axiom of completeness. Questions Are Power. https://questionsarepower.files.wordpress.com/2016/03/many_roads_from_the_axiom_of_completeness-2.pdf

Nightingale, A. (2018). Vagueness as the Embodiment of Inquiry: An account of vagueness as it pertains to logic, and a study of teaching vagueness to young Thai (P4) students. https://questionsarepower.files.wordpress.com/2018/09/nightingale-dissertation-vagueness-as-the-embodiment-of-inquiry.pdf

Peirce, C. S. (1933). Collected Papers — IV Chapter 3 — Existential Graphs., pp. 4.397-4.417, edited by Charles Hartshorne and Paul Weiss, Harvard University Press, Cambridge.

Peirce, C. S., Hartshorne, C., & Weiss, P. (1934). Collected papers. Cambridge: The Belknap Pr. of Harvard University Press.

Peirce, C. S., Hartshorne, C., Weiss, P., & Burks, A. W. (1960). Collected papers of Charles Sanders Peirce. (Vol 5). Cambridge: Belknap Press of Harvard University Press.

Rowland, Tim. (2000). The Pragmatics of Mathematics Education: Vagueness and Mathematical Discourse. Taylor and Francis.

Russell, B.A.W. (1923). ‘Vagueness’, Australasian Journal of Psychology and Philosophy, 1, pp. 84-92.

Russell, G., (2008), “One true logic?” Journal of Philosophical Logic, 37(6): 593-611.

Russell, G. (2019). “Logical pluralism.” In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2019 Edition). https://plato.stanford.edu/archives/spr2019/entries/logical-pluralism/

Shapiro, S. (2008). Vagueness in context. Oxford University Press.

Shapiro, S. (2014). Varieties of Logic. Oxford University Press.

Tarski, A. “On the Concept of Logical Consequence.” In Logic, Semantics, Metamathematics: papers from 1923 to 1938, chapter 16, pages 409-420. Clarendon Press, Oxford, 1956. Translated by J. H. Woodger.

Tye, M. (1994). Sorites Paradoxes and the Semantics of Vagueness. Philosophical Perspectives, 8, 189.

Ullmann, S. (1972). Semantics: An introduction to the science of meaning. Oxford: Blackwell.

Verlaine, P. (1884). Jadis et naguère. Paris: Léon Vanier.

Varzi, A. C., 2002, “On logical relativity,” Philosophical Issues, 12: 197-219.

Weber, Z. (n.d.). Inconsistent Mathematics. Retrieved from https://www.iep.utm.edu/math-inc/#H6

Wittgenstein, L., Bosanquet, R., & Diamond, C. (1976). Lectures on the foundations of mathematics: Cambridge, 1939. Hassocks: Harvester.

Wyatt, N., & Payette, G. (2019). Against logical generalism. Synthese.

(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!

https://nightingale108.itch.io/logic-puzzle

youtube.com/watch?v=TCorybi0VdQ

Zeno and Grunbaum (revised)

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One of Zeno’s famous paradoxes is his challenge to the mathematicians view that any finite line segment can be divided by a point. If so, the resulting lines can be subdivided. Zeno’s question is then “If lines can be divided and subdivided, what would the size of the lines be after fully dividing the line?” The absurdity does not lie in the question, but in any answer a mathematician could give. If a mathematician says there are lines with sizes as an outcome, then the sum of the infinite number of lines would make the original line infinite (we said originally that it was finite), and so a contradiction. Otherwise if the result is points, or lines of zero length, the sum of the lengths of points of course would be zero (but we said the line was of non-zero length), and so another contradiction. There are many other possible answers to the question. The reason the question is so famous is that all the answers so far have been unsatisfactory or absurd. I can only guess at why this question has been so mistreated over the 2500 years since it was asked (for example, authors often put any absurdity with Zeno himself, when the questions were intended to show absurdity in his opponents. Authors also accuse Zeno of all sorts of foolish intentions for his questions, such as that motion is impossible). Zeno’s intention, however, is not in question: he was a student of Parmenides and was simply making arguments to defend his teacher’s doctrine that there was only One thing in the world. Regardless, Zeno’s question has outlasted any answer.

Adolf Grunbaum is eminent in the position that Zeno’s paradoxes are refuted by modern mathematics. In his essay “Modern Science and Refutation of the Paradoxes of Zeno”, he began by making exactly the mistaken claim that “Zeno attempted to demonstrate the impossibility of motion” (p. 165 Zeno’s Paradoxes). Grunbaum goes on to introduce the common notion among mathematicians, introduced by Cantor in the 19th century, that there are different kinds of infinities and so we are faced with choosing a particular kind of infinity for the result of the infinite subdivisions of a line. He argues that the kind of infinity of points on a line is “super-denumerable” and cannot be added the way Zeno proposed. Normal addition is reserved for the familiar denumerable infinity that proceeds like the natural numbers (1, 2, 3, etc). However, the divisions Zeno proposed begin as denumerable (one division or point, then another, etc), and the number of divisions is the same as the number of points. Grunbaum, like Cantor before him, argues that the result of this process, while denumerable at each finite stage, results in something that is not denumerable but “super-denumerable.” As we divide we are working with line-segments of some size, and in the limiting case what results are not segments of zero size (which could be summed), but something with no concept of size whatever. The super-denumerable point is not the result of the division process — it’s a different kind of thing asserted to exist at the place where the process would have to end. The inductive evidence (each division produces a smaller segment) is simply declared irrelevant at the limit. Grunbaum doesn’t show that Zeno’s question has an answer. He introduces a category that sits outside the framework in which Zeno’s question is meaningful. Zeno asked what you get at the end of infinite division. Grunbaum says: at the limit you get something to which the concepts of size and summation don’t apply. But that is exactly what Zeno was pointing at — that the process leads somewhere the mathematician’s tools can’t follow. Naming that place doesn’t illuminate it. The question isn’t defeated. It’s excluded. And Zeno, being a good questioner, would simply ask the next question: what is this thing with no concept of size, and how does it compose a line we can measure?

But there is something deeper going on here than this mathematical play of words. A line segment can be defined by its endpoints. Indeed, the endpoints are all that is needed to make a formula for a straight line. The “stuff” between the endpoints, using Aristotle’s terminology, would be the substance. Now we have come to the question “What is a line segment?” Aristotle would say that first and foremost it is its form (or formula), which is to say, its endpoints. Aristotle added to this of course, including in any “what” also its cause or “why” (Aristotle’s Metaphysics). Such expansive thinking has long gone out of use, but Grunbaum would have us believe that the substance of a line-segment is a thing so different from its form as to be completely incommensurable- that Aristotle’s conception of being (which was a marriage of form and substance) must be utterly divided, leaving us with much deeper problems than what we had before with Zeno’s question. Do we abandon form (endpoints) in favor of substance, since the “stuff” of the line-segment would ultimately be a collection of Grunbaum’s “super-denumerable points.” In that case, what would addition or any other mathematical concept be, since we must compare endpoints to measure, and use the result to add, without these formal concepts there is little left of mathematics at all.

Aristotle’s substance, at least one of his definitions of it, was a subject that could not be predicated. Another definition was form or essence, and form and substance are deeply connected. In searching for substance, predicates divide the subject. “The human is a man” (or “Woman”) is a division of human. Metaphor, in this view, is destructive to a search for substance. It expands words: “Humans are stars” makes a mess of things, only adding possible predicates. Unless you search for the intersection of “human” and “star”, in which case you are dividing both. Regardless, seeking knowledge of substance is a process of division- so too with a process of division of line segments. And what is the result of this infinite division, this search for substance in the excised world of pure form that is mathematics? It is merely the division itself – the point – which is what we started with when we were looking for substance. What is our point? Or is it changed into a super-denumerable point? Or some other kind of point or division of a line? What is this scalpel? Have I used it violently in searching for it?

OP: https://questionsarepower.org/2015/02/13/zeno-grunbaum-and-order/

Vagueness in Mathematical Terms (reworked and more accessible)

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High Up (On Vagueness and Mathematics)

Imagine walking down Olympus Mons — the largest volcano in the solar system, on Mars. Its slopes are so gradual that you might walk for hours barely noticing the descent. At the top, you are clearly “high up.” At the bottom, you are clearly not. But somewhere in between, the description falters. You become unsure whether “high up” applies or not. That fuzzy middle zone is what philosophers call vagueness.

We can make this precise — or try to. Map the mountain onto a number line: 0 at the summit, 1 at the base. The set of points where you count as “high up” has a boundary. Call it sup — the least upper bound, the last point before “high up” runs out. The set of points where you count as “not high up” also has a boundary: inf, the greatest lower bound, the first point where “not high up” begins.

Now suppose — reasonably — that sup itself counts as “high up” (since every point below it does), and that inf counts as “not high up.” What happens when we ask how sup and inf relate to each other?

There are only three possibilities, and each one produces a contradiction:

If sup = inf, then that single point is both “high up” and “not high up.”
If sup > inf, then the real number line — which is dense, meaning there’s always another number between any two — guarantees a point z sitting between them. That point would be both “high up” (since it’s below sup) and “not high up” (since it’s above inf).
If inf > sup, the same logic applies in reverse.

The standard response is to blame the vague word. “High up” is imprecise — a folk term, not a technical one. Strip it away and mathematics, supposedly, is safe.

But stripping the vague term doesn’t solve the problem. It moves it.

Consider the wave theory of light. Its mathematical core — the equation governing refraction:

sin(α) / sin(β) = μ

— looks clean and precise. But the philosopher Mary Hesse pointed out that the equation, on its own, is ambiguous: it can be interpreted in multiple, entirely different ways. The symbols don’t come pre-labeled. Perhaps α and β aren’t angles of light at all — perhaps they’re the angles between the Pole Star and two planets at midnight. The mathematics would fit. Which interpretation is correct? The equation doesn’t say. Meaning doesn’t live in the symbols alone.

Vagueness and ambiguity are usually treated as distinct problems. Ambiguity means a word or expression has more than one possible meaning. Vagueness means a word has unclear edges — cases where it’s genuinely uncertain whether it applies. But consider: what if a word were both at once?

Thai has a word, krup, that technically means “yes” but functions more like a polite acknowledgment — because outright agreement can feel presumptuous, as if you’re confirming what the listener already knows. It occupies a middle space between assertion and non-assertion.

Now invent a word: snook. It means “tall” in some contexts and “not tall” in others. When applied to someone of borderline height — someone exactly at the edge of where “tall” is uncertain — is snook ambiguous, vague, or somehow both? Is there a vagueness between vagueness and ambiguity? If so, what does that do to the apparent clarity of mathematical symbols?

Even pure mathematics — mathematics with no interest in mountains or light — is soaked in vagueness. The discipline’s foundational concepts carry it: continuity, completeness, integral, limit. These are not casual approximations. They are the load-bearing terms of analysis, the branch of mathematics that underlies calculus.

And they are vague. Any careful textbook in real analysis will show you functions that slip through the formal definition of continuity — technically satisfying the definition while still behaving in ways the definition was meant to exclude. The definition doesn’t quite capture the intuition. The intuition doesn’t quite surrender to the definition. The gap between them is not a failure waiting to be fixed. It is where thinking happens.

Without words like “continuity” and “completeness” — words that mean something intuitively before they mean something formally, and that keep some of that intuitive life even afterward — mathematics would be unlearnable. Students would have nothing to grab onto. The vagueness isn’t what mathematics tolerates in spite of itself. It’s part of what mathematics thinks with.

The fantasy of a perfectly precise formal world, unsullied by the messiness of natural language, is just that — a fantasy. Vagueness goes all the way down.