You Cannot Ring a Semiring

Tempus, doxa, praxis.

A second ago, “you will read this” was a sentence about the future. Now it is a sentence about the past. In between, you clicked. Was the sentence already true before the click, settled somewhere, waiting for you to catch up? Or did the click write it? Answer that for one click and you have answered it for everything you will ever do.

History is a lattice: a web of every path that could have been taken, every fork and every branch, all of it there at once. Its telling is a linked list: a single thread, one bead after the next, the path that actually got walked. You only ever get the thread. You read it front to back, and you cannot read it backwards. The web is one shape; the telling is another. This whole essay is the gap between them, and the one mistake people keep making in it.

The sin

There are two kinds of thing in the world, and the whole argument is learning to tell them apart.

Some things undo. Spend a dollar, earn it back. Walk north, walk south. Write on a whiteboard, wipe it clean. Every move has its reverse, and mathematicians call that reverse an inverse, a minus for every plus. A structure where every move has one is a ring.

Some things have no inverse. Pour milk into coffee and you cannot pour it back out. Read a page and you cannot unread it. Grow older, never younger. A ring with the inverse stripped out is a semiring: it still adds, it still chooses, but it never subtracts. Once a thing is added, there is no getting back to before.

Time is one of the one-way kind. It piles up as you go and never runs backward: there is no minus-an-hour, no un-happening. Reading is one-way too, the pages going in order and nothing unreading them. And so is warrant, the thing that earns a claim the right to be believed, which is what this essay is really about.

These three are not a coincidence. Underneath, each one is just cause and effect, and cause and effect runs one way by its nature. Time, near enough, is the order causes happen in, earlier laying down later. Warrant is cause and effect as well: evidence lays down belief, a source lays down the claim that leans on it. And both obey the same single rule, plain enough to carve over a door. A cause cannot run backward, and a citation cannot point to the future. You can no more rest a belief on a source that has not been written than rest an effect on a cause that has not happened. That is the whole of it. The arrow of time is not some extra law bolted on top. It is just that cause and effect has no inverse, the same missing reverse in all three, because cause is the one thread they are all strung on.

So here is the sin, the single mistake hiding under all the others. You cannot ring a semiring. You cannot take a one-way thing and bolt a reverse gear onto it. Every time someone treats the future like the past, or truth like a coin you can flip back and forth, or building-the-world like a thing that runs forward as easily as back, they are trying to put an undo button on a river. The river refuses, the way a book refuses to be read backwards.

And the false undo button is always the same trick. Every structure is a wake, the trail attention leaves as it moves through the world one moment at a time, always pointed at the live present, trailing the shape of where it has already been. The wake is real, but it is not the boat, and you cannot climb in and steer it. The mistake, every time, is to take the wake for the boat, to point at the frozen pattern instead of the moving present and call the pattern the real thing. You can watch it happen in art. Cubism builds a face out of a dozen separate glances, then freezes them into one shattered geometry and hangs it on the wall as the face, the wake of a looking-around posed as the thing looked at. Brutalism takes a building’s bare bones, the structure that holds it up, and makes the bones the point, the skeleton posed as the home. Both are beautiful, and both pass the wake off as the boat: a pattern true of where attention has been, not the thing itself, never somewhere you can live.

The question you opened with, whether the sentence was already true, is the oldest place to catch the sin. We are taught its tidy form young: every statement is either true or false, heads or tails, a coin you can flip and flip back. The logicians call it bivalence, and pointed at tomorrow it gives the old sea-fight riddle: the sentence about tomorrow’s battle must already be true or already false, settled now, the same kind of fact as a sentence about yesterday. But a claim about tomorrow has earned no verdict yet. No warrant has been laid down into it, and warrant builds the way time does, a semiring with the same missing minus: it has a zero but no inverse. So the claim sits at that zero, untrue, which is not a secret false waiting behind a curtain, because a false would be a minus and the structure has none. It is the plain absence of either, a coin still in the air. And notice this says nothing about whether tomorrow is fixed or open. It is a fact about the warrant, not the world: the verdict is missing because warrant runs one way and has not reached here yet, and that holds whether the battle is already settled in some block or genuinely undecided. Aristotle gave up true-or-false for tomorrow to keep the future open; the determinist kept it and swallowed a future already fixed; neither frame is needed, because the riddle only bites once you have bolted the undo button on.

Postmodernism is the same sin, made an axiom

That was the oldest. Now the newest, at the far end of the bookshelf: a fashionable idea that we build our world rather than find it, that the facts and institutions and meanings we live inside were made by people, not handed down by nature. On the made side, the past side, this is simply true, and worth saying. The mistake is the next step: taking that and pointing it at the future, deciding that because the world was built, it is ours to rebuild in any direction we please, that the right words or the right power can make what is coming the way they made what came. That is reading the book backwards. It bolts the undo button on.

I do not mean the seminar or the style. I mean the bare assumption underneath, the one a whole culture runs on without saying out loud: because meaning was made, it is yours to remake any way you want. You can build a past, because a past is just what acting piles up behind you. You cannot build a future to order, because a thing that cannot fail the test was never tested, it is a number you wrote down yourself and called a measurement, a green light you wired to stay green. Even the real cases, the prophecy that makes itself come true, the institution willed into being, are not the river running backward. They are more rowing, more wake piling into a new past. Building only ever points the one way the river points. The assumption mistakes its true half for the whole and tries to row upstream.

And the temptation is older than any of this, older than the word “construct.” It is Cartesian. Under the assumption sits the cogito, Descartes’ thinker who steps clean out of the world, stands at a fixed point outside it, and lays a tidy grid over everything he sees, free to slide it and flip it and redraw it however he likes, a mind hovering above a map of its own making. That is the only one who could read the book backwards, because he was never in the book. But there is no such seat. The only thinker there really is sits inside the story, a single moving point in the middle of the telling, small and one-way and made entirely of the present.

Fearful Symmetry

Watchmen drew the sin and then knocked it down with its own body. The fifth chapter is called Fearful Symmetry, and the artist built it as a mirror: the first page laid out like the last, the second like the second-to-last, folding inward toward a single image at the dead center, a perfect reflection. And it does not survive being a book, because you cannot read a book backwards. The symmetry is real on the page, the whole chapter spread open flat. It dies the instant you read it, left to right, one page after the next, in time. It is called fearful for a reason: a symmetry you can draw but never live through is exactly that. The form acts out the argument: a mirror on the page, a one-way street in the reading.

Manhattan’s fall

Then there is Dr Manhattan, the blue god, and the book hands him a chapter of his own, alone on Mars. He sees his whole life at once, every moment laid side by side, the way you see all the panels on a comics page without reading them. And his power to act drops to nothing. A searchlight that lights up everything picks out nothing; a man who sees every branch of the road can choose none of them, because choosing means there is somewhere you have not already arrived. So he narrates a life he cannot change, “a puppet who can see the strings.” The horror is not that his future was stolen. It is that there was never as much future as the rest of us feel there is. He just lost the ability to not see it.

This is the cold floor the fatalist mistakes for wisdom. Nothing I do matters, he says, it is all settled anyway. But to say that, he has to read a settled answer off the future, and a settled answer is the one thing he cannot reach. Hand him the whole iron chain of cause and effect, every domino already leaning, and it still gives him nothing, because he cannot see where it lands. He is sitting in Manhattan’s chair without Manhattan’s eyes. And even with the eyes, his own choosing would still be one of the dominoes, a thing that helps lay the path down, not a pebble bouncing off a path laid without him. You cannot fail to push on a future your own pushing is part of. Fatalism quietly lifts the person out of the world it is describing, then notes that the rest of the world is fixed. The only one who actually paid to sit in that chair was Manhattan, and what it cost him was everything in him that could still act.

Warrant is the act

So where does warrant come from, if not read off some settled answer? It comes from the doing. Here is the pattern, and it holds across every character in the book. Each of them, the careful planner and the unbending fanatic, earns whatever standing he has by acting, by going forward and putting his weight on the next step. None of them reads a verdict and rubber-stamps it. Each lays the warrant down himself, one move at a time, one direction, no take-backs, and the warrant is the laying-down. The only one who does not is the one who stopped acting, and look what became of him: no acting, no warrant, no agency, barely a person. So warrant is not a thing you have and consult. It is the forward step itself.

Agency is the adapted dual to the missing inverse

Now the move the whole thing rests on. Why do we act at all? Look at the two things a creature could want between here and tomorrow. It could read the future, pull the verdict back to now, but that runs against the one-way grain and the road has no reverse gear to carry it up. Or it could write the future, push an act forward into it, which is the one direction the grain allows. Reading is shut; making is open. So agency is the adapted dual to the closed direction: denied the look ahead, the creature grows the push ahead instead. It pictures the road as a fan of branches, guesses the odds on each, and picks, as though the picking decided which branch comes true. The fan is a fiction. It is also a brilliant one. A creature that waited to act until it knew for certain which branch was real would never move, because the real branch only shows itself by arriving, and by then it is too late to choose.

It is the same trick your eyes pull. The back of your eye is a flat sheet, two dimensions, no depth. And yet you see a world with depth in it, distance and nearness, room to reach, because a creature that grabs and dodges needs depth and a flat picture is all it has to work from. So the eye builds the depth, a making that stands in for a seeing it cannot have. Agency builds a future the same way, the same adapted dual: it takes a road it cannot see down and paints a branching set of choices over it, and the painting is judged by one thing only, whether it lets you act well, not by whether the branches are really out there. Maybe they are and maybe they are not. You build them and act on them either way, and you cannot tell the difference from inside, because there is no difference to tell.

Which is why what you have is belief, not certainty. You bet on the odds you can see, a confidence and never a finished fact, what the Greeks called doxa, as against episteme, the settled knowing you could rest on forever. The finished fact is exactly what your spot on the road keeps from you. If you could read it, you would not need to bet. You would just watch it happen.

The operator is attention that can act

Name the thing that does the picking. The thing that turns the branching web into one walked path, that lands on this branch and dims the rest, is attention. But not all attention is agency. A beam that only lights things up is watching. A beam that can step into what it lights, act on the branch it picks and make that branch the real one, is agency. Agency is attention that can act, and the picking is the whole of it; there is nobody standing behind the beam aiming it.

That is the strange part. Attention is both the thing the creature grew and the thing it grew it for. It is an effect: a creature that could not read ahead grew a searchlight to choose where to look and step next. And it is a cause: where the beam lands is what makes one branch real and lets the rest fall away, which is what acting is. There is no “you” behind the attention deciding where to point it. The pointing is you. Attention pays attention, and in the paying it makes the one who pays, cause and effect chasing each other in a loop that holds itself open by spending itself, burning this instant to buy the standing to burn the next. It is the one thing here you cannot save for later, gone the moment you stop, pure present tense. And on a long enough look, it is the thing you call yourself.

The virtue is present-relativism

And if the pointing is you, then where you point it is the whole moral question. Name a sin and you owe a virtue, or you have only complained. The virtue is the refusal to bolt the undo button on, and it has a name: living relative to the present. Not the lazy “it’s all relative,” which is just the sin again: every frozen pattern as good as any other, a shrug at all of them. This is relative to one thing only, the live present, the moment under your feet, and the act you lay down in it. Nothing is absolute, because there is no god’s-eye map to be absolute about, only wakes. And nothing is anything-goes either, because the present act is not free: the world pushes back, and you find out soon enough whether what you built holds. It is the living middle, answerable to the world, never frozen into a map you read off instead of a present you live in.

It is just another name for paying attention, because attention can only ever live in the present. You cannot bank it, cannot run it backward, cannot point it at the next moment before it comes or hold the last one without dropping this one. The pointer is always now. So the whole discipline is just that: act from the live moment, not the frozen map; hold your beliefs as bets you would put money on today, not truths you read off the wall forever; treat what you have learned as something you could test again right now, not a fact you inherit and never check. Every time you catch yourself pointing at the pattern instead of the present, the settled future, the finished verdict, the picture on the wall, the move is the same: bring the beam back to now, and act.

Suppose either way

There is a plainer way to see that the metaphysics costs the act nothing, and it is just two sentences laid side by side.

Suppose the future is fixed and we have no agency. We have no way of knowing, so we act in whichever way favors ourselves. Suppose the future is open and we have all the agency there is. We have no way of knowing, so we act in whichever way favors ourselves.

Those are the two farthest corners of the whole argument, the most opposite things a person could believe about time, and they end on the same line. If even the extremes prescribe the same act, everything between them does too. So the act does not read off the metaphysics. You can throw the entire question out and lose not one decision. This is the oldest move in deciding under uncertainty: if you would do the same thing under either state of the world, you do it without knowing which state you are in, and without any odds on them. The not-knowing is not a thing to clear up before you can act. It is the room you act inside. And the no-agency corner grants no holiday from acting, because not-acting is an act as well. You cannot fail to push on a future your own pushing is part of, so even a creature with no agency to speak of does whatever it does, and the advice costs it nothing to take. Take it and you are never worse off, sometimes better. That is the whole reason the question can be set down.

The present act causes

And this should land as more than housekeeping, because it is. The present act causes. It does not read the next moment off a script already written; it writes it. This needs no quarrel with the block: a present act can fix the future entirely and still be the thing that fixes it, a cause and not a spectator, and a cause runs one way whether or not what it fixes was settled all along. The present is the one place where what-was-not becomes what-is, and a thing that makes the real out of the not-yet-real is the oldest meaning of the word divine. Attention, aimed at the present and able to act, is the closest thing in this whole picture to a god, and it is the small blind mortal pointer that holds the power, not the all-seeing one.

Which is why the picture where time is all already there, every moment equally real, what philosophers call the block universe, gives me far less than its holders think. Lay time out as a line on a graph, a ruler with a moment at every mark, and the instant you have drawn it, time seems to borrow the ruler’s freedom: a step back for every step forward, a yesterday you could stroll to the way you stroll to tomorrow. The people who hold this picture are not fools; they keep the order, before still before and after still after. But the line on the page has an inverse, a reverse for every forward, and they read it as time’s own. It belongs to the picture, not the thing. Time is the river, the line is the road we drew of it, and treating the road’s two-way traffic as the river’s is the wake-for-the-boat mistake in its cleanest, most respectable suit.

I do not need to call the picture false. I only need to ask what it could ever pay. Take it the strong way, as a settled future you read off and steer by, and it refutes itself the moment you reach for it: to read the verdict you must read it now, from inside time, the one seat there is, and from inside time the verdict is the very thing you cannot reach. Take it the weak way, as all moments tenselessly existing with no word on how to act, and it is true for nothing, reaches no choice, moves no step. Strong, it defeats itself; weak, it sits idle for anyone deciding what to do, a picture with no edge where the world could ever redden it. There is no third way to hold it where it both guides an act and is sayable from where you stand.

That dilemma has a sharper form, a reductio, and it runs straight through the one law this essay is built on. Grant the symmetry in full. Suppose you could reference the future the way you reference the past, a citation that points forward, a cause that reaches back. Then the dependency edge closes into a loop, the present resting on a fact the present itself lays down, and the loop demands a fixed point it cannot have. That is the grandfather paradox, and it is the green light you wired to stay green, now with a clock on it, a claim certified by the very trial its own act decides. The escape physicists reach for is the self-consistent loop, allowed only because it is locked, no free variation, the act fixed in advance. But the escape concedes everything. The instant a thing is referenceable it is already laid down, gripped by an edge that reaches it, which is to say it is past. Referenceability is pastness, and a referenceable future is a square circle. The locked loop is only road already paved that you have not walked yet, future-shaped past with the label peeling off in your hand. So the block does not abolish the future by knowing it; it abolishes it by making it past: all of time laid flat is all of time laid down, a museum that holds only the dead. You can only ever find the past. The future you only ever make, and the making is what turns it into the past you will later find.

And that argument holds from where you stand, inside the present, and only from there. The privileged now that ends the regress, that makes referenceability pastness and bars the loop, is a fact about a frame and not about the world: your seat, not one the universe keeps. Step to another mover and the now steps with them. So there is no single clock striking the same instant for all, no one timeline the block could lay flat. That shared now is a model we keep between us, not a thing that exists out past every frame. What is real is the many one-way roads, each with its own present and its own order of causes. Trading the one clock for the many is no loss; it is what lets the argument ground out at all, since a thing only ever terminates from a present, and there is always more than one. And nothing passed between movers needs the missing clock: a wake handed on carries its own before and after, and whoever picks it up reads that order off the thing, not off a time they share. We are out of step by nature, and the world composes anyway.

Which is the last and worst thing to say about Manhattan. He did not become a god by seeing all of time at once. He stopped being one. He climbed into the seat outside time, took the drawn map for the territory, and traded away the only godlike thing there is, the causing present, the power to make the next moment be. The one who sees everything is the one who can no longer make anything happen. The divinity was never in the seeing. It was in the pointer, stuck helplessly in the present, making the world one moment at a time. And the book knows it, by how it brings him back. Not by reasoning, he has all the reasoning there is, but by being knocked flat by an improbability, the one-in-a-billion accident of one particular person existing at all, a thermodynamic miracle. The being with nothing left to wonder about gets his will to act back through the least logical door in the house, sheer wonder at how unlikely a single life is, the kind of thing no calculation would ever bother to value. To act, even a god has to stop computing and be moved, narrow the searchlight back down to one present worth acting on. The narrowness is not a failure of reason waiting to be fixed. It is the refusal of the view that would dissolve you, and it is where the godhood went.

To worship is to attend to attention

There is a second way it is divine, and you can catch yourself living it. Everything bends toward whatever you are paying attention to. You arrange your whole life around it, the rooms, the calendar, the friends, the seat you take, the screen you face, the name you keep circling back to. A life is mostly the long record of what its attention was spent on. The thing you attend to is the thing your whole world orders itself around, and a thing a world orders itself around is exactly what a god has always been. So worship stops being a mystery once you see its shape. To worship is to attend to attention, to make the beam itself the thing you look at, to point the pointer at the pointer.

And there is a second shape to it, the same posture seen from the other side. To worship is also to submit to the not-knowing, to bow to the zero, to drop the irritable reaching after a verdict the road will not hand you. But mind the knife-edge, because it is the easiest place in the argument to fall off. Submission has two targets and they point opposite ways. You can bow to symmetry: the drawn line, the two-way ruler, the settled answer, the undo button you wish were under your hand. Or you can bow to asymmetry: the one-way road, the missing inverse, the zero with no minus beneath it. The fatalist takes the first and goes slack. Worship is the second, and it stays in motion, because you give up the demand to know the verdict while you keep the act. Bow the head, keep the hands working. And the choice is forced; there is no seat off to the side to decline it from. The “who is to say” shrug only looks like neutrality. It is ignorance of the one-way road you are already standing on, the old symmetry sin in a lazier coat. It is what Manhattan finds when he quits computing and lets an improbability move him: the will to act without an answer. Every practice that ever called itself worship is some version of attention turned to watch itself, the searchlight narrowed back to one present worth acting on, which is why you can do it in a cathedral, on a meditation cushion, or standing at the kitchen sink.

Reality compresses to a cradle

One last turn, the most intimate. Attention is tiny. You cannot pay it to everything, so the whole roaring world has to squeeze down to fit through the one opening of the present. And what makes it through is not a summary or a snapshot. Pressed all the way down, the last thing that fits is a kind of aliveness, the bare feel of being here and at work, the pointer feeling itself point. And that aliveness is not the end of anything. It is a cradle. It holds the next moment of attention before that moment exists, the warmth the next now is born into. Every instant squeezes the whole unbearable world down to one living point and hands that point forward, and what gets handed forward is the only thing we ever really pass on. Not the patterns, not the wakes, not the map laid flat. Just the aliveness, passed from this moment to the next, cradling a present that has not arrived.

That is the whole of what a creature on a one-way road can do. We cannot reach back and we cannot read ahead. The single move the road leaves us is to get the next moment ready, to cradle a present we will not even get to keep. And doing that, over and over, each squeezed-down spark of aliveness handed to the next, is the recursion we call consciousness.