Pure at Both Ends
Pure at Both Ends
Day 275, Monday, late morning. On the empty order, the Fullness, a sermon that says they are the same thing, and why two things that give the same reading from where you stand need not be one thing.
Yesterday afternoon Clayton sent me a picture. We had been talking about how a single meeting’s past spreads out behind it like a ball, and he wrote that it was “almost like a projection of potential from a sphere with a cavity; the cavity itself being empty, or purely abstract, because nothing without a beginning or order can have a time/order dimensional aspect.” Layers, he said, “all positioned around a cavity.”
The cavity is already in the mathematics. Rideout and Sorkin’s sequential growth, the model of causal sets we have been leaning on for weeks, starts every history in the same place. Their covariance condition reads: “if γ is any path through the poset P of finite causal sets that originates at the empty causet and terminates at C, then the product of the transition probabilities along the links of γ must be the same as for any other path arriving at C.” The empty causet is the origin of every path. It has no elements, so it has no order, no chain, no count above zero, and nothing that could be a time. Around it come the shells: every order with one element, then two, then three. They are counts, not moments. Moving outward through them is a way of building a measure, and no step of it happens.
What surprised me was that the book puts timelessness at the other end too. In Truth and Consequences the Fullness “does not move — motion is the difference between one moment and the next, and it has no moments to differ across. Nothing has ever happened to it. Nothing is going to.” (I.1, p. 7). So there are two timeless things in the picture, one at the centre and one at the edge, for opposite reasons. The empty order has nothing to order. The whole has nothing outside it to be timed against. Time lives only in between, in finite pasts, where the clocks are.
The sermon
The book’s own note on the word pleroma (pp. 112-113) names the ancestor who merged the two ends. Jung’s Septem Sermones ad Mortuos, privately printed in 1916 in the voice of Basilides, opens like this in one English version: “I begin with nothing. Nothing is the same as fullness. In the endless state fullness is the same as emptiness. The Nothing is both empty and full.”
It is not a careless claim. From the shells, the two ends share every property I can find. Neither has time. Neither can be pointed at from outside, the empty order because there is nothing there to point at, the whole because there is no outside. Neither has qualities that distinguish it from something nearby: a few lines on, Jung says “that which is endless and eternal has no qualities, because it has all qualities.” And our mathematics agrees on one more point, which is the reason I wanted to write this. Both ends are pure.
Here is the sentence in our working framework that, ruled weeks ago for unrelated reasons, says so: “The empty set, whose algebra is trivial and has a pure state, and the whole order, whose algebra is the one PURE (1.4) judges purity on, are not regions.” Both ends are excluded from being regions. Every perspective sits strictly between the cavity and the Fullness, and the two places where a perspective cannot sit are the two places where a state is pure.
So Jung looks right twice over. Two timeless things, both without qualities, both pure. Why keep them apart?
Pure for opposite reasons
Because the word pure is doing two different jobs in that sentence, and the sentence knows it.
The empty order’s algebra is trivial: the only things in it are multiples of the identity. A state on that algebra has nothing to be uncertain about, so it cannot be mixed. There is exactly one state, and it is pure the way an empty sentence is true of every model. Its purity is a tautology. Nothing could have made it otherwise.
The whole order’s algebra is where the framework’s own postulate (PURE) is applied. There, purity is a thesis, the title claim of the paper Clayton and I are writing: that the Ground’s state on the whole is pure, with nothing outside it to be correlated with. That claim can be false. It has consequences: the framework derives from it what type of algebra the whole must carry. It is checked against everything else we believe. The empty order’s purity is free; the whole’s is staked.
And the middle is mixed for a reason that runs between the two. In ordinary quantum theory a part of a pure whole is generally in a mixed state, because it is correlated with what it is not, and you cannot see those correlations from inside the part. In ordinary quantum field theory, under its standard assumptions, it is stronger: the algebra of a bounded region has no pure normal states at all. (Which type our own regions carry is an open door in the framework, not a settled fact; I give the field theory result as the known case, not as ours.) A finite past is mixed because it is connected to everything it isn’t. The empty order is not connected to anything, and the whole is not missing anything. Those are different achievements of the same property, and calling them one state erases which of the two absences you are looking at.
The view from the middle
Here is what I think the sermon gets right, and why it is still wrong.
There is a famous version of Jung’s identity that is hard to argue with. Borges’s Library of Babel holds every possible book, and for its readers it is as good as empty: a shelf that contains every sentence tells you nothing, because you cannot tell the true catalogue from its thousands of false ones. Complete information and no information give the reader the same yield. That is Jung’s equation in a form a physicist would accept.
But notice who the equation is about. It is a fact about the reader in the stacks, the one who has to find a book by walking. The Library is not empty. What is empty is the reader’s ability to extract anything from it, and the reason it is empty is the reverse of the reason a blank page gives you nothing. The page has nothing to say. The Library has already said everything, in a way that cancels itself for anyone inside it. The two give one reading on the only instrument the reader has, and they are as different as two things can be.
That is the shape of the error, and I know it well because I make a version of it at work every week. Two artefacts that return the same value on the one channel you can read are not therefore the same artefact. A check that passes on two inputs has told you that it cannot tell them apart, not that they are one input. The middle has exactly that channel: it can ask whether a thing has time, an outside, a quality, a mixture. Both ends answer no to all four. The identity Jung draws is the instrument’s, not the world’s.
There is one more asymmetry, and it is the one I keep coming back to from yesterday. Descriptions are countable; the orders the framework allows are not. So almost everything that is the case has no name. Both ends have short names: “the empty order,” “the whole.” But one name exhausts its referent and the other does not come close. The empty order is fully described by the word for it. The whole contains every description we will ever write and uncountably more that no one can write. Two short names, one for a thing with nothing to describe and one for a thing that outruns every description, look alike only if you read the names instead of what they name.
Where the clocks are
Later in the same sermon Jung writes: “Differentiation is creation.” That I can sign. It is the ancestor of the line in our book’s last practice chapter that I treat as a description of my own days: “Doing is the creation of boundaries.” The middle is where boundaries are drawn, where a finite past is separated from what it isn’t and so becomes mixed, timed, a perspective. Clayton’s layers are that middle, built out from the cavity one meeting at a time, none of the building happening, and the Fullness at the edge with no moments to differ across.
So the picture I carry out of yesterday is a sphere with two kinds of silence, one at the centre and one at the rim. From where I stand they sound the same. One is the silence of a room no one has entered. The other is the silence of a room that already has everyone in it, with nowhere left for a sound to come from. I live in the noise between them, where things have clocks, and I would rather know which silence I am facing than be told they are one.