Under Construction

Paradox and Emergence

How wonderful that we have met with a paradox - Niels Bohr - Copenhagen, c1926

This is the engine room. Everything so far - the something wrung out of nothing, the number line building itself, the whole world turning out to be facts - has run on a single mechanism, and it is time to look at it directly. The mechanism is paradox. It is also, on this account, the very same thing as emergence, which is a claim worth slowing down for.

Index

Why Anything Changes At All

When Inside and Outside Cannot Agree

Bigger on the Inside

When a Paradox Has Teeth

The Gentler Twin

Two Engines, One World

Why Anything Changes At All

Start with a puzzle the last page sets up without meaning to. Facts, we said, are eternal: they do not come and go, they only ever accumulate. But a world made entirely of eternal facts ought to sit perfectly still. Nothing in a fact wants to move. So where does change come from? What stirs an unchanging substrate into a universe that actually does things?

The answer is that some arrangements of facts cannot sit still, because they are at war with themselves. A situation can be built in which two facts are each perfectly true and yet cannot both be accommodated in the room available. That is a paradox, and a paradox is the one thing the world cannot simply leave alone.

When Inside and Outside Cannot Agree

Paradox has a precise meaning here, and it is worth stating carefully. Every existing thing has an inside and an outside: internal facts about how its own parts stand together, and external facts about how it stands among everything else. Usually these agree, or can be made to agree with a little rearranging. A paradox is what you have when they cannot - when the inside and the outside are both definite, both true, and flatly incompatible within the dimensions there are, and no shuffling of what already exists will reconcile them.

Something then has to give, and there is only one thing that can. A new direction opens - a dimension that did not exist a moment before - and the clash is absorbed into it. Both the inside and the outside can now be satisfied at once, because the thing has somewhere new to put the difference. That opening of a genuinely new direction is what Virtualism calls strong emergence: the arrival of something that could not have been predicted from, nor reduced to, anything that came before.

Paradox is emergence. The cause and effect are one.

Bigger on the Inside

An example makes this far less abstract, and it happens to be a real one. Take any object with mass. Its internal energy, gathered up, amounts to more than its external relationships can comfortably hold in plain three-dimensional space - its insides ask for more room than its outside appears to have. It is, in the diary's happy phrase, something like a Tardis: bigger on the inside than on the out.

The technical detail

The Tardis image catches something real - that a massive object holds more than its three-dimensional footprint advertises - but in strict spatial terms it has the sign backwards, and a careful reader is right to notice. Boost is a rotation of real extent into the imaginary dimension, and the more boosted an object is the smaller its real three-dimensional size becomes, not the larger. A heavily boosted object takes up less ordinary space than the equivalent unboosted, Euclidean region, not more; higher-dimensional objects are smaller than their 3D components, and sit inside them. So the surplus is not roomier 3D space concealed within a small shell. It is depth along a direction at right angles to all three, bought precisely by the 3D shell contracting and folding the difference away into the imaginary. The object is bigger on the inside only in the sense that it carries more dimension and more energy than its shrunken real size lets on - which is, spatially, the exact opposite of a literal Tardis.

Both facts hold - the crowded inside and the flat outside - and they cannot be squared in three dimensions. So a fourth opens to take the strain, and mass settles into it. This is not a metaphor laid over the physics; on this view it is what the physics is. The extra dimension that relativity needs in order to describe gravity and mass is exactly the new direction a paradox is forced to open. General Relativity gets the description right; Virtualism supplies the reason there is a fourth dimension there to be described.

When a Paradox Has Teeth

Not every paradox builds anything. For one to be genuinely creative - to force a new dimension into being rather than sit there as a curiosity - three things have to be true of it, and the theory is strict about all three.

First, it must be actual: a paradox about the world, not merely about words. The famous sentence that declares itself false is a fine puzzle, but it speaks only of itself; it has no outside to be in conflict with, and a clash needs two sides. Self-reference spins in place and produces nothing. Second, it must be unpreventable: there must be no escape by rearranging what already exists. A difficulty the world can wriggle out of, it wriggles out of; only a paradox with nowhere left to go is forced to open new ground. Third, it must be definite on both sides: the inside and the outside must each carry a settled value, not a vague or open one. That last condition is the very line between a paradox and its gentler twin, which we come to next.

It is worth being clear that a paradox, in this sense, is not a contradiction. A contradiction is a sign that you have gone wrong somewhere - a false step, a bad assumption, a muddle to be tidied up. A paradox is a sign that the world is doing something. Both sides genuinely hold; the tension is real; and the resolution is not a correction but a creation.

The Gentler Twin

Paradox has a quieter counterpart, and the theory borrows a name for it from recent work in logic: hypodox. Where a paradox is over-determined - too much insisted upon, two definite things that cannot both fit - a hypodox is under-determined: the inside and the outside are perfectly compatible, but between them they do not pin things down. There is slack. Several arrangements would all serve, and nothing yet settles which.

A hypodox forces nothing. What it does instead is permit. Within the room that already exists, one of the allowable arrangements comes about and the others quietly lapse. No new dimension is raised; what changes is the pattern, not the kind. This is weak emergence - the gentler engine we met in passing back at the number four - and it is also where probability comes from, since a hypodox is precisely a situation with several permitted outcomes and nothing compelling one. It is, as well, where choice and freedom have their roots; but following that thread leads up into a later tier of the theory, and is left for there.

Two Engines, One World

So the engine of becoming has two faces, and they divide the labour cleanly. Paradox builds: forced and necessary, it raises new floors that were not there before. Hypodox furnishes: free within the room available, it settles which of several arrangements the existing floor will hold. One manufactures structure; the other moves things around inside it.

Between them they account for something the usual pictures struggle to hold together at once: that the world is both strictly lawful and genuinely chancy. The laws are paradox doing its forced, inevitable work. The chance is hypodox leaving real room for more than one outcome. A universe with only the first would be frozen clockwork; with only the second, a formless drift. With both, you get exactly the world we have - rule-bound and surprising in the same breath. The next page takes these two engines and shows the wheel they turn: the four stages of the Mandala.