Chapter 27
The Learner’s World Already Has a Physics
The learner of the preceding part is not a mind contemplating a world. It is a term sitting in parallel with the terms it wants to know about, made of the same material, subject to the same rules, paying the same prices. That was the point of building it that way. But it has a consequence which has so far gone unremarked, and this part is about the consequence.
If the learner is a term in a cost-accounted theory, then it is already inside something that deserves to be called a physics. It has states, and laws that take one state to another, and a currency it must spend to move. Nobody put a physics there. The physics is what the theory looks like from inside, and every cost-accounted interactive theory has one.
The question this part asks is therefore not “how do we get a physics out of computation” — that was the question the earlier arrangement of this material asked, and it presupposed a knower standing outside, building. The question is: which physics, and why that one? A learner opening its eyes in a \(\catciGSLT\) finds itself in a world with some structure and not other structure, and the structure it finds depends on which axioms the theory happens to satisfy. Most objects of \(\catciGSLT\) satisfy very few of them. The world humanity actually turns out to live in satisfies a surprisingly long list, and each item on that list cuts the category down to a subcategory.
27.1 Physics as a sequence of subcategories
The organizing device of this part is a ladder. At the top sits \(\catciGSLT\), containing every cost-accounted interactive theory whatever. Each chapter below imposes one further axiom, and each axiom names a full subcategory whose objects are the theories satisfying it. One chapter is not a rung: the last of them spends the ladder rather than extending it, returning to the learner of Part Part III to say what holding a level costs in a world of this kind. A learner in a theory low on the ladder inhabits a world that looks increasingly like the one physicists describe; a learner high on the ladder inhabits a world in which most of what physics says is simply false — not because the physics is wrong, but because its hypotheses do not hold there.
{1.3}
| Axiom imposed | What the learner then has | Ch. |
|---|---|---|
| none | states, laws, a budget | — |
| every rewrite has an inverse | a time-symmetric world, with the arrow of time in the boundary condition rather than the laws | 28 |
| histories are recorded | a causal order, and a proper time | 29 |
| rewrites carry a semiring value | a decoration — the weighting of Chapter 13 with its codomain chosen: nondeterminism, cost, probability or amplitude, according to the semiring | 30 |
| conversion is arbitrage-free | a single conserved quantity, and a first law | 31 |
| the decoration is internalized | dynamics as a fact in the logic, not merely about it | 32 |
| values are complex | interference, and an obstruction | 33 |
Two things should be said about this table before it is unpacked.
The first is that the axioms are genuinely independent in the sense that matters: there are theories satisfying any given one and failing the next, and Chapter 34 exhibits some. The ladder is not a chain of definitions dressed up as discoveries.
The second is that the ladder does not descend all the way to our physics, and this part does not claim that it does. It descends some distance and then stops at an obstruction which Chapter 33 states precisely: the complex instance gives amplitudes and gives interference, and it does not by itself give the Born rule. What is offered is a family of possible physics with our own somewhere inside it, not a derivation of ours.
Chapter 58, in the Prestige, describes a different ladder — a tower of oracle extensions, in which each position extends the ones below it and the physics available at a higher position is strictly richer. It is easy to confuse the two, and they are perpendicular. The ladder of this part moves down within a fixed computational power, adding axioms and thereby shrinking the class of theories under consideration. The ladder of Chapter 58 moves up, adding computational power and thereby enlarging what any given theory can express. A theory can be low on this ladder and low on that one — a very classical world inhabited by very limited agents — or high on both. The two coordinates are independent, and several confusions about what computation can and cannot ground come from collapsing them into one.
27.2 What the earlier arrangement got wrong
It is worth recording, since some readers will have seen an earlier form of this material.
That presentation carried two maps rather than one, called them a Lagrangian and a Hamiltonian, and claimed that everything else followed by construction. The first was an unnecessary duplication: one decoration into a semiring suffices, and the choice of semiring does the work the two maps were introduced to do. The second was a misnaming. The third was an overstatement, and the most interesting of the three errors, because the parts that do not follow by construction turn out to be exactly the parts that carry content. Conservation holds only under a condition; that condition has a cohomological measure; and the identification of the conserved charge with the generator of time evolution remains a conjecture. The table below is annotated accordingly, and the annotations should be taken seriously.
27.3 The Rosetta Stone
The following previews the correspondence developed in this part. The third column records the status of each row: con marks a construction, true by definition of the objects involved; thm marks a theorem with a hypothesis that can fail; cnj marks a conjecture we have not proved. A fully annotated version appears in Chapter 34.
{1.35}
| GSLT | Physics | Status |
|---|---|---|
| Term | State / initial condition | con |
| Rewrite rule | Law of motion | con |
| Rewrite path | Trajectory | con |
| Synchronization tree | Causal graph | con |
| Minimum path length | Proper time | con |
| Equations \(\eqs\) | Gauge equivalence | con |
| Reversible envelope \(\GSLT^\dagger\) | Time-symmetric theory | con |
| Decoration, tropical | Action functional; least action | thm |
| Decoration, complex | Path amplitude \(e^{iS/\hbar}\) | con |
| Sum over paths | Feynman path integral | con |
| Interference of paths | Quantum interference | cnj |
| No-arbitrage conversion | Existence of an energy function | thm |
| Virtual token \(\nu\) | Energy, as a Noether charge | thm |
| Holonomy of the energy form | Failure of energy to be a state function | thm |
| Dissipation \(\theta\) | Second law; free energy | thm |
| Ledger law \(\sigma+\kappa=\sigma_0\) | Potential plus kinetic | thm |
| Erasure bounded by holonomy | Landauer’s principle | thm |
| Located authority | Entanglement of space and time | cnj |
| \(\nu\) generates reduction | Hamiltonian, generator of evolution | cnj |
| Symmetry of the decoration | Noether current | cnj |
The reader who has come through Part Part II will notice that several rows of this table have already been met wearing other clothes. The no-arbitrage row is Theorem 15.1, introduced in Part Part I as a fact about exchange rates and about to be read as a fact about energy. The ledger law is the conservation identity the mortal scientist lives under: what it has not yet spent, plus what it has spent, is what it started with. The second law is why a scientist that is merely right can still starve. This is neither coincidence nor analogy. It is the same accounting, and the reason the learner had to be built first is that the accounting is easier to believe once one has watched something live under it.
27.4 Organization of this part
Chapter 28 constructs the time-symmetric envelope \(\GSLT^\dagger\) and locates the arrow of time in the boundary condition. Chapter 29 identifies the synchronization trees with causal graphs in the sense of Bombelli, Lee and Sorkin, and reads a proper time off the minimum path length. Chapter 30 takes the weighting endofunctor of Chapter 13 — one map into a semiring — and asks what the choice of semiring delivers, locating the least-action principle in the tropical instance. Chapter 31 asks what is conserved and at what price. It is the longest chapter in the part, and it contains the first law, the second law, the ledger law and the Landauer bound; it also re-derives the virtual token of Chapter 15 in its physical reading, which is the one place in the book where a single theorem is deliberately proved twice. Chapter 32 extends the modal operators to carry the full dynamical state of a transition, so that the dynamics becomes a fact stated in the logic rather than merely about it. Chapter 33 lifts the decoration to complex amplitudes, states the obstruction honestly — there are two of them, and only one is about bisimulation — and finds that a presentation has a second place to put coherence, namely the equations it declines to quotient. Chapter 34 collects the ladder into one picture. Chapter 35 then spends it on the previous part: a learner holding a setpoint against friction is paying a bill in the currency Chapter 31 priced, and the chapter says what the bill comes to and who it is charged to. Chapter 36 says what this part does not establish.