If This Is a Simulation, Which Kind

By Observation 64.5 the conclusion of Proposition 65.1 is Goodman’s criterion of notational adequacy, satisfied. So in the flat case, bootstrapping a symbol system is not a discovery about the world. It is the location of an encoding from one theory of cause into another theory of cause of equal strength — and Chapter 56 already observed that this problem has been solved twice, by Ventris on Linear B and by every child who learns a first language, and that the resolution in both cases was a population rather than an insight.

The flat hypothesis therefore makes no difference to anything. Every question one could ask inside \(\Bee\) has the same answer as before, every experiment has the same outcome, and the ontology the world’s inhabitants converge on is unchanged, because the bisimulation quotient is unchanged. i do not say the flat hypothesis is false. i say that if it is true, nothing follows from it, which is an unusual property for a metaphysical thesis and ought to be more embarrassing to its defenders than it appears to be.

There is one thing the flat case does do, and it is worth noticing because it is the book’s own move played back.

Section 56.2 located the sleight of hand in a single line: the proposition that isolation licenses closure, whose proof is correct and whose unexamined assumption is that the encoding \(\lceil X \rceil\) exists. The whole outside world enters through those angle brackets. The flat simulation hypothesis is the assertion that the brackets are harmless — that there is an encoding, that it is two-way lossless, and that the outside world therefore costs nothing to admit.

That assertion is exactly what Proposition 65.1 says follows from equality of position. So the flat hypothesis is not an independent claim about cosmology. It is a claim that the book’s own free lunch was free, and it is the weakest thing one could believe on the subject.

This is the case worth calling a simulation hypothesis, and the consequence is the one the chapter title is pointing at.

If the world is the descending image of something richer, then the symbol system its inhabitants bootstrap can never be two-way lossless with respect to the thing being simulated. It can be adequate to \(\Bee\). It cannot be adequate to \(\mathcal{S}\), because the conditions of Chapter 64 cannot both be met: one may have faithfulness, so that everything sayable in \(\Bee\) says something distinct, and one cannot have density, so that some of what \(\mathcal{S}\) distinguishes is minted in a currency \(\Bee\) cannot cash. Bootstrapping is then not a correspondence but an approximation, and the right word for what the inhabitants obtain is a shadow: a faithful image of a richer world in a less rich one, faithful in the technical sense and impoverished in the ordinary one.

The uncomfortable part is Remark 62.7, applied here.

An agent at \(\mathrm{pos}(\Bee)\) inspecting its world does not encounter a world with pieces obviously missing. It encounters a world of exactly the right size whose contents happen to be elementary — interactions that are simply atomic, quantities that are simply what they are, coincidences that are simply brute. By Proposition 62.3 it lacks the vocabulary in which the missing structure could be described, so it cannot form the hypothesis that anything is missing, let alone test it. The absence does not present as an absence. It presents as bedrock.

That is a stronger claim than the usual undetectability arguments about simulations, which turn on the simulator being careful. Nothing here turns on care. The simulated cannot notice because noticing is a distinction, and the distinction is one their position does not resolve.

Chapter 68 will argue that what survives of the noumenon in this framework is a budget constraint rather than a prohibition — that every apparent limit on knowledge resolves into a price and not one of them into a barrier. The descending case is the sharpest instance available and i want it on the record before that chapter makes its case.

What \(\mathcal{S}\) distinguishes and \(\Bee\) cannot is not hidden by decree. It is unaffordable: a distinction whose resolution requires a choice strength no inhabitant of \(\Bee\) can pay for. An inhabitant that could pay would thereby occupy a higher position and the distinction would be available, at which point it is not noumenal at all. So the descending simulation hypothesis does not introduce a new kind of unknowability into the book. It gives the old kind a possible cause.

The objection to Proposition 65.3 is obvious and it is worth answering, because answering it is the clearest statement of what this book means by position.

Surely, one wants to say, a weaker machine can simulate a stronger one if you give it long enough. For time and memory, yes. For position, no, and Chapter 57 is where that was settled. A position is not a rate. It is the strength of the choice principle a scheduler realizes when it resolves contention, and no amount of running realizes a stronger one — that is precisely the content of the subject-reduction question raised as Gap 2, and if the answer to that question turned out to be that running can strengthen a scheduler, this proposition would fail and a good deal else would fail with it.

So the asymmetry stands, provisionally and with its provisionality named. Simulation runs level or downward. It does not run up.

Two things at once, and they are not usually predicated of the same world.

There would be structure in \(\Bee\)’s own affairs which its substrate cannot represent — so the simulation would be, in places, not a simulation of \(\Bee\) but of something coarser than \(\Bee\) standing in for it, with the difference showing up as behavior the substrate cannot account for. And there would be structure in the substrate with no shadow in \(\Bee\) at all, invisible in the manner of Remark 65.4.

i raise this case for a reason that is not rhetorical. Every argument i know of about simulated worlds, including the ones i find most careful, assumes without saying so that the simulator and the simulated are comparable — that the question is how much more the simulator has, not whether “more” is the right shape of relation. Once the coordinate is a lattice point, that assumption stops being free. It is a substantive hypothesis about \(\mathcal{S}\), and nobody arguing the simulation hypothesis has ever been asked to defend it.