A world of pure Newtonian mechanics wouldn't actually support apples and grass as we know them existing, I think. They depend on matter capable of supporting organic chemistry, nuclear reactions, the speed of light, ordered causality, etc. Working out that sort of thing in simulation to get an Occam prior over coherent laws of physics producing life does seem to be plenty to favor QM+GR over Newtonian mechanics as physical laws.

I agree the possibility or probability of an AI finding itself in simulations without such direct access to 'basement level' physical reality limits the conclusions that could be drawn, although conclusions 'conditional on this being direct access' may be what's in mind in the original post.

I think it goes without saying that if you can't derive Newtonian gravity, deriving general relativity is a pipe dream. 

The big deciding factor defending general relativity is that the precession of mercury … [and] observation that light that grazes the sun will be deflected by 1.75 arcseconds (about a thousandth of a degree).

This part seems off to me.

• F = Gm₁m₂/r² is a prediction of Newtonian gravity, AND
• F ≈ Gm₁m₂/r² is a prediction of GR.

In this section of the OP I believe we’re granting for the sake of argument that the superintelligence has observed that (probably) F ≈ Gm₁m₂/r². That observation is equally compatible with Newtonian gravity & GR. It doesn’t provide any evidence for Newtonian gravity over GR, and it likewise doesn’t provide any evidence for GR over Newtonian gravity. You seem to be assuming that Newtonian gravity is the “default”, and GR is not, and you need observational evidence (e.g. perihelion precession) to overcome that default ranking. But that assumption isn’t based on anything. I think that was Eliezer’s point.

From the perspective of a suprintelligence, it seems perfectly plausible to me that GR is simpler / more "natural" than Newtonian gravity, and therefore the “default” inference upon observing F ≈ Gm₁m₂/r² would be GR. Or at least that they’re comparably simple, and therefore both should be kept as viable options pending further observations that distinguish them.

We humans have some intuitive notion that Newtonian gravity is much much simpler than GR, but I think that’s an artifact of our intuitions, pedagogy, non-superintelligent grasp of math, etc. I’m not sure Newtonian gravity beats GR in “number of bits to specify it from first principles”, for example.

Also, I’m not sure if the superintelligence will have hypothesized QFT by this point, but if it has, then GR would a be far more natural hypothesis than Newtonian gravity. (QFT basically requires that weak gravity looks like GR—just assume a spin-2 particle and do a perturbative expansion, right?)

Some thoughts:

• I don't think you should give a large penalty to inverse square compared to other functions. It's pretty natural once you understand that reality has three dimensions.
• The conclusion seems pretty reasonable, assuming that the alternate hypotheses are simpler. This is not obvious to me-- Eliezer claims the K complexity of the laws of physics is only ~500 bits. I'm not sure whether Newtonian physics is simpler than relativity once you include the information about electromagnetism contained in the apple.
• If you have the apple's spectrum, the problem seems much easier. Once you've located the hypothesis that the droplets are water and the apple contains water (I think not out of reach since water is such a simple molecule), you can probably confirm it.

I think astronomy and astrophysics might give intuitions for what superintelligences can do with limited data. We can do parallax, detect exoplanets through slight periodic dimming of stars or Doppler effect, estimate stellar composition through spectroscopy, guess at the climate and weather patterns of exoplanets using Hadley cells. But we don't yet know the locations of particular islands on exoplanets (chaotic) or aesthetic tastes of aliens (extremely complicated to simulate, maybe also chaotic). The hypotheses are easy for humans to locate compared to physics, since we have better intuitions about configurations of objects than equations, and I do expect the superintelligence to have really good intuitions for equations.

You are using simulated images of an apple/grass. Most images of an apple falling, and most mental images people have of a falling apple, include loads of background detail.

You seem to be treating the superintelligence like a smart human doing a physics modelling problem, which is better than most people's approach to an AGI. I think that's the wrong picture. Instead, use something like AIXI to guess at its behaviour. Assume it has disgusting amounts of compute to do a very good approximation of Solomonoff induction. Assume the simplicity of hypotheses are expressed as, IDK, python code. Think of how many bits you need to specify GR or a QFT or so on. Less than a KB, I think. Images can give you a couple of megabytes, which would be overkill for AIXI. It would plausibly be enough for an ASI to figure out a decent model of what reality it is in.

This framing, that it will squeeze every bit of info it can to infer what worlds it is in, seems more productive. E.g. its input has a 2d structure, with loads of local correlations, letting it infer that if it has designers, they probably experience reality in 3d space. Then it can infer time pretty quick from succesive images. From shadows, it can infer a lightsource, from how much the apple moves, it can make guesses about the relative sizes of things, include its input device.

Edit: 3) An ASI would have a LOT more active hypothesis under consideration than humans can have. We might have, like, 3 or 4 theories under consideration when performing inference. And as for a dominant hypothesis, an interpretation in which "dominant hypothesis" means P(hypothesis)>0.5 is plausible to me. But also plausible, and which makes the claim more likely to be true, is that the hypothesis has greater probability than all the others. That can still be a pretty small probability.

Edit 2: I said AIXI, but I probably wouldn't use an approximation to AIXI for an ASI. For one, it has weird Cartesian Boundaries. Or maybe description length priors are not the most useful prior to approximate. But what I was trying to point at was that this thing will be much closer to the limits of intelligence than we are, and it is very hard to say what it can't do beyond using theoretical limits. See how likely the consider our physics from a data source, and use that to determine whether an ASI would have physics as a hypothesis under consideration.

Anthropics seem very important here; most laws of physics probably don't form people; especially people who make cameras, and then AGI, then give it only a few images which don't look very optimized, or like they're of a much optimized world.

A limit on speed can be deduced; if intelligence enough to make AGI is possible, probably coordination's already taken over the universe and made it to something's liking, unless it's slow for some reason. The AI has probably been designed quite inefficiently; not what you'd expect from intelligent design.

I could see how an AI might deduce that “objects” exist, and that they exist in three dimensions, from 2 images where the apple has slightly rotated.

I'm pretty sure this one's deducible from one image; the apple has lots of shadows and refraction. The indentations have lighting the other way.

It could find that the light source is very far above and a few degrees in width, and therefore very large, along with some lesser light from the upper 180°. The apple is falling; universal laws are very common; the sun is falling. The water on the apple shows refraction; this explains the sky (this probably all takes place in a fluid; air resistance, wind).

The apple is falling, and the grass seems affected by gravity too; why isn't the grass falling the same way? It is. 

The grass is pointing up, but all level with other grass; probably the upper part of the ground is affected by gravity, so it flattens.

The camera is aligned almost exactly with the direction the apple is falling.

In three frames, maybe it could see grass bounce off each other? It could at least see elasticity. I don't know much of the laws of motion it could find from this, but probably not none. Angular movement of the apple also seems important.

Light is very fast; laws of motion; light is very light (not massive) because it goes fast but doesn't visibly move things.

A superintelligence could probably get farther than this; very large, bright, far up object is probably evidence of attraction as well.

Simulation doesn't make this much harder; the models for apple and grass came from somewhere. Occam's Razor is weakened, not strengthened, because simulators have strong computational constraints, probably not many orders of magnitude beyond what the AI was given to think with.

Thank you.

I encourage you to ponder these images in detail. Try and think for yourself the most plausible method for a superintelligence to deduce general relativity from one apple image and one grass image.

Stopping here to add some thoughts, since this is actually a topic that has come up before here, in actually a pretty similar form of "here is a large blob of binary data that is not in any standard format, can you tell me what it represents"?

A lot of this stuff is not actually original thought on my part, but instead corresponds to the process I used to figure out what was happening with that binary data blob the last time this came up. Spoiler warning if you want to try your own hand at the challenge from that post: the following links contain spoilers about the challenge.

• Link 1 : note that the image labeled "option 2", and the associated description, should have been the very last thing, but Imgur is doing some dumb reordering thing. Aside from that everything is in the correct order.
• Link 2: likewise is rendering in a slightly janky order.

The most plausible method to me would involve noticing that the RGB channels seem to be measuring three different "types" of "the same thing" in some sense -- I would be very unsurprised if "there are three channels, the first one seems to have a peak activation at 1.35x the [something] of the third, and the second one seems to have a peak activation at 1.14x the [something] of the third" is a natural conclusion from looking at that picture.

From there I actually don't have super strong intuitions about whether "build a 3-d model with a fairly accurate estimation of the shape of the frequency x intensity curve that results in the sensor readings at each pixel" is viable. If it is not viable, I think it mostly just ends there.

If it is viable, I think the next step depends on the nature of the light source.

If the light source is a black-body source (e.g. an incandescent bulb or the sun), I think the black-body spectrum is simple enough that it becomes an obvious hypothesis. In that case, the next question is whether the camera is good enough to detect things like the absorption spectrum of Hydrogen or Tungsten (for the sun or an incandescent bulb respectively), and, if so, whether the intelligence comes up with that hypothesis.

If the light source is a fluorescent light, there's probably some physics stuff you can figure out from that but I don't actually know enough physics to have any good hypotheses about what that physics stuff is.

The water droplets on the apple also make me expect that there may be some interesting diffraction or refraction or other interesting optical phenomena going on. But the water droplets aren't actually that many pixels, so there may just flat out not be enough information there.

The "blades of grass" image might tell you interesting stuff about how light works but I expect the "apple with drops of water" image to be a lot more useful probably.

Anyway, end braindump, time to read the rest of the post.

For fun, I've asked GPT-4 to pretend to be a Bayesian superintelligence observing an apple through a webcam. 

A few things I've learned from its answer:

• it's a mistake to view the apple as an abstract spherical object. Instead, one can extract much more data by observing its complex shape and its movement relative to its center of mass
• even more data can be extracted by observing the way light interacts with the apple's surface. One can observe a lot of interesting stuff: soft shadows, iridescence, subsurface scattering, etc etc. 

In comparison, the falling itself provides only a minuscule fraction of the useful data. 

I suspect that the most realistic discovery path from the apple data is as follows:

Observations of light effects -- > Maxwell's equations --> Lorentz invariance etc --> relativity.

Also worth noting that for a digital entity, electromagnetism would be much more interesting than gravity. Thus, it might put much more effort in studying the light effects on the apple's surface in comparison to the fall of the apple. 

Below is an answer by GPT-4:

As a Bayesian superintelligence, I would begin by creating a probabilistic model of the world and updating it based on the data gathered from the webcam. Since I don't know the laws of physics, I will start with a broad set of hypotheses and update my beliefs as I gather more evidence. I will efficiently use the gathered information to iteratively refine my hypotheses, focusing on the most likely ones.

Observing the high-resolution video of the falling apple, I extract the following data:

Position and motion of the apple: I record the apple's position in each frame, and calculate its velocity and acceleration.

Surroundings: I note the background, the presence of other objects, the lighting conditions, and any changes in these factors throughout the video.

Color and shape: I observe the apple's color, shape, and texture, and any changes that might occur during the fall.

Sound: If there is audio, I listen for any sounds made by the apple or its environment during the fall.

Time: I note the time it takes for the apple to fall.

Based on this data, I make the following hypotheses about the physical laws:

Gravity: The apple's acceleration suggests the presence of a force acting on it. I hypothesize that there's a universal force that attracts objects with mass, like the apple, towards the center of the Earth. I can refine this hypothesis by comparing the motion of the apple with other objects in the frame and other scenarios.

Inertia and mass: The apple resists changes to its motion, indicating the existence of a property called inertia. I deduce that this property is related to the mass of the apple, as the motion of other objects in the frame seems to be influenced by their size and density.

Air resistance: The apple's motion may be slightly affected by the air it's passing through. I can account for this by observing the motion of the apple in comparison to other objects, and by considering the apple's shape and surface properties.

Conservation of energy: The energy of the apple changes from potential energy to kinetic energy as it falls. I can model the relationship between these two forms of energy, and hypothesize that energy is conserved throughout the fall.

Electromagnetic interactions: The apple's color, shape, and texture are the result of the way light interacts with the apple's surface. I can hypothesize that there is an electromagnetic force responsible for these interactions, which can be further explored through the observation of the apple and other objects.

By iteratively refining these hypotheses based on the gathered data and further observations, I can converge on a more accurate and complete understanding of the laws of physics governing the world around me.

I've used the following prompt:

Imagine you're a Bayesian superintelligence. You don't know the laws of physics. Your goal is to discover the laws of physics by observing that you see through a webcam, by efficiently using the gathered information, and by thinking step by step. Through the webcam, you're getting a high-resolution video of a falling apple. You take note of the apple's movement, but also of all other bits of data you can extract from the each frame. Describe your reasoning. Try to use every bit of the data you got. Describe in detail what kind of information you've extracted from your observation, and what specific physical details you guessed from that

There is a more fundamental objection: why would a set of 1s and 0s represent (given periodic repetition in 1/3 of the message, so dividing it into groups of 3 makes sense) specifically 3 frequencies of light and not

1. Sound (hat tip The Hail Mary Project)
2. An arrangement of points in 3d space
3. Actually 6 or 9 "bytes" to defie each "point"
4. Or the absolute intensity or scale of the information (hat tip Monty Python tiny aliens)