Undusting Dust Theory

Could sand save us from heat death?

Warning: Major spoilers for Egan’s novel ‘Permutation City’ ahead.

In the set of all natural numbers (1, 2, 3…), how often should we expect to see an odd number?

Why Dust Theory Matters

Earlier this year, I felt I had hit an intellectual wall - it seemed to me that I had become familiarized with the majority of outstanding fundamental problems in the philosophy of science. Most of what I was reading felt like a restating of ideas I had read elsewhere, and I started to grow frustrated with the lack of novelty.

To switch things up, I decided - after a 5-year hiatus - to give fiction another try. And what better place to start than with the hardest of the hard in the sci-fi genre: Greg Egan - a mysterious Australian author with an online presence limited to an archaic website and his Twitter account: no pictures, no video interviews, no podcasts. No one knows who’s behind the pen.

Giving Egan’s work a try was absolutely the right move - after reading his groundbreaking novel Permutation City, I haven’t been able to stop thinking about the profound philosophical problems that stem from the book’s crowning achievement: Dust Theory.

Admittedly, explaining Dust Theory is no easy task - the novel assumes a lot of background knowledge from readers (which is why I highly recommend reading David Deutsch’s The Fabric of Reality first), but for the purposes of this section, it suffices to say that the Theory matters because it seems to provide a pathway for humanity to escape the eschatological implications of modern physics - all while remaining perfectly compatible with our best scientific explanations.

Of course, whether this is something desirable or repulsive will fully depend on the reader’s philosophical and ethical inclinations. What cannot be denied, however, is the potentially profound reach of the theory’s assertions. Hence, in today’s (rather long) entry, I’d like to share with readers an overview of what Dust Theory states, why its creator objects to it, why I feel his objection does not work, and what I ultimately think about the legitimacy of the Theory.

Building-Up Dust Theory

I will start by building-up the central claim of Dust Theory in seven steps. Please note that the version I’m presenting here is not identical to the one Egan advances in his novel. My version is more explicit in its assumptions and has been modified in various ways for ease of explanation and alignment with the latest scientific discoveries - especially for steps 4 - 6. Hence, any error is my own.

1. First, assume that mental states (i.e., minds) can be simulated by simulating their corresponding brain states. That brain states could be simulated is uncontroversial, as brains are made of matter, and all known matter behaves in a simulation-friendly manner. To complete the first step, one only needs to add the assumption of mind-brain supervenience; that is, the idea that mental states cannot be divorced from the physical states of the brain: whenever a brain state is instantiated, so is its corresponding mental state.

2. Next, consider that for simulatees, instantiating a sequence of their brain states in order or disorder makes no difference. This is to say that if we manage to perfectly simulate ten seconds worth of Cristiano Ronaldo brain states, from the POV of the simulated Ronaldo, it will make absolutely no difference whether these brain states are instantiated in chronological order or not. Why? Because every single one of Ronaldo’s brain states carries in it the ‘memories’ of the previous states. Thus, Ronaldo’s mind at t = 5 will tell him that he has already gone through the experiences corresponding to the first 5 seconds (even if the first brain state that the computer instantiates is the one at t = 5).

3. Moreover, consider that for simulatees, instantiating a sequence of their brain states on a single piece of hardware, or across various devices would make no difference either. Going back to the Ronaldo example, this means that from the POV of simulated Ronaldo, it won’t make a difference whether his brain states are all computed on the same device, or if part of the computations take place on a different piece of equipment - as long as all the brain states are instantiated, he will feel no difference.

4. Next, assume that never-ending environments that I) contain people (i.e., minds) and II) are simulatable on Universal Turing Machines can be described mathematically. Doing so would require specifying a theoretical environment governed by a set of rules that allow brains to be instantiated, an infinite number of experiences to occur, and infinitely many memories to be stored. Additionally, the environment would have to be theoretically simulatable on a Universal Turing Machine (UTM); that is, a computer with unbounded energy and memory, capable of simulating any and all motions of any and all physical objects (note that computers like the ones we use today are approximations to UTMs, with the only difference being finite memory and energy). The never-ending nature of such an environment contradicts no known law of physics, given that it is merely a mathematical description - one can supply the environment with infinite energy or exempt it from any energy requirement by fiat. Physically instantiating such an environment, however, would require an infinite amount of structure to actually be available in our universe.

5. Expectedly, the next step is to view the universe as indeed containing sufficient structure (‘stuff’) to instantiate an infinity of UTM states. One way to defend the view that the universe contains infinite structure comes from the field of physical cosmology. According to our current best explanations, the universe is spatially flat (in terms of curvature). If we add to this the widely adopted assumption of a simply connected topology, and pair it with the cosmological principle, we arrive at the conclusion that the universe is spatially infinite, with non-zero average matter density throughout it (i.e., infinite matter). This in turn allows an infinity of states of Universal Turing Machines to be physically instantiated, as each such state can always be mapped onto some of the infinitely available matter.

6. Dust Theory then crucially asserts that all mathematical abstractions compatible with physical reality are instantiated in it. This is equivalent to saying that the pyramids in Giza instantiate abstract pyramids, soccer balls instantiate abstract spheres, and - perhaps more controversially - a block of marble simultaneously instantiates all the statues that could be carved from it.

7. With the previous steps in mind, the central claim of the Theory can be stated: Never-ending virtual environments containing people can be instantiated in physical reality. To clearly see how, let me walk you through how it would work:

   • First, using an approximate Universal Turing Machine (e.g., a MacBook Pro), we write a program that simulates another computer within itself - we’ll call this the Virtual MacBook. This Virtual MacBook will be programed to require no energy source at the simulation level (so the computations will happen in much the same way a virtual chess piece moves: without drawing from any in-game power supply) and its memory will double at each computational step. This means that our Virtual MacBook has no power or memory constraints, and is thus set up to behave as an actual Universal Turing Machine (like the one described on Step 4), rather than an approximation to one.

   • Next, we start running a program of a never-ending virtual environment containing people (like the one from Step 4) on the Virtual MacBook. Note that the Virtual MacBook cannot continue running on the MacBook Pro indefinitely, as the latter will eventually run out of resources, given its finite memory storage (and entropy).

   • However, as mentioned above on Step 3, from the POV of the people on the never-ending virtual environment, it makes no difference whether they are simulated on the same hardware or across various devices. Thus, once the environment can no longer run on the Virtual MacBook running on the MacBook Pro, it will find itself running on some other hardware (e.g., on a Virtual MacBook whose states are instantiated on grains of sand on the beach, on dust accumulated on a shelf, or some other collection of matter).

   • How do we know the above will happen? Because the subsequent computational states of the Virtual MacBook running said environment correspond to a mathematical abstraction; and, given that I) reality contains infinite structure (as asserted in Step 5), and II) all mathematical abstractions compatible with physical reality are instantiated in it (as suggested in Step 6), there will always be some hardware (i.e., ‘stuff’) onto which the Virtual MacBook’s next state will be mapped. This, in turn, will allow the people on the never-ending virtual environment running on it to always find the subsequent instants of the simulation (and thus themselves) instantiated somewhere.

   • What about time constraints? Even if there is always more structure onto which the Virtual MacBook will find its next computational step mapped, surely, we’ll eventually run out of time, right? Currently, the eschatological scenario implied by our best cosmological theories is the heat death of the universe, where the universe expands forever - implying we won’t ‘run out of time’. However, even if the allegedly refuted big crunch scenario (where the universe ends by contracting and eventually collapsing) made a comeback, time would not be an issue. As stated on Step 2, from the POV of the simulatee, the chronological order of the instantiation does not matter - meaning all the states could be simultaneously simulated, doing away with any worry about time constraints.

What is Egan’s Issue with Dust Theory?

To summarize the previous section, Dust Theory holds that it is possible to physically instantiate never-ending virtual environments containing people. Why? Because I) it’s possible to mathematically describe the states of computers running such environments, II) there is enough structure in the universe to make such computational states compatible with physical reality, and III) all mathematical abstractions compatible with physical reality are instantiated in it.

While the argument might have some philosophically questionable premises, from the scientific/empirical point of view, it seems to be fairly unproblematic, as it violates no known laws of nature. Readers might thus be surprised to learn that Egan’s primary objection against Dust Theory is empirical. Quoting him directly, his objection reads as follows:

“I think the universe we live in provides strong empirical evidence against the “pure” Dust Theory, because it is far too orderly and obeys far simpler and more homogeneous physical laws than it would need to, merely in order to contain observers with an enduring sense of their own existence. If every arrangement of the dust that contained such observers was realised, then there would be billions of times more arrangements in which the observers were surrounded by chaotic events, than arrangements in which there were uniform physical laws.”

Unpacking this objection will require some additional work. I’ll break it down into three steps:

1. First, Egan reminds readers that we live in an extremely orderly universe, where the laws of nature seem to be unchanging: the gravitational constant has always been (and, presumably, will always be) 6.6743 x 10^-11m³/(kg*s²), energy can never (and will never) be created or destroyed, the time-evolution of physical systems will always be unitary, and so on and so forth.

2. Next, he points out that if Dust Theory was true, then every possible arrangement of physical structure (“dust”) containing people (“observers”) would be realized. Using the language of the previous section, this means that under Dust Theory, any and all virtual environments compatible with physical reality would be physically instantiated. As pointed out by Egan, this would include environments where people’s surroundings were chaotic and absolutely incomprehensible (e.g., a universe with laws of nature identical to ours up to the year 2027, where all constants of nature change every ‘day’ thereafter).

3. Taking the first two steps in conjunction, Egan concludes that there are “billions of times more” environments in which people’s surroundings are absolutely incomprehensible than environments with uniform physical laws. Hence, the fact that we live in an improbably orderly environment (i.e., our universe) suggests that something is fundamentally wrong with Dust Theory. Why? Because if the hardware on which a mind is instantiated does not matter, then we should not be surprised to find ourselves instantiated in our brains at one moment - which are governed by uniform laws of physics - and in the next, on a wholly different physical system: say, a virtual computer whose states happen to be mapped onto grains of sand on the beach, running a program that has, up to that moment, been identical to our universe’s history, but will become disorderly on the next. In fact, Egan argues that we should expect this to be so, as - according to him - there are more disorderly environments than orderly ones. Although he doesn’t explicitly defend this last claim about the higher abundance of disorderly environments, it likely comes from the seemingly intuitive idea that to specify an orderly universe, one must keep all constants of nature unchanged throughout time, and for each such orderly universe, one can conjure an infinity of disorderly variants - each differing in the value some constant changes to, or the moment at which it does so.

The Problem with Egan’s Objection

There is, however, a serious issue with Egan’s argument. To understand the problem, we need to step out of Dust Theory for a minute and briefly delve into the world of set theory. To do so, let me bring back the question I asked at the start of this entry:

In the set of all natural numbers (1, 2, 3…), how often should we expect to see an odd number?

The answer seems to be intuitive and straightforward: We should expect to see an odd number half the time. This, however, only holds if we count the elements of the set sequentially. If we were to rearrange the numbers as follows:

1, 3, 5, 2, 7, 9, 11, 4, 13, 15, 17, 6…

the answer seems to change. You might think that I’m cheating with this rearrangement. If I was doing this with the set of all natural numbers from 1 to 100, I will eventually run out of odd numbers, and the even numbers would take over the end of the sequence - bringing back the answer to ‘half the time’. However, given that we are dealing with an infinite set, this difficulty will never arise; there will always be additional odd numbers to keep the new arrangement going.

Of course, things would be different if there was a rule stating that the elements of the set of all natural numbers must always be counted sequentially. However, that is an additional assumption not contained in the pure definition of the set. Thus, any claim about the abundance of odd or even numbers will be underdetermined.

You can probably see where this is going.

According to our best scientific theories, the set of physical structures that instantiate our future mental states is uncountably infinite. For readers with some background on quantum theory, here is why: under unitary quantum theory, systems don’t just evolve into a single state, but into many - a state of affairs known as a superposition. Some of the features that vary across these states are continuous degrees of freedom like position, momentum, and field strength. Given the continuum-many values that these degrees of freedom could take on, the number of states that the overall system can evolve into is also uncountably infinite. If we apply this logic to our brains (i.e., if we consider them as physical quantum systems), it follows that there are uncountably-many brain states where we could find ourselves instantiated, all while still being surrounded by orderly laws of nature.

With this in mind, we can explicitly state why Egan’s objection to his own theory does not work: The subset of possible futures where the laws of nature remain the same, and the subset of possible futures where the universe becomes disorderly are both (uncountably) infinite - and absent a rule telling us how to weigh them, it is impossible to determine which type of future is more abundant, or likely. Hence, it is erroneous for Egan to claim that the orderly universe that we find ourselves in presents an empirical problem for Dust Theory.

My Thoughts on the Theory (Closing Remarks)

Do I, then, believe in Dust Theory?

Although I have no conclusive argument against the Theory, I think it rests on two shaky premises: one that I’m skeptical about, and one which I believe to be false.

Scientifically inclined readers might think I’m talking about the assumption of infinite structure being genuinely available in our universe, as there seems to be no consensus on whether the topology of the universe is simply connected or multiply connected. However, I believe there are good philosophical reasons to prefer the former over the latter (which I might defend on some future blog entry), much in the same way that the theory that there is an external world is preferable over the theory that we are brains in a vat.

Instead, the premise I’m skeptical about is the one stating that “all mathematical abstractions compatible with physical reality are instantiated in it”. I have no trouble asserting that soccer balls instantiate spheres and Giza instantiates pyramids. However, I’m not sure if it is the case that a block of marble simultaneously instantiates all the statues that could be carved from it. On one hand, it feels absurd to make such claim - as surely something must be done to the marble to bring a statue about. On the other hand, what if we don’t touch the block but instead make a 3D-scan of it, and then we identify the portion of the block that will correspond to the statue? Would we then say that the statue is not there? Wouldn’t some artists claim that we are merely freeing it from its crystalline bounds? I honestly don’t know, but I’m glad that this is not the only point of contention I have with the Theory.

The assumption that I’m almost certain is false is the one asserting that from the POV of simulatees, “instantiating a sequence of their brain states on a single piece of hardware, or across various devices would make no difference”. I think there is something about causal interactions that plays an important role in consciousness, so that a sequence of brain states with no physically causal links between them would not be up to the task of instantiating a mind - even if they encoded what would constitute the next instance of the previous brain state. I couldn’t tell you exactly what motivates this hunch, but my intuition about this assumption feels stronger - and I think most people would agree. Perhaps it is a prejudice stemming from the fact that the only minds we have access to (ours) have always been instantiated on the same hardware.

That said, despite my reservations about the legitimacy of Dust Theory, I can’t help but feel grateful with Egan for concocting such an intricate and beautiful theory - all for the purpose of writing an interesting novel. It has certainly given me a lot to think about, introducing me to new areas of knowledge to dig deeper into, while also providing me a chance to make use of the concepts that I became familiarized with during the last few years, and supplying me a lifeline to escape my reading drought.