I received a small grant from the ACX Grants, allocated by the Long-Term Future Fund to write about Constructor Theory, the kind of problems that it attempts to solve, and whether it can give any insights into the AI alignment problem. This post is mostly based on the paper Constructor Theory by David Deutsch. Unless otherwise specified, quotes are taken from this paper.
This post is a sequel to Information, Distinguishability, and Causality which itself is a sequel to Information is a Counterfactual Property. It is not essential to read those posts before this post, but they might be useful.
This is only my third post on LessWrong, so I would appreciate feedback.
In the previous posts, I argued that the concept of 'information' is counterfactual. That is, to describe the informational qualities of a physical system, we cannot just refer to the state of the system, we must also consider the possible states that the system could be in and the possible transformations which could occur on the system. In what we called the ‘Prevailing Conception’ (PC) of physics, systems are described in terms of their evolution from some initial state according to dynamical laws. Descriptions solely according to the PC do not allow for counterfactual statements to be made: they only tell you what does happen, not what could happen. I have already argued that counterfactuals may be useful if we wish to incorporate the concept of information into physics. In this post, I will describe some more reasons why exploring counterfactuals in physics may be useful. Here, I will describe some cases where the PC seems to be inadequate, but where we have hints that other approaches may work. In particular, I will compare the PC to counterfactual descriptions of reality, and Constructor Theory - a proposal which seeks to express the laws of physics in terms of which transformations are possible or impossible. To keep the examples familiar and simple, most discussions here we will be of classical physics, though the principles discussed extend to all areas of physics.
Counterfactual descriptions can be more simple and general than PC descriptions of the same system
Suppose an engineer presents you with the blueprint for a complicated clockwork device. It has thousands of springs and interlocking gears. They tell you that, by turning a crank on the machine, the gears will be set in motion and will start turning a wheel which, once moving, will never stop. They claim that this wheel could be hooked up to dynamo and provide an unlimited supply of electricity. They are claiming to have built a perpetual motion machine. To disprove them, you could go about modelling the system using the PC. You could specify the initial conditions, including the positions of all of the gears and the impulse that you impart to the crank. You could then meticulously go about calculating the evolution of each component of the machine using the laws of classical mechanics. By the end of these long calculations you would find that the torque on one axle is not large enough and the wheel would eventually stop spinning. In this way, you would have disproved their claim and you can tell them that their machine will not work.
But in reality, you knew from the moment they presented the blueprint that the machine would not work. We know this because there is a broad counterfactual law in physics that says ‘it is impossible to build a perpetual motion machine’. It is far easier and more elegant to appeal to this principle than it is to calculate the torque on every axle of the machine. The engineer may not be happy with your explanation. They may say ‘well, yes, that principle applies in most cases, but I am claiming the principle is wrong. It is true a lot of the time, but not all of the time. How can you be so sure that it applies in every case, even those you haven’t seen before?’. Even those who do not believe that the engineer have built a perpetual motion machine may be inclined to agree with him on this point. After all, the history of science is filled with examples of ‘universal principles’ which turned out not be true. The engineer says that he will only be satisfied if you go through the full calculations, using the PC, and prove to him that the torque on the final axle is not large enough. This brings us to the next point I wish to make:
There is no reason to view PC explanations as more fundamental than counterfactual explanations
By accepting the PC explanation, but not the counterfactual explanation, the engineer is implicitly assuming that the PC is more legitimate than the counterfactual mode of explanation. He is claiming that our counterfactual principle has not been shown to work in all possible cases, and therefore doesn’t accept it as a valid explanation. I think that this is a mistake. First, notice that if, as the engineer claims, his machine is an exception to the principle that it is impossible to build a perpetual motion machine, then it also must be an exception to the dynamical laws which we use in the PC. The counterfactual principle places constraints on the dynamical laws. If we are willing to accept that the counterfactual principle could be wrong, we must also be willing to accept that the dynamical laws might be wrong.
So where does this leave us regarding the engineer’s machine? Does this mean that we should accept neither the counterfactual explanation, nor the explanation in the PC? Most of us would agree that we do not believe that the ‘perpetual motion machine’ will work, but we cannot rule out the possibility that this machine constitutes an exception to the currently-known laws of physics. How can we justify this belief? Answering this question will involve a slight diversion, but I think it is worthwhile.
This discussion is closely related to various classic questions in the philosophy of science, including the problem of induction. They can all be broadly framed as: ‘How can we be sure that laws which apply in one situation generalise to other situations?’. These questions are solved by taking Karl Popper’s approach to science, where scientific theories are viewed as conjectures, tentatively put forward to solve problems. Under this view, we cannot be certain that the engineer’s machine will not work, but, unless the engineer provides us with an new, better explanation of how physics works, we have no reason to abandon our current best theories. Of course, one way the engineer could get our attention would be to build the machine and demonstrate that it does generate perpetual motion. But, unless the engineer provides us with a good explanation as to why this would happen, we have no reason to expect that he would succeed, and no reason to abandon our current principles. This issue, including what constitutes a ‘good’ theory, has been discussed at length elsewhere, so I will not labour it here. The important point I wish to make is that science does not place any restrictions on the kind of explanation that is allowed. If counterfactual explanations or dynamical laws help us to explain and understand the world, there is no reason to believe that one kind of explanation is more fundamental than the other.
Aside: Noether's Theorem
Up until this point, readers familiar with physics may object to my characterisation of physics. They may object that it is artificial to draw a distinction between the dynamical laws of physics and the counterfactual principle that it is impossible to build a perpetual motion machine. They may argue that, while the two modes of explanation seem different, they are actually very closely linked. After all, we have Noether’s theorem , which provides a bridge between dynamical laws (which govern how a system evolves) to conservation laws (which specify that certain quantities of the system will be conserved over time). In particular, Noether’s theorem states that certain symmetries in a physical system correspond to conserved quantities. These symmetries are found by considering a property of a system known as the ‘action’. The action of a system also characterises the dynamical laws governing the evolution of the system. Furthermore, our claim that it is impossible to build a perpetual motion machine is an example of the principle of the conservation of energy. Some may say that building a perpetual motion machine violates the conservation of energy as specified by Noether’s theorem. Thus, (the argument goes) this demonstrates that counterfactual principles flow from dynamical laws and there is no need to separately invoke counterfactual principles. I disagree with this claim and I will explain why.
Noether’s theorem tell us that there are certain quantities of a system which stay constant throughout its evolution. For example, applying Noether’s theorem to our machine would tell us that the sum total energy of the system would stay constant. In other words, energy would be conserved. If you calculate the energy you impart on the machine by turning the crank, it it would be finite, yet the engineer claims that they can extract an infinite amount of energy from the system. This violates the conservation of energy, as enforced by Noether’s theorem. Suppose you tell this to the engineer and he confidently replies: ‘Oh, you’re absolutely correct, energy is conserved as a consequence of Noether’s theorem. But that doesn’t stop my machine from working. You see, my machine has a special component, whose energy decreases, exactly in line with the amount of energy that is harnessed from the machine. The sum total of the energy of the system stays exactly the same. The more energy that is yielded from the machine, the further the energy of the component decreases.’ In one sense, the engineer is correct, having one component whose energy can arbitrarily decrease while the energy of the other components increase does not violate Noether’s theorem. This is analogous to the fact that the principle of conservation of momentum does not prevent a component of a system initially at rest having a arbitrarily large momentum in one direction, provided that another component has an equally large momentum in the opposite direction.
What are we to make of this explanation? It seems that we cannot use Noether’s theorem and the conservation of energy to rule out the possibility of machine which takes an arbitrary amount of energy from an infinite 'well', whose energy becomes more and more negative as the amount of energy extracted from it grows.
In physics, this problem is solved by the fact that energy is 'bounded from below', meaning that energy cannot decrease below a certain point. This fact means that it is impossible to have a component whose energy can decrease to an arbitrary level, as the engineer claims to have. Combined with the conservation of energy, which tells us that the total energy of the system is unchanged, this leads us to conclude that the engineer’s machine cannot work. Does this mean that we have disproved the engineer without invoking counterfactuals? No! Because the claim that energy is bounded from below is itself a counterfactual principle. It tells us that it is impossible for a system to have an arbitrarily decreasing energy. It is a claim about what kind of physical phenomena are possible or impossible. Thus, we are unable to claim that the machine will not work purely using the principle of conservation of energy alone – we must invoke counterfactual principles.
Readers familiar with thermodynamics may object to the fact that this analysis has ignored the second law of thermodynamics, which would also forbid the engineer's machine from working. The reason for this is that the second law is also a counterfactual principle which cannot be expressed in the PC. This will be discussed in the next section, after a quick historical aside.
Historical Aside: Stevinus
I will end this section with an interesting historical note, which informally supports the use of counterfactuals principles in physics. In the early 17th century, the Flemish mathematician Stevinus investigated the problem of two masses, A and B, on an inclined plane, connected by a pulley:
The inclined plane is characterised by a right-angled triangle with side lengths 3, 4, 5. Stevinus was interested in what the ratio of weight would have to be between masses A and B in order for the system to be in equilibrium (ie. if A weighed 1kg, how heavy would B have to be in order to balance it out). As usual in physics problems, he ignored friction between the masses and the plane, and assumed that the weight of the pully was negligible. To solve this problem, Stevinus used two things: his knowledge that perpetual motion is impossible, and a thought experiment where he imagined adding more masses to A and B, to form a 'wreath' around the plane :
If you haven't seen this before, its worth taking a few minutes to try to figure out how he solved the problem. I won't go through the full solution here, but it can be found, along with more discussion of the conservation of energy, in the Feynman Lectures on Physics, Chapter 4, section 4-2.
The point I am trying to make is that Stevinus used the counterfactual principle that building a perpetual motion machine is impossible to solve practical problems over 100 years before Émilie du Châtelet postulated the conservation of energy and over 300 years before Noether proved her theorems. Even if you are not convinced by the argument that counterfactual principles are fundamental to physics, I hope that this would convince you that they are at least useful.
The Second Law of Thermodynamics
There seem to be certain laws and principles of physics which are emergent, meaning that they deal with complex, high-level systems and are not ‘fundamental’ in a reductionist sense. It is often difficult to treat these laws using the PC.
One example of such a law is the second law of thermodynamics. The fundamental difficulty of reconciling the second law with the PC is that all known laws of motion are time-reversible (meaning that taking a physical evolution and ‘reversing’ the direction of time, leads to an equally valid physical evolution). The second law, on the other hand, is irreversible in time, meaning that it specifies a ‘direction’ in time and ‘reversing’ the direction of time of a physical evolution leads to an evolution which is invalid. To attempt reconcile this fact with the PC, many proposals resort to coarse-graining or averaging over physical states. For example, one can describe the second law in terms of the increase in entropy and entropy as a measure of uncertainty. This solution is very elegant, but also makes entropy (and hence the second law) subjective and somewhat arbitrary.
As it stands, reconciling the irreversible second law with reversible dynamical laws in the PC is impossible without resorting to some sort of averaging or coarse-graining. Nonetheless, there is a straightforward way of stating the second law as a counterfactual statement about what physical transformations are possible. Namely: 'it is impossible to engineer a cyclic process which converts heat entirely into work'. Admittedly, this statement is a bit vague and informal, but it is a starting point. Deutsch writes: 'it seems reasonable to hope that the informal statements of all the laws of thermodynamics, as well as concepts analogous to heat, entropy and equilibrium, could be made precise within constructor theory by formalising existing informal statements such as ‘it is impossible to build a perpetual motion machine of the second kind '.
The Turing Principle
Another example which Deutsch gives of a law which cannot be explained using the prevailing conception is the 'Turing Principle'. Deutsch’s formulation of the Turing principle is that ‘a computer capable of simulating any physical system is physically possible’. We often take for granted that physical systems can be described mathematically in terms of computable functions. Deutsch claims that this fact needs an explanation, since there are vastly more non-computable functions than computable ones . We have discovered striking regularities in the world: not only can our laws of physics be simulated by a computer, but our laws of physics allow such a computer to be built in this world. Furthermore, Deutsch claims that the Church-Turing thesis is not sufficient to explain this as, on his view, mathematics and logical rules of inference used to describe it are not independent of the laws of physics. He argues that ‘[d]ifferent laws of physics would in general mack different functions computable and therefore difference mathematical assertions provable’. Thus, statements about computers and computability are really statements about physics and the Turing principle requires a physical explanation.
Explaining the Turing principle using the PC is difficult. In, fact it is difficult enough to state the Turing principle in the language of the PC. One could write down a set of equations characterising the initial state of a computer and laws describing how the potentials in the transistors change over time. Then, one would somehow have to express the fact that it is possible for there to be some kind of mapping between the pattern caused by these changing transistor potentials and every other possible physical system.
On the other hand, if one formulates physics in terms of tasks which tasks are possible or impossible, the Turing principle has a natural expression (if not an explanation, per se). I would like to come back to this in more detail in a later post, but those who are keen can read sections 2.8 and 3.7 the paper 'Constructor Theory'.
The Initial State problem
In the PC, the state of any system can be described as the result of an evolution, according to dynamical laws, starting from an initial state. The initial state of a system must be provided by fiat. If we wish to explain why the initial state is how it is under the PC, we can only explain it as the result of evolution from a previous initial state. If we wish to explain that state, we again must explain it in terms of evolution from an even earlier state. Tracing this process back leads us to the initial state of the universe . But what is the initial state of the universe, and why is the way that it is? Under the PC, the initial state of the universe must be dictated by fiat. It cannot be explained in terms of evolution from an earlier state, as there was no earlier state. I will call this the ‘initial state problem’. If we wish to explain the initial state of the universe, we will need a different mode of explanation.
In a counterfactual approach to physics, like constructor theory, the initial state of the universe does not hold the same level of significance as in the PC. The states of the universe at each time would emerge as a result of deep counterfactual principles about which tasks are possible or impossible. It may be possible to explain the initial state of the universe as the state that it required to allow (or forbid) certain tasks happening at other states of the universe’s lifetime. If this was true, then a counterfactual approach to physics may solve the initial state problem. If not, then some other mode of explanation will be needed.
A well-known problem in physics is the fact that our best theory of gravity (general relativity) is incompatible with the best theory of everything else (quantum mechanics). Normally, for practical purposes, this incompatibility does not pose problems, since the effect of gravity is often negligible in the small-scale regimes where quantum mechanics most obviously finds applications. Conversely, the parts of quantum mechanics which contradict general relativity become negligible in large systems such as galaxies and planets where general relativity is often applied.
The term 'hybrid systems' refers to cases where we cannot treat one or the other as negligible. For example, if we had a system which allowed us to place a mass in a quantum superposition of two different positions and measure its gravitational effect on another mass, what would we see? The PC does not tell us, as we have two contradictory sets of dynamical laws which both claim to apply to the situation. However, by appealing to general counterfactual principles, which cannot be stated in the PC, we can make predictions about such systems, even if we don't know the form of the dynamical laws. For example, we can predict that such a hybrid system will not allow one to make a perpetual motion machine. Similarly, other counterfactual, information-theoretic principles are used in this paper to investigate properties of interactions between two quantum systems, even if the form of the interaction is not known. This can be applied to make predictions about the form that a gravitational interaction between quantum systems can take, even if we do not have a full theory of quantum gravity.
In this post I have attempted to summarise and flesh out some of the key arguments in the paper Constructor Theory. There are two kinds of arguments in this post. One kind is negative arguments, which argue that the prevailing conception of physics is inadequate for solving problems. The other kind are positive arguments, which argue in favour of counterfactual, constructor-theoretic approaches to physics. Even if the constructor theory approach turns out to be wrong in some way, some other theory will be needed address the problems in the prevailing conception.
In the next post, I'll describe a bit more of the formalism of constructor theory.
See The Beginning of Infinity by David Deutsch, Popper by Bryan Magee or The Logic of Scientific Discovery by Karl Popper
At least, his machine cannot work forever. The machine may have some finite well of energy that can be used up, giving the illusion of perpetual motion for a little while, but it will eventually be depleted.
Public domain image from Wikipedia
The number of computable functions is a countable infinity, but the total number of functions is uncountable.
Another logical possibility is that there is no initial state of the universe and we can keep tracing the evolution backwards in time forever. This approach has its own problems, but I will not go into them here.