2.3.1 Varied stable objects

2.3.1.1 Life as we know it requires a world which is intelligible and predictable, but also rich enough to support the complex mechanisms needed for cognition. Furthermore, given the definition of physicalism in 2.1, the world must organize itself into this rich state from a simple starting configuration.

2.3.1.2 How does the world do this? The statement is so broad that giving a comprehensive treatment is impossible. Instead I'll be focusing on a property of the world that seems to be a prerequisite for its richness -- at least, for the way that rich complexity develops in our universe. The property is the existence of stable richly-varied objects which move around and interact with each other.

• 2.3.1.2.1 What do I mean by this? To go through the phrase in order, an object is some physically contiguous part of the world that we are considering separately from the rest. Typically an object's parts will be more closely correlated with each other than with the rest of the world, e.g. the parts of a cup have correlated positions because they move together. A sub-part of an object could itself be considered an object(but doesn't have to be).
• 2.3.1.2.2 Objects are stable if they can maintain their existence in a variety of environments; e.g. a container ship maintains its integrity in a variety of ocean conditions; more mundanely, objects such as tables maintain their shape despite being buffeted by random air molecules.
• 2.3.1.2.3 Objects are varied because there are many types of distinct objects. There are human bodies, rocks, trees, molecules, planets, pens. Different types of objects have different properties and interactions. Furthermore, objects are richly varied; there is not merely a large but random assortment of objects, but intricate hierarchies of sub-types with greater and lesser degrees of resemblance. There are also relationships of inclusion between objects and their sub-parts, e.g. organs making up an animal, cells making up an organ.
• 2.3.1.2.4 Finally objects interact with each other in many different ways. A hammer applies pressure to nails; proteins bind to each other; planets gravitationally attract each other. And of course, objects can be moved around.

2.3.1.3 Now the existence of such objects may seem extremely obvious. However, the Game of Life does not have them!

• 2.3.1.3.1 But wait, what about the patterns from the previous section? Aren't they a rich and varied collection of object-like things?
  • 2.3.1.3.1.1 There are two issues, stability and movement. GoL patterns are not stable to environmental perturbations. Patterns such as the spaceships only function in the fixed all-empty background; encountering another pattern will destroy them(unless they were specifically created in order to interact with that pattern). Similarly, altering a small part of a pattern will destroy it. The lack of stability makes it difficult for objects to interact.
  • 2.3.1.3.1.2 And with regard to movement, while patterns such as the glider can move, the issue is that the structure of a pattern is linked to its movement. A glider must move diagonally once every 4 iterations; whereas stationary oscillators must stay still. In our world objects can be moved independently of their structure.
• 2.3.1.3.2 The above issues also seemingly make it difficult to create a simple initial state which naturally leads to such objects forming. In our universe, objects arise and continue to exist in chaotic initial conditions, whereas in GoL patterns cannot survive in such conditions.

2.3.1.4 All this raises the question of what properties of our universe's physics differentiate it from the GoL and make it possible for such objects to exist here.

• 2.3.1.4.1 Even this question leads to endless complexities. Giving a full account of the existence of objects would involve detailed analysis of quantum mechanics, cosmology, chemistry, etc. and would(and does) span entire tomes.
• 2.3.1.4.2 You could rather think of the goal here as constraining the search space of possible universes that could contain objects like ours, by analyzing some high-level mathematical properties which seem especially important in supporting their existence.
• 2.3.1.4.3* In particular, the main properties I will be focusing on are measure-preservation and chaotic dynamics. These are important because they give the universe an inherent directionality under abstraction: dynamics tend to move towards regions of larger entropy(= logarithm of state-space volume)

2.3.2 Measure-preserving dynamics

2.3.2.1 In this section I will be analyzing the laws of physics at a classical level, as this approximation is good enough to capture the dynamics of interest.

2.3.2.2 Classical physics describes the world as a collection of real-valued variables(possibly a continuum thereof in field theory). In the Hamiltonian formulation the variables come in pairs; positions and conjugate momenta. The space of possible configurations of the variables is known as phase space. For example, a system of N particles in 3D space has a 6N-dimensional phase space.

2.3.2.3 Hamiltonian dynamics have the crucial feature that the phase-space volume, or measure, of a region is conserved by time evolution. If we take a region of phase-space with a non-zero volume and evolve it forward, it will have the same volume.

• 2.3.2.3.1 A simple discrete analogue of measure-preservation can be found in reversible cellular automata such as Critters. Since the dynamics are reversible, if we have a collection of M Critters states, evolving them all forward in time again gives M distinct states.

2.3.2.4 Now, the significance of measure-preservation comes in the way that it interacts with abstraction, chaos, subsystems, and local minima. Let's consider each in turn.

2.3.2.5 Firstly, abstraction. Consider that macroscopic objects such as tables and the human body contain vast numbers of degrees of freedom -- macroscopic objects contain on the order of 10^25 atoms per kilogram. Thus in order to build any sort of theory it is necessary to coarse grain -- consider a relatively small number of significant large-scale variables we are interested in tracking(such as average temperature, pressure). Then consider the reduced system consisting of only the significant variables.

2.3.2.6 This raises the question of how to choose the significant variables to coarse grain over, and how to understand the dynamics on the reduced system. Here, the notion of chaos is useful. The idea is that for very complex dynamics, a decent approximation of the long-run trajectory of the system is to assume that it moves randomly throughout the subset of the state space determined by its collection of conserved quantities. So the probability(averaged over time) that it is in a given coarse-grained region of this subset is proportional to the region's volume. The study of these probability distributions is known as statistical mechanics.

• 2.3.2.6.1 Since the regions of state space can vary exponentially in volume, we can sometimes make the further simplification that the system is in or near the largest-probability region. This is known as entropy-maximization -- entropy being the logarithm of the state-space volume of a region.
• 2.3.2.6.2* For some systems, we can give a mathematical argument that in the long run, the system should spend time in regions proportional to their volume. These systems have the property that for any region of non-zero volume, for almost all points in the region, the forward trajectory of the point spends time in any given region proportional to its volume. Since we never have infinitely precise knowledge of a classical physical state, this essentially implies any trajectory we expect to see will sample the space uniformly.

2.3.2.7 An important wrinkle in the above is that when doing statistical mechanics, we are almost always thinking of a subsystem, not the universe as a whole. The above arguments about the highest probability region being the region of largest volume apply to the joint system of the subsystem and its environment. Thus when considering the probability of a given configuration of the subsystem, we must also consider the state-space-volume of the associated region of states of the environment.

• 2.3.2.7.1 Subsystems are usually chosen to be relatively weakly-coupled with their environment, so we model the environment having an independent distribution -- except for globally conserved quantities such as energy. That is, for a given state of the system, the environment is in an independent random state conditional on the values of energy, volume, particle number etc. exchanged with the system.
• 2.3.2.7.2 Since the environment's state-space volume increases exponentially with the energy it gets from the system, systems are encouraged to settle on configurations of minimal energy, subject to keeping their internal entropy high. This tradeoff leads to phenomena such as phase transitions at different temperatures(temperature being the inverse of the exponential factor determining how the environment converts energy to entropy). The negative logarithm of the overall environment+system state-space-volume associated with a configuration of the coarse-grained variables of the subsystem, multiplied by the k_B T to convert to units of energy, is known as the free energy of that configuration. Thus systems tend to configurations of minimal free energy.

2.3.2.8 One final crucial factor is that, on relevant timescales, the dynamics of reality are not globally entropy-maximizing, but rather entropy-maximizing within a subspace defined by (free) energy barriers.

• 2.3.2.8.1 As a simple example, consider a log. The overall entropy of the universe could be increased if the energy within its cells were released into the environment; however, this will not happen on its own because the chemical bonds in the cells are at a local energy minimum. But when burned, the temperature becomes high enough for the system to break these bonds and release their energy, increasing global entropy.

2.3.2.9 We can summarize the above discussion by saying that, under chaotic measure-preserving dynamics, subsystems tend to local minima of free energy.

2.3.3 Objects as Local Minima of Free Energy

2.3.3.1 So what's the relevance of the above discussion for our original objective, accounting for the existence of stable varied objects in our universe?

2.3.3.2 The basic idea is: objects are configurations of subsystems corresponding to a local minimum of free energy.

2.3.3.3 The key property of local minima is that they are definitionally stable to small perturbations; since systems generically minimize free energy, a small perturbation from a local minimum will quickly return to the minimum.

• 2.3.3.3.1 As a simple example, consider an iron rod. How does it maintain its rod-like shape? The atoms inside it sit in a polycrystalline lattice, which is composed of numerous small crystal regions with differing alignments. This configuration locally minimizes the free energy of the iron atoms.
• 2.3.3.3.2 If you were to push one end of the rod, the atoms at one end would initially be displaced from their neighbours. However, this displacement causes a force to be exerted on the neighbours, accelerating them a bit. These atoms in turn exert a force on their neighbours, ultimately converting the local push into global rotational movement of the rod. If the rod is in contact with an external system which can absorb kinetic energy, say a rough surface, this rotational energy will in turn be dispersed to the environment.
• 2.3.3.3.3 Note that minima will in general be manifolds, not points. For instance, a metal rod can be rotated, meaning at least 3 degrees of freedom can be changed while remaining in the local minimum. So we really consider the object to correspond to the entire locally minimal manifold.

2.3.3.4 A variety of objects is possible because the energy landscape has many disconnected local minima.

• 2.3.3.4.1 Again consider an iron rod. The same iron could be forged into a multitude of shapes, e.g. a donut, a triangle, a statue of a bird. That there are various polycrystalline structures which are local minima of the iron's free energy is what makes this possible.

2.3.3.5 Other key properties of objects are that they move around and interact with each other.

2.3.3.6 Objects can move around because of (a) the position-momentum/velocity variable pairing of Hamiltonian/Newtonian dynamics (b) the invariance of the laws of physics under translation, rotation and (approximately) change of velocity.

• 2.3.3.6.1 A local minimum of free energy is defined by the relative positions of the constituents of an object. By giving every part of the system the same velocity, we can move it while maintaining the local minimum. The various invariances allow this to happen without changing what configurations are local minima.

2.3.3.7 As for interaction, this automatically happens because the laws do not specifically single out objects as separate from each other, so naturally they interact when close. Actually, a more interesting question is how the boundaries between solid objects are maintained -- why don't different objects merge together when in contact? The answers vary.

• 2.3.3.7.1 For instance, in the case of iron rods, the local crystal patches will not generally be in alignment, the surface of each rod will generally be rough and difficult to bring in alignment with its counterpart, and each rod will have a coating of oxides. Interestingly, in a vacuum two smooth metal surfaces brought into contact will sometimes merge.
• 2.3.3.7.2 Generally speaking: the local minima of different objects cannot generically be combined into a local minimum of the joint system, at least not without overcoming free-energy barriers.

2.3.3.8 This covers the properties of objects defined in 2.3.1. Given our definition of physicalism, we are still left with the question: how does the rich variety of objects come to exist from a simple starting condition?

2.3.4 Creation of Objects: Work and Temperature

2.3.4.1 Let's work backwards from the existence of an object to the conditions that caused it to exist(and allow it to continue to exist).

2.3.4.2 Take (again) an iron rod. Its atoms are in a stable polycrystalline lattice.

• 2.3.4.2.1 How did they come to be this way? Let's say a blacksmith made it in a forge, by heating the iron and hammering it into a rod shape.
• 2.3.4.2.2 What makes this possible? (a) The heat of the forge makes the iron more pliable. (b) The force from the hammer deforms it.
  • 2.3.4.2.2.1 In terms of the free energy landscape, the forge has a higher local temperature than its surroundings. This heats up the iron, which both changes the free energy landscape and makes transitions across barriers more likely.
  • 2.3.4.2.2.2 Whereas the force of the hammer provides a direct injection of energy pushing in a particular direction, doing work on the metal.

2.3.4.3 Thus we see that directed work and temperature differences are key enabling factors. "Stable" objects are really only stable in certain conditions; when exposed to sufficient force, or in regions of high temperature, they are unstable, and this allows a variety of local minima to be moved between.

2.3.4.4 Now, how are temperature differences created? By the definition of temperature(inverse of derivative of entropy with respect to energy), systems with different temperature should exchange energy until they are in equilibrium. Typically a gradient is created by burning fuel(stored free energy) of some sort. For instance blacksmiths might burn wood to create the heat of their forge.

• 2.3.4.4.1 Likewise, work is performed by burning some sort of fuel, combined with a mechanism for concentrating the resulting force in a small area. Blacksmiths use the energy in their muscles(stored as ATP) to swing their hammer. This energy is ultimately derived from food.

2.3.4.5 Wood, food, and other fuel sources derive their free energy from plants, which derive their free energy from sunlight(and in particular the fact that sunlight is hotter than the rest of the sky), which derives its free energy from fusion. Fusion happens because stars consist of lighter elements which are not the most energetically stable, so free energy can be produced by merging them.

• 2.3.4.5.1 Fusion happens in stars and not elsewhere(thus producing the temperature difference between stars and space which is crucial to the free energy gradient) because matter is densely concentrated there, which in turn is ultimately due to the early state of the universe -- mostly homogeneous with small scale-free perturbations(a very low entropy state from the perspective of gravity), which collapsed to form regions of varying density.
• 2.3.4.5.2* Nucleosynthesis in stars(and supernovae, and in the early universe) is also important in giving rise to the wide diversity of chemical elements from the simple building blocks of protons and neutrons. Both the stability of nuclei and the slow burning rate(yet high energy) of nuclear fuel are downstream of the nuclear force being stronger yet shorter-range than electromagnetism.

2.3.4.6 So we see that objects are downstream of applied work and temperature differences, which are downstream of flows of free energy, which are downstream of slow-burning "fuel" sources, ultimately downstream of the low-entropy initial conditions.

• 2.3.4.6.1 The above story is not a sufficient explanation of the existence of objects, but it does seem to be an necessary part of the explanation, at least in our universe.

2.3.4.7 As a somewhat separate matter, "fuel sources powering flows of free energy" is also a ubiquitously used pattern by life for getting stuff done, beyond its role in supporting the creation of stable objects. e.g. ATP molecules power chemical reactions inside cells.

• 2.3.4.7.1 Human engineering, of course, uses flows of free energy, in the form of electrical current, to power myriad devices.
• 2.3.4.7.2 Why are free energy flows useful? To simplify a bit, because of their ability to overwrite the state of an existing system, since "state" essentially consists of a local free energy minimum for reasons discussed above. In a messy universe, being able to push the state of your environment in the direction you want is essential. The Landauer limit quantifies the precise free-energetic cost of overwriting the existing state of a system at a given temperature.
• 2.3.4.7.3 The fact that free energy can be used in this way, as a "common currency" of irreversibility between different systems, is perhaps the most important consequence of measure-preservation.

2.3.5* Why don't we live in Critters?

2.3.5.1 The discussion above explains why we don't see varied stable etc. objects in the Game of Life: because GoL is not measure-preserving, it has no equivalent interchangeable currency of irreversibility like free energy. But what about Critters? Critters is reversible, so it preserves the counting measure on collections of states.

2.3.5.2 There's a few factors. Firstly and most obviously, Critters does not have the velocity-position pairing of classical mechanics, so objects cannot be moved around independently of their structure.

• 2.3.5.2.1 It's not totally clear how to implement something like this in cellular automata. Second-order CAs move somewhat in this direction, as do lattice fluid simulations.

2.3.5.3 A perhaps more severe problem for the story above is that there is no obvious analogue of temperature, which was a crucial factor in the explanation of how stable objects come to exist(formed in high-temperature regions where energy barriers are traversable then cooled). Recall temperature is the inverse of the derivative of entropy with respect to energy, a conserved quantity.

• 2.3.5.3.1 Critters does have the conserved quantity "number of critters(living tiles) on same-parity timesteps", and in theory one could define an equivalent inverse derivative of entropy. But it does not seem to give rise to the same rich dynamics. A random Critters state consisting of a perturbed square of solid white typically has the critters expand outwards to a circle of uniform density, emitting a handful of walkers which walk off to infinity or march back to the circle(if the critters are on a torus). The dynamics seem similar to kinetically constrained models.
• 2.3.5.3.2 In general, Hamiltonian dynamical systems have many rich properties besides measure preservation. One important one is Noether's Theorem: for every symmetry of a set of physical laws, there is an associated conserved quantity.
• 2.3.5.3.3 Energy as a conserved quantity is rather special. It's the conserved quantity associated with time translation symmetry. This makes it well-suited as the "common currency" of entropy for various different processes, since they all participate in time evolution.
  • 2.3.5.3.3.1* Due to quantum mechanics, energy also controls the time frequency of processes, and the space frequency(of waves) via the energy-momentum relation. This is why we need particle accelerators to probe very short distances.

2.3.5.4 Critters does not have as many (apparently) base-level entities and forces as our world. As noted in 2.3.4.5.3, the hierarchy of forces in our world seems important in giving rise to a multitude of building blocks from simple parts, and also in providing a slow-burning source of free energy.

2.3.5.5 It would be interesting to see to what extent the sorts of dynamics talked about above can indeed be implemented in a CA or other discrete computational structure.

2.3.5.6 Of course, other parts of our world are more sharply incompatible with such discrete structures, namely continuous space and time and quantum mechanics. Do these things have a role in the existence of life?

• 2.3.5.6.1 Continuous space and time seem to be important for velocity-position pairing, although as far as I know there's no reason similar discrete analogues couldn't be made to work.
• 2.3.5.6.2 What about quantum mechanics? Of course, in practical terms much of the physics discussed above in fact depends on quantum mechanics. But is there a more structural reason that universes with QM are much more likely to support life? I think it's plausible that the answer is yes. However, the case is not nearly as straightforward as for measure-preservation. Thus in the next section I will simply be giving an overview of (my preferred way of) interpreting what quantum mechanics even is. Later sections of the text will give the case for how this is linked to life/sentience.