3. Reprimitivization, Physical Sets, Partonomies, Levels, and Sem-linking

Since the three bonding forces vary so dramatically in strength, timing, and scale, there are domains in which each can be analyzed almost entirely in isolation. The domain of the strong force is nuclear physics, the domain of electromagnetism is chemistry, and the domain of gravity is cosmology. The bonds within each of these depend entirely on the force of the domain, and it’s useful to imagine the entities being bonded as primitives in their own right within that domain, in a process I call reprimitivization, giving a new pile of Dust. In nuclear physics, no reprimitivization is necessary because quarks are in fact primitives. The electromagnetic Dust for chemistry is electrons and nuclei, and only the latter need to be reprimitivized. In cosmology, the new primitives are stars, planets, black holes, moons, and other celestial bodies. Each of these domains has frayed edges—the frays between nuclear physics and chemistry being in contexts like nuclear fission and neutron star collapse, the frays between chemistry and cosmology in contexts like planetary formation and comet tails.

Within each domain, pairs or groups of primitives that are bonded can be bonded further, usually with bonds that operate at a different scale of strength, distance, or timing. In the nucleus, triplets of quarks bond through the strong force into nucleons, i.e. protons and neutrons, and anywhere from 2 to well over 100 nucleons bond together through the weaker residual strong force into entire nuclei. In chemistry, electrons bond to one nucleus to form neutral atoms and charged ions, to two or a few1 nuclei to form covalent molecules or polyatomic ions, or to indefinitely many to form metals. Atoms bonded to other atoms through covalent bonds can form molecules in a range from minimal clumps like carbon monoxide or water, to indefinitely long chains like polymers and plastics, to indefinitely flat sheets like those in talc and graphite, to indefinitely bulky 3D matrices like quartz and diamond. Ions bond to form salts. Molecules bond through Van der Waals forces to form bulk materials. In cosmology, primitives are bonded either in bundle of orbits like in a solar system, or in a mostly unstructured cloud of gravitational interaction like in a star cluster or a galaxy.

The act of bonding creates a relationship between two or more primitives. To characterize such relationships, let’s take a short detour through mathematics to acquire some relevant vocabulary. The “set,” a foundational notion in math, is defined as a collection of distinct objects.2 Examples of sets include the even numbers, the odd numbers, the multiples of 13, negative numbers. Of course, sets aren’t restricted to collections of numbers, so there are more examples like collections of linear functions, finite groups, differentiable manifolds, and the rest of the moonshine that mathematicians study. The notation for sets consists of two curly braces as bookends, with some stuff in between, separated by commas. Let’s say we want a set that contains the integers between 4 and 10, excluding 4 and 10. Then the braces with the “stuff in between” is: {5, 6, 7, 8, 9}. Five elements—pretty easy. There’s also some special notation for the set containing no elements, the empty set: Ø, which could otherwise by written as {}.

Now things get interesting. We can use sets to make sets! Let’s make the set {5, 6, {7, 8}, 9}. Here we have a set containing 5, 6, a set containing 7 and 8, and 9. The set has 4 elements, one of which is a set containing 2 elements. It may be confusing why it has 4 and not 5. In addition to the conventional notion of  “element,” let’s entertain the notion of a “mathematical primitive” and think of it as a mathematical object that isn’t a set.3 Then we can assert that though the set contains 4 elements, it contains 5 mathematical primitives. How about the set {5, 6, Ø, 9}? This set also contains 4 elements, but curiously contains only 3 mathematical primitives! There’s a sense in which the empty set creates something from nothing.4 More vividly, we can ask if {Ø} = Ø? Certainly not! {Ø}, on the left, isn’t empty; it in fact contains Ø. Ø, on the right, is empty; it contains nothing. So we then observe that while Ø contains nothing, it itself is not nothing. Finally, we can observe that {5, 6, 7, 8, 9} does not equal {5, 6, {7, 8}, 9} because the elements don’t match.

An important notion in applied math is the “tree,” which is a specific type of graph. A graph is a mathematical object constructed from points or “vertices” together with the lines or “edges” that connect them.5 A tree is a graph that is connected and has no loops in it; that is, there is always exactly one path along edges and vertices to get from any one vertex to any other without backtracking. The file structure of all computers as far as I know is a tree, which you can see in most file viewers by expanding all folders and subfolders if you’ve ever had enough time to waste. The leaves of a tree are those vertices that are connected to only one edge and are in a sense on the “tips” of the tree. In a computer file system, that would be all of the individual files in whichever folders. If you consider mathematical primitives as leaves, then every set has a corresponding tree, where the primitives and sets are vertices and the edges connect sets to their contents, either other sets or primitives. Such a tree can also be called a nested hierarchy, or just a hierarchy.


Figure 1. A few sets and their trees.

But what does this have to do with physical bonding? The bonding patterns of each force within its domain can be regarded as sets. A nucleus, for instance, is a set of 1-100+ sets of 3 quarks. The nucleus of helium-4 as an example would be {{qu1, qu2, qd1}, {qu1, qu2, qd1}, {qu1, qd1, qd2}, {qu1, qd1, qd2}}. A molecule, say water, would be {nucH1, e1, e2, e3, e4, {e5, nucO1, e6} e7, e8, e9, e10, nucH2}, although a conventional Lewis diagram would be clearer.6 The solar System would be {Sun, {Mercury}, {Venus}, {Earth, {Moon}}, {Mars, {Phobos, Deimos}}, {asteroids}, {Jupiter, {Metis, Adrastea, Amalthea, Thebe, Io, Europa, etc.}}, etc.}. Each of these would be a physical set, and more specifically a strong nuclear set, an electromagnetic set, and a gravitational set, respectively. The tree or nested hierarchy of each physical set is called its “partonomy,” since it breaks down the parthood relationships within the set,7 and each nesting creates a level, with the Dust at the lowest level and each reprimitivization on its own level.

Reprimitivization is quite interesting because it’s fundamentally fraudulent—there is only one real Dust heap—while still being very useful. Reprimitivization, however, is merely a special case of the more general fraudulent process of treating a collection of primitives or a collection of collections just like an individual primitive. Notice the ease with which you can talk about protons and electrons in the same sentence. Further, reprimitivization operates at levels where the responsibility of a particular bonding force for the collecting is always very clear, but often we will have collections where there is no such clarity. As in the first chapter, a power strip is a collection of sockets and wires and other parts, but which of the bonding forces is responsible for keeping them together? Which of the bonding forces is responsible for keeping a tree together? An answer isn’t terribly difficult to deduce (hint: electromagnetism), but it would be difficult to talk about the attraction and repulsion of Dust in the elevator-pitch version of the answer.

I call that process sem-linking, for reasons that will be clear later. Reprimitivization sem-links nucleons into nuclei as chemical primitives and enormous quantities of chemicals into stars and planets as cosmological primitives, but sem-linking itself covers even more ground, taking us from three quarks to a proton and from half a dozen sockets to a power strip and from branches and leaves to a tree. Basically, every node in a partonomy that is not a leaf is the sem-linking of some of the nodes in the level below it. Further, the physical sets constructed by sem-linking will have interactions of their own, which of course are predicated on the interactions of the Dust, the four fundamental forces of the Standard Model.

Footnotes

1. As in benzene or phenyl groups, for instance

2. In a set, the objects must be distinct. If they’re not, you have a multiset. If a set or multiset is ordered, then you have a tuple. If a tuple only contains numbers, then you have a vector. If you order numbers along two or more dimensions, then you have a matrix or a tensor.

3. Also known as an “ur-element.”

4. This is less fanciful than it sounds. Mathematical systems can construct the numbers and beyond from just the empty set, like the standard Zermelo-Fraenkel axiomatization of set theory. Integers would not be “mathematical primitives” in this case, but rather carefully constructed sets in their own right. The mathematical primitive there is just the empty set and the nothing it contains. See G. Spencer-Brown’s Laws of Form.

5. A graph also means a picture used to plot information, often in the Cartesian plane, but that is separate meaning of the word.

6. But it would also hide the two non-valence electrons in the first shell around the oxygen nucleus.

7. Also called a “meronomy,” but the Special Composition Question can wait.

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2. The Condensation of the Dust

I don’t care to wed my arguments to any specific rendition of fundamental physics, but all of my arguments’ illustrations will be of things in our Universe painted by such physics, and we can better appreciate the illustrations if we know something about the style. There are four fundamental interactions that the physical primitives engage in: the strong nuclear force, the weak nuclear force, the electromagnetic force, and gravity. The exact profile of interactions reflects the type of primitive. Quarks interact through all of them, for instance, and electrons through all but the strong force. All of the forces but the weak force participate in bonding, one of the most important kinds of interactions. The weak force mostly functions as an intermediary between the strong force and the electromagnetic force, supporting the interface between the intranuclear and extranuclear environments during radioactive decay. The bonding forces vary drastically in their bonding strength, with the strong force by far the strongest and gravity by far the weakest, with electromagnetism in between.

Just as relevant as the relative strengths of the bonding forces is their valency and their relationship with distance. The strong force and gravity are only attractive, and thus we can call them monovalent, and no two primitives will be repelled through these forces. Electromagnetism, on the other hand, is both attractive and repulsive and thus divalent. If two primitives have the same electromagnetic charge, they are repelled, and if they have the opposite charge, they are attracted. A consequence of this is that if you have three primitives together, they cannot all be attracted to each other electromagnetically. With many more than three, what eventually happens is that the primitives distribute themselves evenly such that no large region has more positive charge than negative charge and thus most objects do not attract or repel each other through the electromagnetic force. When two primitives are attracted, they move towards each other until they no longer can, either because they latch into quantum mechanical structures (like electrons and protons into hydrogen atoms) or because they meet and transform into other particles (like electrons and protons into neutrons and neutrinos, with help from the weak force).

The electromagnetic force and gravity get weaker with distance. Since gravity is monovalent, larger and larger objects only attract each other gravitationally even more as the force cannot cancel locally like electromagnetism. At very large scales like the solar system, we then see that gravity ends up doing all the work of keeping everything orbiting together while the strong force and electromagnetism do none of it. The strong nuclear force, despite being monovalently attractive, comes with three types of charge, known as “color charge,” each of which can be positive or negative. Unlike gravity and electromagnetism, it does not get weaker with distance.1 Just like electromagnetism and unlike gravity, charges end up cancelling almost entirely at small scales, now within protons and neutrons, with some slight leakage beyond them that reaches the scale of an atomic nucleus known as the residual strong force, which does get weaker with distance. Protons and neutrons are the quantum mechanical structures that quarks latch into because they’re prevented from collapsing all the way by the Pauli exclusion principle, which also gives atoms larger than hydrogen their layered electronic structure.

The valency and distance-dependence of the three bonding forces combine so as to produce various domains of material density separated by quantifiable boundaries. Immediately after the start of the Big Bang, the Dust was distributed almost entirely homogeneously throughout the Universe in a very dense quark-gluon plasma. After several minutes of explosive expansion, the strong force acted to preserve tiny pockets of high quark-gluon density in the face of the expansion as protons and neutrons and nuclei like those of hydrogen and helium. Then about 380,000 years later, still a cosmological eye blink, the nuclei wrapped themselves in as many electrons as they had protons, locally cancelling electromagnetic charges and making the Universe transparent to photons, the first of which are visible as the cosmic microwave background radiation, the quintessential piece of evidence for the Big Bang. The atoms were still distributed homogeneously, however, and since the strong force and the electromagnetic force had already created their bonds, the only force left to do anything was gravity.

And gravity was ready. Since gravity wants to pull everything together, the state of the Universe where gravity has the most work left to do is the state where everything is apart, diffuse, and homogeneous. Unfortunately, a homogeneous Universe has no preferred locations where matter can gravitationally coalesce.2 Fortunately, the Universe was only almost homogeneous, however, with shallow ripples created by the spontaneous meows of probabilistic interactions dead and alive.3 The ripples in the Dust created regions in the primordial gas that were slightly denser than others, regions where gravity could take hold and pull things together. Without them, the Dust would have remained smooth, and no gravitational coalescence would occur, nor would any further bonding be produced among the atoms, and nature would have produced no more structure.4 Gravity, though not yet in conflict with either the strong force nor the electromagnetic force, was already in conflict with something else: heat. Local spontaneously-produced high-density regions in the gas could be disbanded by heatwaves pulsing through them. Ultimately, the interference of heat merely postponed the further coalescence of matter because local regions of sufficient density and gravitation to overcome the outward pressure of heat would eventually materialize in the vastness of space. This instability of gas to gravity is known as the Jeans instability, and it is the first limit between the homogeneous cosmic gas and its opposite: the black hole.

Once matter has coalesced enough, further action by gravity cannot be blocked by heat, and the gas collapses in free fall. The collapse stops when gravity comes into conflict with the electromagnetic force, when electrons can get no closer together due to Pauli exclusion, creating an outward pressure known as electron degeneracy pressure, which is much stronger than the pressure from heat that gravity has previously conquered. This is the density regime where most stars and all planets and moons lie, squeezed together by gravity but upheld by electromagnetism. If even more matter has collapsed together, then gravity is strong enough to transform the primitives, each proton vacuuming up an electron with the weak force, flipping one of its up-quarks into a down-quark and making it a neutron, while ejecting a neutrino. With the electrons and their degeneracy pressure out of the way, the matter once again collapses in free fall until the neutrons can get no closer together due to their own Pauli exclusion.

This is the density regime of neutron stars, with the amount of mass required to overcome the electron degeneracy pressure known as the Chandrasekhar limit, the second limit between cosmic gas and black holes. Our sun’s mass is only 72% of the way to this limit, so it will never become a neutron star or a black hole. Heat again acts to postpone collapse to the neutron star regime; a star above the Chandrasekhar limit will only become a neutron star once it has cooled sufficiently. With even more mass, the neutron degeneracy pressure can also be overcome, melting the neutrons together into a quark-gluon plasma. The amount of mass required for that is known as the Tolman–Oppenheimer–Volkoff limit. Finally, with even more mass, the quark degeneracy pressure is also overcome, but the limit for that has not been named because its details are even murkier than those for the Tolman–Oppenheimer–Volkoff limit. Heat, electromagnetism, and the strong force all vanquished by gravity, the sufficiently massive object implodes into a black hole, which is beyond the description of our current physics.

It’s interesting to summarize how the relative strengths of the forces map themselves to events in Universal history and in the development of local pockets. The strong force is stronger than electromagnetism which is stronger than gravity. The first bonds that were formed in the Universe were strong bonds followed by electromagnetic bonds and then gravitational bonds. When gas clouds collapse to a black hole, the first limit lies at the frontier of heat and gravity, the next at the frontier of electromagnetism and gravity, where every human has lived and died, and the last at the staggered frontier of the strong force and gravity. Just like a humid atmosphere filled with water vapor spontaneously develops flakes of snow and droplets of water, so did the Universe filled with Dust spontaneously develop flakes of planet and droplets of star in swirling galactic storms.

Footnotes

1.  The strong force is very strange for many reasons. Physicist still haven’t characterized it completely.

2. See Buridan’s ass.

3. See Schrödinger’s cat.

4. See “clinamen.”