1. Divisibility, Repetition, and the Dust

If you take a casual walk or snatch a few glances at your surroundings, you will probably not notice something trivially obvious. You will probably not notice that basically anything you see can be divided into parts. Of course, in retrospect that’s a forthcoming fact, but did you actually think it? Regardless, if we consider the idea of such division, one question arises immediately: can we continue dividing forever? If you were to pick something to test the question, you’d probably pick something brittle roughly your hands’ size. As you divide it and divide the divisions, your hands would quickly fail you after the first dozen or so recurrences, and your eyes shortly after as the parts become too small to identify, let alone handle. Technical difficulties would get in the way of an answer for you just as it did for the entire human species for millennia.

The question was a hot topic among philosophers, even long before certain philosophers became known as “scientists” and then “physicists.” The ancient Greeks held a variety of views. Plato and Democritus argued that if you divided far enough, you would eventually find a flurry of geometric particles like earthy cubes or fiery tetrahedrons stacking on and flowing past each other. Aristotle pointed out that this would require a void between the particles, and since “nature abhors a vacuum,” he postulated that things in the World were made of continuous elements like earth and fire that blended together in various proportions and were imbued by “form” only accessible at larger scales. The ancient Indians also participated in the debate, although it’s unclear whether there was any direct dialogue with the Greeks. In the Nyaya and Vaisheshika schools of orthodox Hinduism, divisibility ended with particles that combined two to a pair and three pairs to a triad. The Jain school, a Hindu heterodoxy, also argued elaborately for particles each with their own smell and color, among other properties.

More recently, philosophers like Leibniz presented the possibility that there might be no end to the divisibility but without it ending in continuity either. “[E]ach portion of matter can be conceived as like a garden full of plants, or like a pond full of fish. But each branch of a plant, each organ of an animal, each drop of its bodily fluids is also a similar garden or a similar pond,”1 he explained, foreshadowing future conjectures that the smallest entities might just be universes in their own right, and our entire Universe a speck in another. The debate could have continued forever in a cacophony of reasonableness or lack thereof, but fortunately our species subdued the technical difficulties, and the actual character of the minuscule trudged an evidenced trail into our minds and theories.

We are lucky to stand on the shoulders of giants. In the twenty-first century, we stand on an entire dogpile of them, or perhaps a dogpile of dogpiles. We definitively learned in the nineteenth century that the divisibility ends with a list of indivisible particles, each a pure element of chemical activity like hydrogen or nitrogen. But wait! If you’re a student of modern chemistry, you should find that statement in egregious error. This error, however, is written directly into scientific vocabulary. When developments in twentieth-century physics showed that the allegedly elementary atoms were divisible into even smaller particles, it became clear that nineteenth-century chemistry had not found the end of divisibility and had misnamed the atom—”indivisible” being its Greek meaning.

We are now aware that atoms violate their etymology and can be divided into nuclei and the electrons orbiting them. We have found that the nuclei can be further divided into protons and neutrons, both of which divide yet again into three quarks of two different types, up-quarks and down-quarks. The two types mix and match so as to give the proton a +1 electric charge and the neutron a 0 charge, and they fortunately can no longer be divided. The electron has an electric charge of equal magnitude but opposite sign as the proton, and since in an uncharged atom there are the same number of electrons in the electron shells as there are protons in the nucleus, their charges cancel out, and the whole assemblage is electrically neutral.

This picture of the minuscule is readily comparable to the fantasies of the Greek and Indian atomists. The details are, of course, radically different, but the overall projects harmonize in intent. We should ask, however, whether the modern picture is the last word; have we actually found the end of divisibility? Physicists think so, which is why they call electrons and quarks “elementary particles” without etymological disclaimers unlike with “atom” or “element.” Their reasons are complicated; in dry experimental terms, evidence hasn’t been found to the contrary, but in more profound theoretical terms, the math works out beautifully if they are indivisible. The list of particles we know fill a list of slots in the mathematical paradigms of the symmetries and regularities of dimensionality and change. To explain further than this would be to delve into theoretical physics, but that is not this book.2

There are several more elementary particles than the ones I have mentioned, but most of them are unstable and decay very quickly into the conventional ones. The ones I have mentioned are also all of a type called “fermions,” which incorrectly but effectively means they stay around a long time and are made to move around and stick together by other elementary particles, known as “gauge bosons,” which carry forces. Some gauge bosons are the glue that keep protons and neutrons and nuclei together. Another gauge boson is the photon, which carries the electromagnetic force responsible for light and the chemical bonding of electrons to and among nuclei in an atom or molecule. An important property of fermions is that they cannot occupy the same location, a property known as the Pauli exclusion principle, whereas bosons can, and this restriction will have implications for how fermions pack together.

We see that our Universe supplies a very specific answer to the divisibility question. The divisibility is finite, and it ends with particles.3 From this point forward, I’ll refer to those elementary fermions that stay around a while as physical primitives, because I don’t care to wed my arguments to any specific rendition of fundamental physics except for particleness; and I’ll refer to the four forces of the Standard Model of physics—electromagnetism and gravity among others—as carried by the elementary bosons4 as the interactions of physical primitives. Importantly, the intuitive, classical picture of discrete particles bouncing around like billiard balls because of whatever reasons is sufficient. So if you know or vaguely remember a few things about molecular bonding, but nothing at all about wavefunctions, then your physics intuition will serve as a perfectly good foundation. I’ll also refer to all the physical primitives in the Universe collectively as the Dust.

If you take another walk or glance, there is another trivially obvious fact that you will probably not notice either. You might miss that basically anything you can see repeats many times in the World—the trees, one after the next, lining the street; the power strips under your desk, violating the fire code. If we divide, we curiously observe that repetition often continues to hold. Each tree has many leaves and branches, each power strip has many sockets. Each division gives a list of parts which may repeat. It could easily be the case that the number of kinds of things grows with each division. Leaves and branches are not sockets, and plant cells are not wires. At the bottom, the physical primitives of trees could be different from the physical primitives of power strips and everything else. It is not a surprise to us that they in fact share physical primitives given our familiarity with electrons and protons and neutrons, but perhaps it should be.

In the end, we see that both things that can and things that cannot be divided repeat many times in the World. Why do they repeat? For the former, it is clear that there is something in the nature of certain arrangements of the Dust that make them good at being in the World. For the latter, this explanation obviously does not suffice, because a primitive cannot be an arrangement of other primitives. This book leaves the latter question as to the proliferation of physical primitives alone, but will try to attack the former question exhaustively in a regimented but flexible framework that will touch many scientific disciplines along the way.


1. The Monadology

2. See Zero to Infinity by Peter Rowlands.

3. You may object that I am ignoring wave-particle duality. Wave-particle duality is encountered below the quantum-to-classical transition, and not above it, so I won’t address it in this book, as my main purpose is not to discuss quantum mechanics but the classical realm. The differences between the quantum and the classical domains are profound, but they are beyond the scope of this work even while being indispensable for my larger project. See Appendix N. For now, suffice it to say that treating fermions straightforwardly like particles distorts their character very little when viewed from above by us classical beings. Alternatively, you may object that I am ignoring String Theory; see chapter 10 and stay tuned for my next book.

4. Or otherwise in the problematic case of gravity.