Smoothness
TLDR: Embracing the principle that reality is smooth, not discrete, leads to more nuanced thinking and better scientific intuitions.
Prerequisites: None
Zeno’s paradoxes are wild. It’s easy, with the benefit of knowledge and hindsight, to see them as trivial, but I claim that there’s something highly surprising about reality that these thought experiments highlight: smoothness.
Consider, by contrast, the worlds of basically all games. This is most obvious in board games, where the space is divided into squares or tiles, but it’s also true in video games. Despite the appearance of smooth motion on the surface, all video games1 represent the positions and velocities of their entities using digital bits that have a maximum precision. This means that in a game, time is divided into discrete steps and space into discrete positions; at any given moment the characters effectively move a whole number of spaces on an extremely fine-grained grid. At the extremes, this can lead to unexpected behavior and glitching, especially when things collide.
By contrast the world is extremely smooth.2 Not only is it true that between any two distinct points in spacetime there is a distinct midpoint, but there are also uncountably many other points in that interval that can’t be reached through any process of arithmetic on integers.
We can see nature’s preference for smoothness over discreteness again and again in everything from the way objects gradually move from place to place, to the way that their velocity changes gradually, to the way that things heat up or cool down gradually, to the way that things can be cut and subdivided, to the way that there is a continuous path between any two shapes or images, to the way in which water can freeze and boil, to the mathematics of circles and other natural shapes. Despite not having access to slow-motion cameras or the internet, even the ancients had hundreds of examples of this pattern.
And yet, they mostly didn’t take it to heart. Leucippus and Kaṇāda independently postulated that the world was composed of eternal, indivisible particles — atoms. And more generally, most people held that there were distinct categories of things — fire as distinct from water, men as distinct from women, rocks as distinct from animals, and so on — and that these categories were true divisions in reality, rather than an imposed and imperfect framework. Even now, most people do not appreciate the principle of smoothness and how far it goes in characterizing the world. There are an uncountable number of mid-points between all true things, regardless of whether they’re genders or “fundamental particles.”
Humans Love Simple Patterns
To operate in the world we look for patterns and structure, especially simple ones. One of the simplest and most useful cognitive strategies is to carve reality up into discrete chunks and categories. “This is a cat. That is a dog. She is Mom. He is Dad. This is the first carrot. That is the second carrot. This stone is not a carrot. I won’t count it. There are two carrots, overall.” This kind of mental motion is so fundamental that it’s hard to even notice it, and I would never suggest giving it up — it’s simply too useful. But I think it’s worth noticing how, when thinking along these lines, we’re making things up. There isn’t actually an underlying, objective truth about how many carrots are present. “Carrot” is an invention that lives in the mind, and only imperfectly matches reality.
Sometimes this imperfect match is clearly good enough. Even if it’s true that there’s no fundamental distinction between carrots and rocks (e.g. they are both made of electrons and quarks), it is extremely rare for one to change into the other or to encounter something that’s halfway between, and so the distinction holds up for day-to-day use.
The simplicity of these kinds of ontologies is part of their utility. If I were to suggest that you should drop the notion of “carrot” as a category, and instead force yourself to constantly notice the distinct shape and character of each object as a unique configuration of matter… Well, I wouldn’t begrudge you from ignoring me from that point on. It is true that carrots exist, and that notion is useful, but, simultaneously, it is not perfectly true.
Searching for Continuity
The principle of smoothness is useful in two ways:
It helps avoid map-territory errors and points to the complexities of reality.
It can steer us towards more realistic theories, both by:
asking what a mid-point between things would look like, and
asking why we don’t see that mid-point in practice.
Consider the notion that cats and dogs are “different species” for example. I claim that an ancient philosopher who truly internalized the principle of smoothness would respond to this idea by saying: “I agree that we can, in practice, distinguish between cats and dogs. But I bet a part-dog-part-cat animal is theoretically possible, even if there’s some important reason we don’t see any such creatures walking around.” And that philosopher would have been right! They’d be extremely, prophetically right! Thousands of years before Darwin, they would be reaching for an origin of species that explains why cat-like creatures split from dog-like creatures.
Similarly, some people think that there is a distinction between living and unliving things. Our ancient philosopher would again be prophetic in guessing that there is a continuity of semi-alive things in between, and that there’s no fundamental or discrete difference between life and non-life. The same insight can lead to appreciating the ways in which humans are a kind of animal, and the ways in which animals can be more like people than is generally recognized. And, more speculatively, it suggests that some (potential) beings are half-conscious, for all notions of consciousness. It predicts there are things that only possess a fraction of a mind or a thought. It predicts it’s possible to be partially dead. If humans “have a soul,” then the principle claims that it is theoretically possible to have half a soul (or to have 1/√2 souls!).
Attending to why mid-points don’t show up in practice can be a particularly rewarding line of inquiry. Why aren’t more things half-solid and half-liquid? Because these kinds of materials tend to only be stable at a particular temperature or with specific substances. What does temperature have to do with things? Which substances? Why those substances, and not others? The philosopher might not know how to answer these questions, but they can see their ignorance more clearly, and know to be searching for answers.
“Fundamental” Physics
Even as we acknowledge the advantages to adopting the principle of smoothness, it should be noted that over-zealous application can be dangerous. If our hypothetical ancient philosopher had been in dialogue with Leucippus or Kaṇāda, for example, they likely would have argued that the atomic theory of matter is wrongheaded, and that at the fundamental level, there are no firm divisions between things. But of course the world is made of atoms! They would’ve been wrong! Or rather, they would’ve failed to predict the discovery of atoms.
But, in fact, it turns out there isn’t a firm division between different kinds of atoms. We now understand that everything from hydrogen to uranium actually can be split into more fundamental stuff, and there are examples of weird, intermediate forms of matter, just as the principle of smoothness indicates.
But perhaps those in favor of atoms and/or elemental building-blocks simply jumped the gun? After all, we now think of electrons and quarks as distinct, fundamental particles. That’s not very smooth! The sharp distinction between different particles is antithetical to smoothness, as is the very notion of a countable “particle” that exists at a specific point. Smoothness says we should expect there to be parts of space that contain “half a particle” or bits of matter that are part-electron and part-quark.
And it turns out, that if you get really deep into theoretical physics, this smoothness is quite likely correct! The notion of “a particle at a point” starts falling apart in quantum mechanics, which tells us that reality is actually a smooth continuity across “configurations.” But the configurations in quantum mechanics repeat the idea of distinct particles at specific locations, and we can correctly infer that this means that quantum mechanics isn’t fundamental. Moving deeper, the best theory of reality that we currently have is probably quantum field theory (QFT), which replaces the notion of discrete particles with smooth, continuous fields.3
I certainly don’t know what the ultimate nature of reality is, but I think the principle of smoothness can continue to guide us in finding even more fundamental theories. For instance, the field configurations in QFT are drawn from a Sobolev space, which (among other things) means they’re smooth. But just how smooth are they? This is an open question in physics. My guess is infinitely smooth! More speculatively, I would guess that “energy is quantized” not because intermediate energy levels are impossible,4 but rather that they’re naturally unstable and/or difficult to reach.
As I reflect on it, I can only think of one possible “exception” to the principle of smoothness — only one place where I see discrete integers in the fundamental fabric of reality and I don’t expect that’s a human limitation: the dimensionality of the universe. Whether we live in three dimensions of space and one dimension of time, or some eleven-dimensional alternative, I expect that it makes sense to to think of these dimensions as discrete and fixed.5
Utopian Smoothness
Scientific and philosophical education in Utopia centers around two complementary approaches. The first is seen in The Onion Test, which highlights that almost all knowledge is tentative and incomplete, and that there’s a deeper layer of understanding waiting to be had. The other is First Principles Reasoning, which seeks to find strong foundations for knowledge and understanding in the form of clean, elegant truths that can be applied over and over again. Utopians debate which principles are worth holding as central, but we can see some of the more important ones showing up as recurring themes on this blog.6
Smoothness is a principle that pulls a lot of philosophical weight, both in physics and elsewhere. It is usually taught as foundational to our view of reality, alongside determinism, relativity, and the relationship between space and time. Thanks to grappling with this idea from a young age, topics like irrational numbers, calculus, and advanced physics seem more natural to students in Utopia.
Smoothness also has a kind of cultural weight and place in Utopian discourse. Even when a natural ontology is obvious (e.g. species, genders, celestial bodies), Utopians like to push at the edge cases and explicitly flag the way that frameworks are imperfect. As a result, questions like whether “there are two genders” or “Pluto is a planet” get less traction at the object-level and instead tend to produce a common response of “yes, thank you for reminding us that all categories are imperfect.”
The Utopian emphasis on numeracy can be seen as another practical consequence of having internalized the principle of smoothness on a social level. Utopians love to measure things, and have a somewhat quirky affinity for real-valued measurements over discrete counts. For instance, it is more typical in Utopia to see groceries priced by mass, volume, or nutrients (eg calories7), rather than by count (even when they’re sold in fixed-size containers).
Okay, fine, Tennis for Two probably count as a video game that is smooth, thanks to being built using non-digital logic circuits. As far as I know it’s the only counter-example.
I’m using the word “smooth” in this essay in an analogous way to mathematics. That is, something is smooth if it’s not just continuous, but infinitely differentiable. Smooth domains are those where a path can be drawn between any two entities that consists of uncountably infinite intermediate states where the changes along the path are themselves smooth (and thus the changes to those changes are also smooth, and so on).
Smoothness also predicts that electrons can turn into photons and vice versa, which is indeed possible, just as we believe that (in a certain sense) all kinds of matter can (with sufficient time/energy) change into all other kinds. As far as I understand it, physicists usually think of this as a discrete change that happens instantaneously. My bet is that it’s not, and that there’s an intermediate state which is half-way through the conversion, both on the layer of the wave functional and on the level of configurations (if those turn out to be the right level of abstraction).
Arbitrary energy is, in fact, achievable in free particles and other systems in motion. (Just dial up their momentum.) As far as I know, the only notion of energy being quantized is when considering bound systems that appear at rest from the outside. One hypothesis it that it’s impossible to hold a system at an arbitrary energy because the excess energy manifests as radiation which can’t be bound. If this is right, I’d count it as a win for the smoothness principle, in that the energy-level discreteness would be more a property of how we categorize bound/unbound systems, rather than an actually fundamental aspect of systems.
Another point which I think is in favor of smoothness is the way in which spectral lines aren’t actually observed at sharp, discrete wavelengths, but in fact have some blurring (from a variety of factors) and can be significantly distorted by the potential gradient imposed by surrounding atoms. In other words, the characteristic quanta can change smoothly as the system changes, and are smooth in practice.
I also could be wrong about this! Perhaps there’s a sensible notion of a universe with 2.738 dimensions? I’m not sure what that would look like, though, and think it’s more likely that smoothness simply doesn’t apply at the meta-level of dimensionality.
Including the principle of being principled! Utopians are far more interested in applying a core set of foundational truths than most people in this world.
The universe, therefore causality, is infinite in time, space, and scale. No one talks about the scale very much, good catch.