science

Here are the rules of the mathematical game we call the Standard Model. The rules aren’t set entirely in stone, and in fact we know that the model we’re working with at the moment isn’t exactly right. But it’s pretty close, and here are the rules of the game.

Start with an empty three-dimensional space. Which one? We’re not sure—remember, topologists have a whole catalog of three-dimensional spaces which look locally like . . . this. Cosmologists have done some work to figure out the shape of the universe, based on their own set of models and assumptions. It doesn’t really matter for our purposes. Let’s just say we’re working with the basic, infinite, non-curved three-space.

So you have a large empty space. At any point in the space you can place an infinitesimally small point-object called a “particle.” The space is continuous, so I really mean at any point. No square cells here. And these things we’re calling particles, don’t think of them as tiny, shiny orbs. They’re literally just points. They take up no space. They are mathematical points with zero size.

Not all particles are created equal: They have slightly different properties which determine how they’ll move. Whenever you create a particle, you have to give it a “mass” (a positive number) and a “charge” (positive, negative, or zero). And you can’t just pick any mass and charge—there are only seventeen legal combinations of mass and charge to choose from. We call these combinations the seventeen fundamental particles, and we give each one a cute name like “charm quark” or “tau lepton.”

When you press play, what happens to the particles? They move through the space and interact. Like any automaton, there are precise computational rules which tell you what each particle will do next. Generally, they move very quickly and in straight lines. The only exceptions are interactions: when a particle decays, or when two particles come really close together. Then we have to consult our handy lookup table of interactions to find out what happens next. Depending on the identities of the interacting particles, they might collide and scatter off in different directions, or combine to form a single particle, or (if they come at each other fast enough) they may spew out a jet of new particles.

If you’re curious, here’s a list of all fundamental particle interactions under the Standard Model:

The first one shows, for instance, an electron absorbing a photon and changing direction. These interactions can also run in reverse, e.g., an electron can also emit a photon and change direction.

I’m being vague on the details, but I don’t have much choice. This isn’t the Game of Life, where you just count up boxes to find out what happens next. The exact rules for particle interactions in the Standard Model are absurd, to be honest with you. The calculations involve continuum-sums and imaginary numbers and coupling constants and all sorts of ridiculous math that grad students in physics lose sleep over. It’s a systematic process, but not a neat and simple one.

To save you some time and tuition money, I’ll just give you the quick rundown. Here’s a rough description of what you’ll see if you scatter some particles across space and run the simulation.

In the first instant, there’s a huge burst of activity. Most of the seventeen types of particles are unstable, and they almost immediately undergo decay interactions, splitting into smaller, stable particles. After this initial explosion, you’re left with only a few different types of particles, and only three you really need to keep an eye on: up quarks, down quarks, and electrons.

Next, as time creeps forward, patterns emerge. You start to see the quarks clump together in threes. There’s no law of the Standard Model that says quarks have to clump in threes, but that’s what happens. Trios of quarks interact with each other in a way that keeps them huddled up like that. As in the Game of Life, stable structures start to form over time by repeated applications of the same base rules.

In fact, this tendency toward trios is so strong that after the early excitement dies down, you pretty much never see a quark alone. Always in threes. Sometimes they’ll be clustered in sixes or nines, or any multiple of three, but most of the time it’s just three, flying together in a straight line. At this point, the word “quark” isn’t that useful anymore. You’re better off talking about the three-quark chunk as a whole, to save breath. So we coin some new terms. Two up quarks and one down quark, that’s a “proton.” Two down and one up? “Neutron.”

Then what happens? More patterns and regularities emerge from the interaction rules.

As you watch, you notice positive charges and negative charges drifting together, while like charges drift apart. Again, this isn’t in the rules. A particle over here doesn’t “know” the charge of a particle over there. They just interact with other particles in their local surroundings, getting rerouted as they do. And these reroutings have a bias that adds up over time: Positives move gradually toward negatives and away from other positives.

This is happening very slowly, though, so let’s speed up the simulation. Now this slow drift looks like a sharp tug. You see an electron (negative) falling quickly toward a “proton” (net positive). It speeds up as it gets closer, so fast that it actually speeds right past the proton. As it zooms away, it’s slowed down, pulled backward now by the proton, until it changes direction altogether and does another flyby. And on and on, buzzing back and forth around the pull of the proton.⁸

You see this happen all over the space, anytime a proton and electron find each other. It’s such a common structure, you might as well give it a shorter name than “electron buzzing back and forth around a proton.” Some kind of name like “hydrogen.”

And sometimes, remember, it’s a bigger clump of quarks, a six or a nine or more. These cases are rare, but they do happen, and these bigger clusters pull even more electrons into their orbit. We can give each of these little systems a name, depending on the total amount of charge in the clump, names like “oxygen” and “chlorine” and “gold.”

Maybe you already know what happens next. Speed up the clock even more, until the electrons are a blur, and you’ll see these entire systems (let’s call them “atoms”) are drifting slowly through space. Sometimes they drift past each other unbothered, but sometimes they stick together and start drifting as a unit. You’ll notice that hydrogen and hydrogen love to drift together as a pair, while oxygen prefers to drift with a hydrogen stuck to either side.

We still haven’t introduced any new rules. This is the same simulation playing, over longer and longer amounts of time. Every time we observe some “new” phenomenon, it’s always still explicable in terms of the base rules. Bonding, for instance? That’s electrons following their interaction rules. When two hydrogens are close together, their electrons naturally start orbiting both protons, holding them together. Only when you zoom out and speed up does it look like a new rule: “Hydrogen travels in pairs.”

You probably see where this is going, so let me get straight to it. These new mega-structures—“molecules,” let’s say—also behave in certain predictable ways, and they sometimes form gigantic mega-molecules: fats, proteins, lipids, ribonucleic acids, a whole zoo. And these each have their own characteristics and behaviors, and they sometimes form even bigger structures called organelles, which combine to form even bigger structures called cells. (Zoom out, speed up.) Some cells are off by themselves, while others interact with each other in units called organs, which in turn interact with each other in units called organisms. Some organisms cluster together in social groups or institutions, which cluster together to form classes or tribes, which interact to form an entire society. When societies interact, that’s called history—and that’s about as far as I can take this story.

Because it is a story, isn’t it? Clearly I’m overplaying my hand here. No one’s ever run a physics simulation that actually generated human societies, or even basic cell structures. How could they? It’s not possible. The number of things to keep data on is literally the number of particles in the universe, so there’s simply not enough room.

It’s a story, yes, but it might be a true story! At the very least, it’s a story with a lot of true elements to it. Each step in this chain is inferred from some very successful scientific model. Chemists believe that water is made of two hydrogens and an oxygen, and that theory hasn’t made a wrong prediction yet. Behavioral economists believe you can explain people’s economic behavior in terms of psychological and neurological factors. It’s like a long relay race, where each field of study takes over for a lap.

Still, it’s entirely reasonable to believe this isn’t the whole story. There are gaps in our understanding, which you may find suspicious. No one can really say they know exactly how human behavior arises from electric flashes in neuron circuitry. Artificial intelligence makes the idea plausible, but we haven’t worked out the precise mechanics. You can take this as an opening to argue that there’s something else going on here, some secret sauce that gets added at the level of human brains, which can’t be explained in terms of the interactions of quarks and electrons.

Most math-oriented people I’ve talked to, though, seem to be broadly of the mind that something very close to this story is true. They believe the gaps are incidental and will eventually be filled in. So much has already been explained in terms of simple mathematical models: the motions of the stars, the diversity of life on Earth, natural disasters and the weather, the formation of the entire solar system. Why should we think the rest is any different?

Philosophers call this worldview “naturalism” or “scientific naturalism” and it’s worth thinking about the implications. If this is true, if scientific naturalism is correct, then all of reality obeys strict mathematical rules. The entire universe must be identical to some carefully calibrated automaton. Everything going on around you, not to mention inside you, is a direct mathematical consequence of the laws of nature plus the initial configuration of the universe.

Which is a pretty trippy thought.

It raises some big philosophical questions, to say the least. If you buy into some version of this naturalist framework, here are three things to wonder about on your next commute.

Are these mathematical rules actual, bona fide Laws of Nature, somehow governing the progression of the universe? Or does the universe exist and change in time as a brute fact, and these “rules” are merely patterns we’ve found in it?

In either case, why these rules? They seem so bizarre and arbitrary. Why should this universe exist and not some other one? Does every conceivable mathematical universe exist in the same way this one does? Or are we somehow special, uniquely selected among the possible worlds to be instantiated, concrete, real?

And even if it is all just mathematical rules, and even if we do live in what’s essentially one giant, ultra-complex simulation, the age-old Big Question remains unanswered. Is there any sort of intention, design, plan, intellect, foresight, desire, warmth, or care in the programming?

I doubt we’ll find answers to these questions anytime soon, and they may not even have “answers” in the standard sense of the word. At the end of the day, all we have are models we’ve invented, and each model is limited to a certain scope.

This model, the Standard Model, certainly aims higher than music theory or economics. It makes numerical predictions with a precision exceeding ten decimal places, which repeatedly come true in experiments. It offers a unifying explanation for nearly all phenomena observed in nature, collating and deepening the various pictures given by other models. And it comes bundled with a comprehensive story, this vision of reality as the smacking together of zillions of tiny dots, which many people find beautiful, humbling, even awe-inspiring.

But it’s not everything. It has blind spots. I mean, the current Standard Model doesn’t even explain gravity! (String theorists are working hard to fix this embarrassing oversight.)

Maybe it’s not surprising that we should be able to find a mathematical object that so closely mirrors our reality. The ultimate aim of theoretical math is to collect and analyze all possible models, all possible structures and shapes and systems, all forms of logic and argument, under the same roof. It attempts to translate every conceivable and unconceivable thing into a common language, one universal set of notations and techniques. It’s a project that on its face seems outrageous and impossible. Its continued success in explaining and predicting the phenomena of our daily lives is a curious blessing we don’t fully understand.

At the very least, it’s an interesting thing to think about.