Let’s Talk About Complexity…

Sean Carroll, entropy, and why American coffee is better than I thought

I really hate going to sleep. Something about the lack of experience seems so empty. So as it was getting late last night I made myself a coffee and put on a podcast – Sean Carroll’s Mindscape.

The topic was complexity. In part, how complexity behaves differently to entropy. And appropriately enough, the metaphor Carroll used? A cup of coffee with cream on top.

Now, I should pause and say: we don’t really do ‘cream’ in coffee here in Australia. I am even using ‘cream’ in scare quotes because as far as I can tell it isn’t really cream anyway. We use milk or just drink it black (I like oat milk, myself). So I had to imagine what he meant - a dark base of liquid, a pale layer on top, and a moment when the two begin to swirl. But even without the specifics of the liquid, the shape was familiar - one thing folding into another, and the moment before it all blends. The reason my interest was piqued, though, was not because of caffeine but due to ideas I’ve been banging on about, most recently in a post I called The Four Dance Moves of the Universe.

In it I described how there are only four basic ways the mind and the world can interact. Not four results - just four directions: the world can act on the world, the mind can act on itself, mind can act on the world and the world can act on the mind. That exhausts the roles: actor and recipient. Four combinations.

In that post I also suggest that each of those four interactions behaves a lot like a basic arithmetic operation. Subtraction, division, addition, and multiplication. And when you run those operations across large sets of numbers, each one leaves behind a distinct statistical distribution - a kind of mathematical texture. Those textures, I argue, don’t just show up in data; they show up in experience.

Carroll doesn’t frame things this way. But when he describes complexity rising and falling as the ‘cream’ and coffee mix (and when he reflects on how entropy always increases, but complexity only sometimes peaks) I realised we were tracing the same territory from different angles.

What struck me is how closely his account of coffee tracks the very structure I’d been working on - and how it helped me realise where our ideas brush up against each other, even if we’d name things differently. Let me walk you through the four moves again, this time using Carroll’s formation. (And to be fully transparent: ChatGPT helped me untangle a lot of this. It’s my favourite interlocutor now that most of my friends are bored of these ideas - but let’s be honest, it’s too flattering. So take any wisdom here with a healthy pinch of salt.)

Subtraction: the mind acting on itself
Carroll starts with a cup of black coffee, and a layer of ‘cream’ on top. This is a low entropy state where there’s a clear structure and an obvious order. There’s no stirring yet, just separation.

This reminds me of what happens when the mind narrows its own frame - when it turns inward. You’re not sensing the world; you’re just existing within a kind of blank attention, a low-information hum. When you simulate subtraction with large sets of random numbers - and ignore direction (only counting the size of the difference) - you get a long-tailed distribution. Most differences are small, but every now and then you hit a massive gap.

This shape is familiar, a slope of decay, telling the story of entropy as it’s felt from the inside. This is the mind, not yet engaged with the world, ticking away in uneven gaps. It’s the kind of simplicity Carroll refers to - where description is easy because structure is obvious. But the twist in my version is that this isn’t just physical. It’s cognitive. For me, subtraction is a form of structured absence, a way that entropy appears to consciousness before the world even gets involved. Really it is mind by another name.

I’m not sure Carroll would fully buy this. He’s famously physicalist - entropy for him is a macrostate feature of systems, not something that 'feels' like anything. He might use a phrase like ‘simple low-entropy states tend to require fewer bits to describe’. But isn’t a description already halfway to a phenomenology?

Division: the mind acting on the world
Now the poured ‘cream’ begins to swirl. Carroll points out that this middle stage - where the coffee and ‘cream’ are mixing but not yet uniform - is the most complex. It’s also the hardest to describe. Where is the ‘cream’ now? How much of it is still distinct? What shape are the tendrils?

This is what I call division - the mind carving up the world into concepts. It's when absence meets presence and tries to make sense of it all. Think of when you divide random numbers by each other, you get a sharp spike near one - most things are roughly similar, and only a few comparisons yield extremes.

That spike shows up in cognition too. Most of our judgements centre on things that fit our internal concepts. Only a few comparisons really stand out. This is also the mode where complexity - in Carroll’s definition - peaks. He even notes that a photograph of the swirling mix takes up more digital storage than either the fully unmixed or fully mixed versions. Why? Because there’s less redundancy, more detail and less compression.

Carroll wants to say complexity is essentially the amount of information needed to describe a macrostate. And division - as an experiential process - behaves exactly like that. It’s when conceptual structure multiplies, even as entropy continues its steady climb.

Addition: the world acting on the mind
Eventually, the ‘cream’ disappears and the coffee turns beige. High entropy again, but also simple, once more. Carroll describes this as low complexity because it is featureless. Description gets easy again because everything looks the same.

This is what I associate with addition - not as accumulation, but as what’s known as ‘modulo’ addition. The world keeps pouring in, but the mind resets after each step, adapting to change instead of storing the whole sum. Like a clock ticking past twelve, then starting again at one.

When you add large sets of random numbers modulo 1, the result is a flat, uniform distribution. That maps surprisingly well to experience. In my last post I spoke of a loud sound that quickly fades into the background and a bright light that your eyes adapt to. The sensation is still coming in - but the mind levels it out.

I assume Carroll would blanch at this way of describing things. For him, the final state is just entropy doing its physical thing - uniformity emerging as the system settles. But I want to say that the uniformity isn’t just out there in the world. It’s the interaction of mind and world - in the way the mind learns not to hold on. In this mode, the mind is passive but not absent. It receives, it wraps, and it lets go.

Multiplication: the world acting on itself
There’s one kind of interaction that doesn’t need a mind at all - and that’s multiplication.

Stars form. Planets coalesce. Life emerges. Culture evolves. And then, eventually, things decay.

This is pure multiplication. Small effects compounding over time. Feedback loops building structure. Multiply a bunch of random variables, then take the logarithm of the result, and you get the bell curve – the normal distribution. That’s the shape that emerges when compounding effects pile up over time. That’s how height, income, word frequency, rainfall, and so many other real-world variables behave. It’s also how emergence works.

Carroll doesn’t call it multiplication. But he describes it perfectly. And here, we’re aligned. Complexity can arise without a mind. It doesn’t need an observer. It just needs the right initial conditions, and time to run.

So where does that leave us?

It leaves us with a shared picture, seen from different sides. Carroll’s account is grounded in physics - entropy, microstates, and information theory. Mine is grounded in phenomenology - presence, absence, and experience. But we both end up circling the same idea: complexity is a real, structured thing that arises when systems interact. It has a shape: it peaks and then it fades.

But when systems interact, it’s a lumpy, bumpy mess. It leaves gaps.

The mind, I argue, is what it feels like to be inside one of those bumps - to live in the swirl between simplicity and collapse. Carroll might not agree. But I think the coffee cup hints at something bigger. That when absence meets presence - when the world stirs itself through thought - we don’t just get more information.

We get consciousness.