What Is Complexity? Notes from a Conversation

We hosted a discussion on a question that turns out to be harder than it sounds: what is complexity? Four contributions, four angles — structural, epistemological, generative, and predictive. What follows is a record of where they agreed, where they didn't, and what remains open.

Henrik: complexity as structure across scales

Henrik's starting point is that reduction has a cost. When you simplify a system to make it tractable, you discard information — and you often cannot know in advance whether that information was the information you needed.

His picture of complex systems is architectural: hierarchical encapsulations at various scale levels, each with its own regularities. This suggests working from higher-level abstractions rather than grinding downward, and it suggests decentralised protocols as a way to attack problems that no single model can hold in one place.

Adam: bounded models in unbounded worlds

For Adam, complexity is an epistemological concept before it is anything else. It constrains economists, mathematicians, and organisms in the same way, because it is a fact about the relationship between a model and its world rather than a fact about the world alone.

He is deflationary about several of the usual candidates. Complexity is not P versus NP, not compressibility. What remains is Gödel and the Halting problem. Most things called complex are merely complicated. Emergence, as a term, is not very useful or definable.

What remains is Gödel: a model cannot fully account for itself. From this follow exogenous effects — things that happen because the model never incorporated them. Every model has a "valid until" date and is bound by resources. Y2K is the mundane version: money spent to pivot machines whose clocks had quietly expired.

The interesting move is what to do about it. Adam points to metamodular systems: strategies and metamodels that let a system detect when its current model has run out and pivot to another. Biology already does this. The Red Queen hypothesis describes co-evolution as an arms race in which extinction risk stays roughly constant over time — no strategy wins permanently, but better strategies exist, and living things have evolved the capacity to step outside their own models. As he put it: homeostasis is a risk exoskeleton.

Utku: complexity as process

Utku shifted the question from what complexity is to what it lets us do. What are the conditions under which emergence happens? His answer is that the ingredients matter less than the process — complexity is not a property of the substrate.

The payoff he is after is life. Understanding complexity is a route to understanding living systems, and from there to the question of how to create life, or at least life-like programs. Kauffman's artificial life work is the reference point.

Arne: complexity as the limit of prediction

Arne came at it from prediction, and from the question of whether a sufficiently powerful intelligence could predict what you will do next.

Start with determinism — Laplace's Demon, complete knowledge yielding complete foresight — and its softer variant, where single outcomes are replaced by probability distributions but the flavour of determinism survives. Chaos theory takes the first bite: in chaotic systems, the information you would need is available in principle and inaccessible in practice. Computational irreducibility takes the second. For systems like Rule 110, the only way to know the state at step n is to run n steps. The model is the system. This gives a working definition of a model — anything that represents a process in fewer steps than the process — and two metrics to judge it by: effectiveness and efficiency.

Then autonomous agents enter, and the character of the problem changes. Taleb's turkey is fed reliably all year and grows more confident in the pattern until Thanksgiving. Its model fits every data point available to it and is catastrophically wrong, because the relevant information lives outside its position in information space. Inaccessibility stops being a practical limitation and becomes a property of the system.

The climate thought experiment pushes this. To predict actual global temperature in 2100, you would have to model human decisions; to model those you would have to measure brains; and even granting the measurement, two problems remain — the observer problem, and the fact that publishing the prediction changes the behaviour it predicts. The only prediction immune to this is a self-fulfilling one, which is not worth making.

From this Arne draws three criteria: the situation changes before you can analyse it in full; you cannot get the crucial information without disturbing the system; and predictions have unpredictable effects on outcomes. Games of information — intelligence agencies, markets, a game of Go — are the general case, each party modelling the other modelling them, regress without a fixed point. Or, in his phrase, meeting Gödel's incompleteness theorem in the streets of Vienna.

Convergence

Adam and Arne arrive at the same place from opposite directions. One starts from the logic of models and finds Gödel; the other starts from the practice of forecasting and finds him too. Both end at bounded models in an unbounded world, with a "valid until" date attached.

Henrik's warning about information lost in reduction is the same observation at the level of scale that Arne's effectiveness–efficiency trade-off makes at the level of computation: compression is what a model is for, and compression is what makes it fail.

And Utku's process-over-ingredients converges with Adam's biology-as-model-pivoting. Both locate the answer in dynamics rather than in substance — in what a system does when its current account of the world stops working.

Disagreement

Utku pushed back on putting the halting problem and Gödel's incompleteness at the basis of complexity, which is precisely where Adam and Arne place them. The objection is worth keeping open: it may be that these results mark the boundary of complexity rather than constitute it.

Are computer viruses complex? Are they alive? Utku: no — they cannot improve themselves. Whether self-improvement is the right criterion went unresolved.

So did the status of "emergence." Adam thinks it barely definable; Utku organises his research question around the conditions that produce it.

And Henrik's position on reduction sits against the room's more casual assumption that reduction is understanding.

Recurring threads

Complex adaptive systems are irreversible and path-dependent; time is baked in, not a parameter added afterwards. Strange attractors as probability density clouds. Taleb twice — the turkey, and the injunction to predict impact rather than outcome. And one proposed empirical test, which we liked: give a set of agents limited compute, put them in an environment where they can make money, and see what happens.

Tentative conclusions

What the room converged on: complexity is best understood as the gap between bounded models and open worlds. Reflexivity, observer effects, and computational irreducibility are the mechanisms that hold the gap open. The practical question is not how to close it — it does not close — but how a system detects that its model has expired and pivots.

What stayed open: whether emergence survives as a concept, whether self-improvement is the criterion for life, and whether decentralisation solves the problem or restates it in nicer terms.