Maps, models & power laws #

Every theory is a map, and every map lies a little. These notes are about that gap — and about why small errors, compounded through a network of models, end in power laws.

Don’t navigate the world with a map #

Maps can be dangerous. They give us false comfort and fool us into taking them for reality, when in fact they are always a reduction and a distortion to some degree: one large object cannot be compressed ten thousand times and still be identical to the original.

Put a toy car next to a real car, or an architect’s maquette next to the real building. You cannot live in the maquette; you cannot go shopping in the toy car. You know the difference — so why does everyone, myself included, keep confusing the model (the opinion of a thing) with the real qualities that can be measured objectively? Why do we believe in dogmas — in economics, in psychology, in the theory of xyz? Perhaps because some things are complex and others simple (the toy car versus a theory of time travel): one is easy to picture, the other not. But in a primitive environment we did not worry about the speed of time travel; we wanted to survive. Put a caveman in a time machine and send him to work on Wall Street, and he would go mad — yet the caveman and the banker carry the same genes, except that the caveman may well be far smarter than some of them (whether humanity is getting smarter is another topic). The caveman would distrust maps and models; he would use the heuristics of reality. We are baffled by the noise, like that caveman, and we make all sorts of mistakes in this complex system.

We use maps and models all the time without knowing it.

By “maps” I mean maps of terrain, but also every kind of mathematical formula: Black–Scholes, the efficient-market theory, Freud’s theories, and the rest. Basically the world is random, and some theories happen to fit for a while, by coincidence.

Cycling with a bike computer that has a map function, I once got a new feature: the device began warning of potholes on the road — useful for road cyclists. But it also warned of heavy traffic and, most bizarrely, of a dog in one particular spot. Someone had marked a dog on the map, maybe two years ago; the dog may be dead by now, yet you still get the warning. That is how much our models and maps resemble one another: we run on economic forecasts and never expect that one day, instead of the dog, there might be an elephant escaped from the local circus — and in a different spot.

Notes #

Nothing is linearly connected #

Nothing in the universe is linearly connected. Everything is R², log n, and the like — from atoms to social units to whole countries.

Categorization #

Categorization always serves to minimize the real complexity of things.

The individual and the group #

Taking examples from life — “X works this way”, in isolation — is like taking the behaviour of a single cyclist for that of the whole peloton. The individual and the group are not the same thing; a simple element and a complex system behave nothing alike.

We don’t see the nonlinear #

We don’t see the nonlinear — but why? Is it because we are ourselves nonlinear? To be developed.

Borrow metaphors from another domain #

Our thinking changes depending on the domain. That is why it helps to borrow metaphors and analogies from another domain.

A Martian on Earth #

A good mental exercise: imagine a Martian lands on Earth and observes. If he measures a hundred people to see how tall humans get, he learns roughly how tall we are. But if he checks bank accounts, the answer depends entirely on whether one very rich person happened to be in the sample. In social domains like “success”, this effect rules.

Advertising: labels upon labels #

Advertising attaches a label to a concept, and so sells it. Take a television advert for Valais with the slogan Gravé dans mon cœur; then another advert, selling cheese, reuses the same slogan — so that any thought of Valais becomes automatically, subconsciously attached to the idea of this mountainous region of southern Europe. That community is part of others (the cantons) that make up a country, so that whoever thinks of Valais has the food and its emblematic product auto-attached. That is the trick of advertising, and perhaps of much else: slyly attaching one brand, one label, to another. In real cases the association becomes so close as to be imperceptible — Egypt and the pyramids, Paris and the Eiffel Tower, Rome and the Colosseum — even though these things are far more than their representations.

A model is always only an approximation of the thing itself. So we attach, or merge, one label to another to build a quasi-system of labels, each with its own personality — probably through nonlinearity, logarithmic growth, power laws, antifragility, convexity, optionality. And these systems, being networks of models, have simple nodes that are themselves just individual models. In the end everything is a model; nothing is really as it is described, because of the model. Even human language, being a model, has the limitations inherent to any model — the difficulty of expressing complex feelings, what one feels before a work of art, or during sex; these cannot be put exactly into words. So it is a limitation of a model, then of other models that are approximations of approximations, and this ends in power laws: each model, prediction or prophecy depends on prior models and histories on which it rests, and so on, each error feeding into the next. Since each node is raised to a power, any small perturbation at the far end of the network can arrive with the force of a tsunami. This might partly explain black-swan events — at least their amplitude, though not yet how they form, how those small irregularities arise, or whether by chance; and whether chance exists at all. These are questions that need deep deliberation, or better, empirical tests to falsify the conjectures, in the Popperian sense.

In short: ads are labels on labels, models on models, illusions on illusions, all raised to a power — each made of others, which still rest on their predecessors. Some of them are even heuristics, and this is the overlapping ground I sense: we may be building new heuristics that reality will falsify, or not. If a heuristic survives the clash with reality — the ultimate test — and survives long, it becomes a rule of thumb, folklore. Unless we take heuristics as innate. A heuristic can be old, but perhaps we build our own: don’t touch the hot stove, as a child. That one could not have been innate — the child had to get burned, and the mother’s warning went unheeded, or curiosity won. So we learn through pain, by trial and error, like the child and the stove. A heuristic can be personal, learned through pain and then followed — like the rules of the Church — or innate, some bodily animal movement, as when you touch something hot and pull your hand back before you even think.

Why power laws rule the world #

Why do power laws rule the world? What is a power law — or “a law”, or even a “word”? It is the representation of a concept. And a concept, or conjecture in Popper’s sense, is always a model, a doxa — perhaps because the world is multidimensional and our brains are three-dimensional, so we cannot process it, just as a worm cannot imagine flying since it lives in a two-dimensional world. So a concept like global warming is made of smaller concepts — CO₂ and its link with the ocean, the land, the temperature — all sub-theories, and so on recursively, perhaps ad infinitum. Each of these theories, being doxa like any human theory, is the cause of another, so they are linked like Christmas-tree lights: the current runs from the first bulb to the next, and so on.

Imagine each bulb shifted the phase — starting at 50 Hz and adding 5% each time: 50 × 1.05 = 52.5, then 52.5 × 1.05 = 55.125, then 57.881 — which can be written as D, the deviation of the conjecture from reality: D = model ^ prob(error). You see that a model is always made of smaller models, and those of smaller ones still. Therefore: (A) no model is a monolith, but rather linear — or a network, though I think linear is closest — so that a change affecting every node’s error rate has an exponential effect. (B) As with humans, where the group and the individual are completely different animals, any model is a complex system, being made of sub-models; even a well-built model fails to grasp the nature of a model, because each level of sub-models is still treated as a monolith, and we rarely know how deep the recursion runs to reach the root, or the principle — which for Popper did not exist. (I write this in 2025, and my views may have changed many times over my life, as I tested new theories, most of which were refuted, and so eliminated by nature.)

Back to the central point: we live in a power-law universe. The universe does not care what we think of it — and we are part of it and it of us, since we are a species of organic creatures living on a planet. What are the odds that you, dear reader, were born on this planet, in your body; that this is the only time you will be born; and that time runs fast, so you have little of it to accomplish your task — or even to find it, which may be most of the task, the execution being the smaller part? So go and look; and if you are one of the lucky ones, go and finish your task, so that after leaving this planet and this body you may have the satisfaction of having used well the time the universe gave you, and of having made the world a little better — a positive contribution to the lives of other human beings. And remember the power law: model times model, model squared — which equals idiocy squared.

One more general point: since a model is the root of the error rate, which root do we take? Five, because we reckon a theory has five causally linked theories? What if they form a complex system, wholly nonlinear? In short, it is an estimate. So the more complex the theory — the more sub-theories, like fifty in some intricate mathematical or medical problem versus three — the more the error rate rises exponentially, because the number of sub-theories serves as the root of the error rate. That, I think, is how power laws work, and why the world runs on them — “the world” being itself a mental concept: a forest to an Indian, a planet to most, the edge of a galaxy to an astronomer. This thing “world” is made of an infinite number of conjectures, theories, laws, lore, heuristics, reflexes. Any law, rule or theory is the observation of a recurrence, of cyclicality — pulsars sending their pulses, or a kick in the ankle always making the victim cry out. How many such basic laws are there in our social universe? An infinite number? All of this makes the world chaotic and power-law-like — though that is only our observation of extreme deviations from what we are comfortable with. The universe does not care whether we have laws: gravity, for one, will always be more real, and win any contest where it can. So the world is a cosmic soup of ideas, beliefs, heuristics, religious and political systems, all interacting in unpredictable ways.