Maps, models & power laws #

Every theory is a map, and every map distorts of necessity: to shrink a thing you must throw detail away, and what is thrown away is chosen by whoever drew it. Our understanding is never perfect, and the imperfection is not laziness — it is the price of the reduction.

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 (an opinion of how a thing is) with the real qualities that can be measured objectively? Why do we believe in dogmas — in economics, in psychology, in the theory of xyz?

My guess is that thinking is expensive. Computation costs energy, and an animal that spends energy it does not need to spend gets removed. A model is a way of not thinking: it buys a usable answer at a fraction of the cost of the real one, and the saving is paid for later, by someone, in the rare case where the model is wrong. On that account the brain is not a poor instrument but a well-priced one — sized for survival, not for truth.

Silicon is not bound by that budget in the same way, which raises a question I cannot answer. Does a machine free of the energy constraint keep building better maps without limit? Or does it arrive at the same conclusion the brain arrived at — that past a certain point the extra accuracy is not worth what it costs — and stop? And there is a third possibility that unsettles me more: that a thing we built in our own image, on our own texts, inherits our distortions rather than escaping them, and is not a new kind of mind at all but our own map, enlarged.

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, religious dogmas, etc. 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. The relations that recur are the inverse square, the logarithm, the exponential — from atoms to social units to whole countries. And nature does not draw Euclidean shapes: coastlines, lungs, river deltas and ferns are closer to fractals, which is why the geometry we were taught in school fits so little of what is outside the window.

Some take this as evidence that the universe is a mathematical object. I would put it more weakly, and I think more honestly: mathematics is the best-fitting map we have found, and a map that fits astonishingly well is still a map. (See Truth, doubt & Popper for why I think mathematics is invented rather than found.)

Categorization #

Categorization always serves to minimize the real complexity of things. It is a Procrustean bed.

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.

But I have to be careful here, because I have said two apparently opposite things: that a crowd is stupid, and (in Antifragility & risk) that the system is wiser than any of its members. Both are true, and the variable that decides which one you get is independence. Many people trying things separately, each bearing their own losses, is the cleverest arrangement we know — that is a market, an evolution, a science. The same people deliberating in one room, watching each other’s faces, converge on whatever the loudest or most senior of them said first, and a committee of savants can reach a conclusion no member of it would have signed alone.

Which is why the good ones work alone before they work together, and why a well-built company is designed to keep judgements independent until they are made — on the assumption that sooner or later an idiot will be in charge, and the system had better survive him.

We don’t see the nonlinear #

We don’t see the nonlinear — but why? Is it because we are ourselves nonlinear, and an instrument cannot easily measure what it is made of? We expect technology to advance by even steps and it advances by compounding, in every direction at once, so that the change of a decade looks impossible from the beginning of it and obvious from the end. A thought 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 one — the trick is not cleverness but relocation, taking a problem out of the room where everybody already knows how it is solved.

It also explains a failure I keep meeting. Where a discipline judges work only by the approval of others inside it, the map stops being corrected by the territory and starts being corrected by the neighbours. Nothing there is wrong with the people; the feedback is simply attached to the wrong thing. A plumber finds out he was mistaken when the pipe leaks. Take away that kind of contact and a very intelligent person can spend a career refining a model that has quietly stopped touching anything, and be rewarded throughout. So the question to ask about anyone’s claim, my own included, is not how credentialed they are but what happens to them if it is false.

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 canton of the southern Swiss Alps. The canton is part of the country, so whoever thinks of Valais gets the food and its emblematic product attached without noticing. 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 one label to another and build a whole system of labels, each with its own personality, and the systems are networks whose nodes are themselves nothing but smaller models. In the end everything is a model, and nothing is quite as it is described.

Language is no exception, being a model like the rest: try putting into words what you feel in front of a painting, or during sex, and you meet the limit directly. And since each model rests on earlier models, every error is inherited by whatever is built on top of it. Ten steps, each right nine times out of ten, leave you with a little over a third of a chance of being right at the end. That is why a small distortion at the far end of a long chain can arrive with the force of a tsunami, and it may partly explain the amplitude of black-swan events — though not how they form, nor where the first irregularity comes from, nor whether chance is a real thing or only a name for our ignorance. These need testing rather than more deliberation.

In short: ads are labels on labels, models on models, illusions on illusions, each resting on its predecessors.

Some of these are heuristics, and that is the overlapping ground I sense. We are always building new ones, and reality will falsify them or let them stand. A heuristic that survives the clash long enough becomes a rule of thumb, then folklore. Some may be innate — the hand pulled back from the hot iron before any thought — but most are bought: don’t touch the stove was not born with the child. The mother’s warning went unheeded, curiosity won, and the burn did the teaching. Which is worth setting beside what I say elsewhere about our learning nothing from mistakes. The burn teaches, and teaches permanently — but it teaches one stove. What it does not give is the theory, or the ability to see the same danger wearing different clothes in another part of life.

Why power laws rule the world #

What is a law — or 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 this is because the world has more dimensions than we are built to picture, as the inhabitants of Flatland could not picture a sphere passing through their plane and saw only a circle that grew and shrank and vanished.

A concept like global warming is made of smaller concepts — CO₂ and its links with the ocean, the land, the temperature — each of them a sub-theory, and each of those made of smaller ones again, perhaps ad infinitum. They are strung together like Christmas-tree lights, the current running from the first bulb to the next.

Now put a number on it. Suppose each bulb in the chain is right nine times out of ten. Two bulbs and you are at 0.81; five and you are at 0.59; ten and you are at 0.35. Written out: reliability ≈ (1 − ε)ⁿ, where ε is the error at each step and n the length of the chain. Two things follow. (A) No model is a monolith. It is a chain, or more honestly a network, and a small change in the error rate at every node compounds — so that a theory resting on fifty sub-theories is not a little worse than one resting on three, it is hopeless, and confidently so. (B) A model is itself a complex system: it has properties its parts do not, and even a good model fails to represent what a model is, because at each level it treats the level below as solid. We rarely know how deep the recursion runs, or whether it bottoms out at a first principle — which, for Popper, it does not.

I should be careful with my own vocabulary here, since I have been calling this a power law and it is not one. Compounding error gives an exponential: reliability falls off in the length of the chain. A power law is a different animal — the distribution of the outcomes, where events a thousand times larger are not a million times rarer as the bell curve promises, but merely much rarer. The two are cousins, not twins. They arise from different machinery, and I have been sloppy in running them together. What the chain of bulbs explains is why long arguments fail; it does not by itself explain why the failures, when they come, are so unequal in size. (I write this in 2025, and my views may have changed many times over, as I tested theories and most 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 what the chain of bulbs does to a model built on models: it does not add the idiocies, it multiplies them.

One more general point, and an honest difficulty. To do the arithmetic above I had to decide how many links the chain has, and there is no principled way to decide it. Five, because a theory looks to me as if it rests on five others? And what if they are not a chain at all but a tangle, feeding back into each other? So the number is a guess, and everything computed from it inherits the guess — the estimate of error is itself a model with an error rate, which is either the neatest illustration of the whole argument or the point at which it eats itself. The direction, at least, I am confident of: the more sub-theories a theory needs, the worse its odds, and steeply.

And “the world” is itself a mental concept — a forest to a forager, a planet to most of us, the edge of a galaxy to an astronomer. This thing “world” is made of an uncountable number of conjectures, theories, laws, lore, heuristics and reflexes. And any law, rule or theory is at bottom the observation of a recurrence — a pulsar sending its pulses, a kick in the ankle always making the victim cry out.

How many such basic rules are there in the social universe? More than anyone can hold, and each of them interacting with the others, which is what makes the whole thing behave chaotically. Though I should admit that “chaotic” is a description of our position, not of the universe’s: it means the deviations are larger than we are comfortable with. The universe does not care whether we have laws at all. Gravity is more real than any of our theories about it and wins every contest in which it is entered. What we have is a cosmic soup of ideas, beliefs, heuristics, religions and political systems, interacting in ways no map of them will ever be small enough to draw.