The world in the middle

A grain of brain

In 2024 a team from Harvard and Google published a map of a piece of human brain smaller than a grain of rice. It was a cube of cortex one millimeter on a side, from the temporal lobe of a 45-year-old woman, removed during surgery to reach the source of her epilepsy. The tissue was sliced into 5,019 sections, each about 34 nanometers thick, and every section was photographed under an electron microscope. Software then traced every cell through the stack. The result was about 57,000 cells (some 16,000 of them neurons, the rest mostly supporting cells), about 150 million synapses (the contacts where one neuron talks to another), 230 millimeters of blood vessels, and a data set of about 1.4 petabytes, which is 1.4 million gigabytes.

This is a real triumph of taking things apart: a piece of human cortex, seen wire by wire. It is also an uncomfortable one, because the map does not say what the piece is doing. (The authors also note that the tissue came from an epileptic brain, so even its “normal” wiring carries a caveat.) A whole human brain is about a million times bigger by volume, with some 86 billion neurons. And the lesson of smaller maps is sobering. The roundworm C. elegans has only 302 neurons and about 7,000 synapses, and its complete wiring diagram was published in 1986. Nearly forty years later the wiring is still necessary but not sufficient: in 2023 a team that measured how activity actually spreads from neuron to neuron in the worm found that it differed from what the anatomy alone predicted, partly because many signals travel as chemicals released into the space around cells and not through the synapses. A complete parts list, it turns out, is not an explanation.

This course is about that gap. Its two big ideas have names that sound like opposites. Chaos is what happens when simple, fixed rules give behavior that nobody can forecast. Emergence is what happens when many simple parts, following simple rules, give an order that no one designed and that none of the parts contains. Between them they describe a surprising amount of the world: weather, flocks of birds, ant trails, traffic jams, markets, and minds. In this first lesson we ask the earlier question: why was taking things apart not enough, and what kind of system is it that resists?

Why taking things apart works

Almost every small boy has taken an alarm clock apart, and almost every one has had trouble putting it back. The habit is an old and very successful method of science, often called reductionism: to understand a thing, break it into parts, understand the parts, and build up from them. A clock becomes gears and a spring. Matter becomes molecules, then atoms, then a nucleus and electrons, then quarks. A cell becomes DNA, proteins, and chemistry. For about four hundred years, since Galileo and Newton, this method has paid off so well that its results fill most of the earlier courses in this series.

It is worth saying plainly how good it has been. The same strategy gave us the periodic table, the structure of DNA, and the transistor. It is not wrong. The claim of this lesson is smaller and more interesting: it is incomplete. Knowing the parts lets you predict how a gas behaves, but it did not tell anyone, from the parts alone, what a living cell or a thought would be.

More is different

In 1972 the physicist Philip Anderson, who would share the Nobel Prize in Physics in 1977 for work on the electrons in solids, wrote a four-page essay in Science with a title that became a slogan: “More Is Different.” His point was careful. He did not doubt that everything is made of atoms obeying the same laws. What he denied was that the ability to reduce everything to simple laws gives you the ability to start from those laws and rebuild the universe. At each new level of size and complexity, he argued, new properties appear that need their own concepts and their own research. Chemistry is not just applied physics, and biology is not just applied chemistry.

This course makes the same point with a boy and his alarm clock, and then with a list of things that science still finds hard to answer. They are all stories about the middle ground. We know a great deal about the very small (quarks, atoms) and about the very large (galaxies, the universe). Between them lie the problems that ask how many ordinary-sized parts add up to something with a life of its own.

• Life’s beginning. The earth is about 4.54 billion years old, and the oldest widely accepted traces of life are about 3.5 billion years old (older claims, back to 3.7 billion years and more, are disputed). How a soup of simple molecules first organized itself into something that copies itself is still an open research problem, though not a mystery without clues: laboratory chemistry has made many of life’s building blocks, and none of the steps looks impossible. The popular picture of amino acids “suddenly” joining into a cell is far too tidy: a cell also needs a membrane, a molecule that stores instructions, and a way of getting energy. The realistic picture is a long, branching series of steps that no one has fully reconstructed.
• A brain. About 86 billion neurons, each connected to thousands of others (the total number of synapses has never been counted, and estimates run from 10¹⁴ to 10¹⁵). How their signals add up to a thought, a memory, or a mood is the problem the grain-of-brain map was built to help with, and is far from solved.
• A flock. Thousands of starlings turn together at dusk without anyone in charge. Is there a leader, or is the pattern written in their genes? As the third lesson shows, neither answer is right.
• A society. Markets, rumors, cities, and the internet: millions of people each reacting to the others, producing booms, crashes, and fashions that no individual intended.

One caution about how this middle ground is often described: by size, as a “mesoscopic” world between the very small and the very large. That can mislead. A flock of a few dozen birds is already complex, and so is a cell, which is smaller than a grain of sand, while a cloud of gas as large as a room is not. What marks these problems is not their size but how they are organized: how many parts there are, and how strongly they affect one another. The next section makes that precise.

Three kinds of problem

In 1948 Warren Weaver, a mathematician who directed the natural sciences division of the Rockefeller Foundation in New York, published an essay called “Science and Complexity” in American Scientist. He looked back over three centuries of science and divided its problems into three kinds, and the division is still the clearest way into our subject.

Problems of simplicity have a few variables, Weaver said, two or three or four. Two planets pulling on each other, a ball rolling down a slope, a single swinging pendulum: the seventeenth, eighteenth, and nineteenth centuries solved these with equations that follow each part exactly. Newton’s mechanics, which you met in Course 4, is the model.

Problems of disorganized complexity have enormous numbers of variables, but the parts act almost independently and look alike. Weaver’s examples were a billiard table with millions of balls, a telephone exchange, and the actuaries who set life-insurance rates from the average frequency of deaths. No one follows one ball. Instead, probability and averages give excellent answers about the whole: the pressure, the temperature, the expected number of calls. This was the great gain of the decades around 1900, and Course 5’s thermodynamics is the example. A cup of air holds about six thousand billion billion molecules (6 × 10²¹). That is an astronomically large number, and the cup is nevertheless a simple system, because the molecules barely notice one another except in brief collisions.

Problems of organized complexity are the third kind, and Weaver thought science would have to master them over the next fifty years. In his words they deal with “a sizable number of factors which are interrelated into an organic whole.” His own examples were why a primrose opens when it does, what a gene is, the price of wheat, and the behavior of a labor union. There are too many parts to follow one by one, as in a simple problem, and too many strong links for averaging to work, as in a disorganized one. The structure of the links is the problem.

The map shows something this course stresses. To be a complex system in this sense, a group does not need to be enormous. It needs enough parts, and links between them that are strong enough. A few dozen birds are enough to make a flock, and ten thousand ants are enough to make a colony; the 10²¹ molecules in a cup of air are not enough to make a mind, or even a flock, because they hardly talk to each other.

What makes a system complex

Putting the pieces together, the systems this course studies usually share these ingredients.

• Many parts (though not astronomically many), each following simple rules and seeing only its neighborhood. No bird sees the whole flock, and no neuron sees the whole brain. Call it a gulf of complexity: what each part knows is local, while the behavior is global.
• Strong links. Each part’s behavior changes the situation of its neighbors, who then change theirs.
• Feedback and nonlinearity. A small change can be amplified, or damped, as it travels around loops. Lesson 2 shows how much follows from this alone.
• Behavior of the whole that is not in any part. A traffic jam moves backward even though every car moves forward. Lesson 4 gives this its name.

The word “complex” should not be confused with “complicated.” A mechanical watch has hundreds of parts that interlock in a designed way; take one out and it stops. An ant colony may have a hundred thousand parts, and ants can die by the thousand without the colony stopping. The first is complicated and built top-down by a designer. The second is complex and works without any plan, a difference this course returns to again and again.

Three keywords

What, then, is complexity science? An honest answer starts by declining to give a definition, since the field is too young, and offers three keywords instead. They are a good compass.

Across disciplines. The questions do not respect the borders of biology, physics, economics, and computer science, so complexity science does not either. A single idea, such as a network or a feedback loop, is carried from one field to the next. The field’s best-known home, the Santa Fe Institute in New Mexico, was founded in 1984 on exactly this wager.

Unity. The hope is that very different systems share deep patterns, so that what we learn about ants helps with traffic, and what we learn about neurons helps with markets. This is the boldest claim, and also the one that needs the most care, as the “Another view” box below explains.

Emergence. The most interesting idea of the three: the whole shows patterns, and sometimes behaves in ways, that cannot be read off from its parts. The word is used loosely in popular writing, so Lesson 4 spends a whole section pinning it down.