Weather is a complex system, and the forecast is very sensitive to the initial value accuracy and the dependence of small changes, so it is sometimes inaccurate.

Melanie is a student of Hou Shida and currently teaches at Portland University. His main interests are analogical reasoning, complex systems, genetic algorithms and cellular automata. Since the publication of Complex in 2009, it has received rave reviews, with Douban score of 9. Moreover, I am still tirelessly publishing papers. If you want to query related content, please query Melanie Mitchell, which is rich in pdx and wiki, so don't think about Chinese.

Although it is a popular science book, it is also painful for a small white like me. After two weeks' study, plus 1 week's notes sorting and data supplement, the book review was finally released. In order not to lose the main content of the book as much as possible, read it as easily as possible. Will be divided into two book reviews:

Book Review 1: Complexity Part I: What is a Complex System (Part I)

Book Review 2: Complexity Part II: Wonderful examples of complex systems (link)

This article is 1, and will introduce what a complex system is from the aspects of origin, evolution, definition and measurement.

Before introducing complex systems, briefly introduce reductionism, which is the most natural way to understand the world. It means:

If you understand all the parts of the whole and the mechanism of' integrating' these parts, you can understand the whole.

However, as people explain how complex behaviors emerge from the large-scale combination of simple individuals, new disciplines such as chaos, systems biology, evolutionary economics and network theory have surpassed reductionism.

The anti-reductionist slogan "The whole is greater than the sum of its parts" has also become more and more influential.

In daily life, in fact, many things we touch belong to complex systems, such as information, calculation, dynamics, chaos and evolution. Here are two common examples in life.

\ ant colony

A single army ant is the simplest creature known. If you put 100 military ants on a plane, they will circle around the plane until they are exhausted and die.

But if millions of animals are put together, the group will form a whole and form a so-called "super-individual" of "collective wisdom".

\ brain

In a sense, the human brain is very similar to the ant colony. Both of them are composed of relatively simple individuals, and they only have limited communication, but on the whole they show extremely complex system ("global") behavior.

In the brain, simple individuals are neurons. In addition to neurons, there are many different cells in the brain, but most brain scientists believe that the activities of neurons and the connection mode of neuron groups determine important macroscopic brain activities such as perception, thinking, emotion and consciousness.

\ immunization

Like the brain, the immune system of different animals is different in complexity, but the overall principle is the same. The immune system consists of many different cells, which are distributed all over the body (blood, bone marrow, lymph nodes, etc.). These cells work together efficiently without central control.

The protagonist in the immune system is white blood cells, that is, lymphocytes. White blood cells can recognize molecules corresponding to possible intruders (such as bacteria) through receptors on their cell bodies.

There is a kind of cells called B cells (B means they were born from bone marrow), which has a peculiar property: the more B cells match an intruder, the more offspring cells it produces. In this way, Darwinian natural selection mechanism is formed, and the matching degree between B cells and intruders is getting higher and higher, thus producing antibodies that can search and destroy microbial criminals very efficiently.

This matter has to start with Apple. It is said that iPhone7 removed the earphone hole, and everyone was forced ... No, no, it was the kind of edible round fruit, mostly red fruit. The kind that hit Newton.

Newton's law is the basis of dynamics, which explains the motion of all objects including planets as a basic concept.

/Great God Newton and Three Laws

The following are the famous Newton's three laws:

For students, it is more difficult than studying physics. Newtonian mechanics paints a picture of the "clock universe": set the initial state and then follow three laws to keep running.

Mathematician Laplace realized that it contained the idea of accurate prediction like a clock: in 18 14, he asserted that according to Newton's law, as long as the current position and speed of all particles in the universe are known, in principle, the situation at any moment can be predicted.

After the invention of computers in the 1940s, it seems that this "in principle" possibility may become a reality.

/Heisenberg, the first one to kick the pavilion.

After a long time, 1927. Suddenly, a pavilion kicker came.

Werner Heisenberg put forward the "uncertainty principle" in quantum mechanics, which proved that it is impossible to accurately measure the position and momentum of particles (mass multiplied by velocity).

The more we know about its position, the less we know about its momentum, and vice versa.

Later, the discovery of chaos gave a final blow to the dream of accurate prediction. What chaotic system says is that if the initial position and momentum are extremely inaccurate, it will also lead to huge errors in its long-term prediction.

Also known as "sensitive dependence on initial conditions".

This is very counterintuitive. In fact, for a long time, scientists thought it was impossible. However, chaos has been observed in many systems, such as heart disorder, turbulence, circuits, water droplets and many other seemingly unrelated phenomena. Now the existence of chaotic systems has become a recognized fact in the scientific community.

/poincare, the second kicker.

The first clear example of chaotic system may be given by French mathematician Henri Poincare at the end of 19. Poincare is the founder and possibly the greatest contributor of modern dynamical system theory, which greatly promoted the development of Newtonian mechanics.

When trying to solve a much simpler problem than predicting hurricanes, Poincare found a sensitive dependence on initial conditions. What he tried to solve was the so-called three-body: using Newton's law to predict the long-term motion of three objects through gravitational interaction.

It was in the process of studying the geometric results of three bodies that Poincare discovered the sensitive dependence on initial conditions.

In other words, even if we fully know the law of motion, two different initial conditions (in this case, the initial position, mass and speed of the object), even if the difference is small, sometimes lead to very different subsequent movements of the system.

Reductionists like linearity, while nonlinearity is a reductionist's nightmare. For example, biologists who study populations often use logistic modl to describe the population growth in this case.

The existence of chaos in the system means that perfect Laplacian prediction is not only impossible in practice, but also impossible in principle, because we can never know the infinite digits after the decimal point.

This is a very profound negative conclusion, which, together with quantum mechanics, destroyed the optimism since19th century-the Newton universe runs along a predictable path like a clock.

In order to define complex systems, we first analyze the * * * properties of complex systems:

Definition 1: A complex system is a network composed of a large number of components without central control. Complex collective behavior and complex information processing are produced by simple operation rules, and adaptability is produced by learning and evolution.

If there is no internal or external controller or leader in the organized behavior of the system, it is also called self-organization. Because simple rules produce complex behaviors in an unpredictable way, the macro behavior of this system is sometimes called emergence. So there is another definition of complex system:

Definition 2: A system with emergent and self-organizing behaviors.

The seemingly chaotic behavior may come from a deterministic system without external random sources.

Because some simple deterministic systems are sensitive to initial conditions, their long-term changes are unpredictable even in principle. Although the specific changes of chaotic systems cannot be predicted, there are some "chaotic orders" in the universality of a large number of chaotic systems, such as the road of chaos, Feigenbaum constant and so on.

Therefore, although "prediction becomes impossible" in detail, chaotic systems can be predicted at a higher level.

When the author's son Nicole is still a toddler, he will let him talk to his grandmother by phone. He likes to call, but he can only say one word-"call". The message he sent to grandma was "dadada dadada ..."

In other words, Nicole's macro state has only one possible micro state ("da" sequence), so although this macro state is interesting, the information content is zero.

Grandma knows what she will hear. My son Jack is two years old. He also likes to make phone calls, but he has a bigger vocabulary, so he will tell his grandmother what he did and often surprise her with her words.

Obviously, the sender Jack has more information, because there are more possible microstates, that is, collections of different kinds of information.

/Hilbert's question

There was a German mathematician, David Hilbert, who raised three questions at the Paris International Congress of Mathematicians in 1900:

These three problems have not been solved for 30 years, but Hilbert is confident that the answer must be "yes" and asserts that "there is no problem that cannot be solved".

/godel solution

Discordantly, at the same meeting where he made the above conclusion, a 25-year-old mathematician announced the proof of the incompleteness theorem, and his discovery shocked the whole mathematical community. The young man's name is Kurt? Del, Figure 4.2).

The incompleteness theorem says that if the answer to the above question 2 is "Yes" (that is, the mathematics is consistent), then the answer to the question 1 (whether the mathematics is complete or not) must be "No".

He gave a mathematical proposition to prove it:

/turing's solution

Godel skillfully solved Hilbert's first and second problems, and then the third problem was killed by the British mathematician alan turing. Again, his answer is "no".

We summarize this article with three methods to measure complexity: