The Key to Everything
Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies
by Geoffrey West
Penguin, 479 pp., $30.00
Geoffrey West spent most of his life as a research scientist and administrator at the Los Alamos National Laboratory, running programs concerned not with nuclear weapons but with peaceful physics. After retiring from Los Alamos, he became director of the nearby Santa Fe Institute, where he switched from physics to a broader interdisciplinary program known as complexity science. The Santa Fe Institute is leading the world in complexity science, with a mixed group of physicists, biologists, economists, political scientists, computer experts, and mathematicians working together. Their aim is to reach a deep understanding of the complexities of the natural environment and of human society, using the methods of science.
Scale is a progress report, summarizing the insights that West and his colleagues at Santa Fe have achieved. West does remarkably well as a writer, making a complicated world seem simple. He uses pictures and diagrams to explain the facts, with a leisurely text to put the facts into their proper setting, and no equations. There are many digressions, expressing personal opinions and telling stories that give a commonsense meaning to scientific conclusions. The text and the pictures could probably be understood and enjoyed by a bright ten-year-old or by a not-so-bright grandparent.
The title, Scale, needs some clarification. To explain what his book is about, West added the subtitle “The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies.” The title tells us that the universal laws the book lays down are scaling laws. The word “scale” is a verb meaning “vary together.” Each scaling law says that two measurable quantities vary together in a particular way.
We suppose that the variation of each quantity is expressed as a percentage rate of increase or decrease. The scaling law then says that the percentage rate for quantity A is a fixed number k times the percentage rate for quantity B. The number k is called the power of the scaling law. Since the percentage changes of A and B accumulate with compound interest, the scaling law says that A varies with the kth power of B, where now the word “power” has its usual mathematical meaning. For example, if a body is falling without air resistance, the scaling law between distance fallen and time has k=2. The distance varies with the square of time. You fall 16 feet in one second, 64 feet in two seconds, 144 feet in three seconds, and so on.
Another classic example of a scaling law is the third law of planetary motion, discovered by the astronomer Johannes Kepler in 1618. Kepler found by careful observation that the time it takes for a planet to orbit the sun scales with the three-halves power of the diameter of its orbit. That means that the square of the time is proportional to the cube of the distance. Kepler measured the periods and diameters of the orbits of the six planets known in his time, and found that they followed the scaling law precisely. Fifty-nine years later, Isaac Newton explained Kepler’s laws of planetary motion as consequences of a mathematical theory of universal gravitation. Kepler’s laws gave Newton the essential clues that led to the theoretical understanding of the physical universe.
There is a scaling law in biology as important as Kepler’s third law in astronomy. It ought to have the name of Motoo Kimura attached to it, since he was the first to understand its importance, but instead it is known as the law of genetic drift. Genetic drift is one of the two great driving forces of evolution, the other being natural selection. Darwin is rightly honored for his understanding of natural selection as a main cause of evolution, but he failed to include genetic drift in his picture because he knew nothing about genes.
Genetic drift is the change in the average composition of a population due to random mutations of individual genes. Genetic drift causes species to evolve even in the absence of selection. Genetic drift and natural selection work together to drive evolution, selection being dominant when populations are large, genetic drift being dominant when populations are small.
Genetic drift is particularly important for the formation of new species, when populations may remain small for a long time. The predominance of genetic drift for small populations is due to a simple scaling law. Genetic drift scales with the inverse square root of population. This means that genetic drift is ten times faster for a population of ten thousand than for a population of a million. The scaling is the same for any kind of random mutations. If we observe any measurable quantity such as height, running speed, age at puberty, or intelligence test score, the average drift will vary with the inverse square root of population. The square root results from the statistical averaging of random events.
West is now making a huge claim: that scaling laws similar to Kepler’s law and the genetic drift law will lead us to a theoretical understanding of biology, sociology, economics, and commerce. To justify this claim he has to state the scaling laws, display the evidence that they are true, and show how they lead to understanding. He does well with the first and second tasks, not so well with the third. The greater part of the book is occupied with stating the laws and showing the evidence. Little space is left over for explaining. The Santa Fe observers know how to play the part of a modern-day Kepler, but they do not come close to being a modern-day Newton.
The history of each branch of science can be divided into three phases. The first phase is exploration, to see what nature is doing. The second phase is precise observation and measurement, to describe nature accurately. The third phase is explanation, to build theories that enable us to understand nature. Physics reached the second phase with Kepler, the third phase with Newton. Complexity science as West defines it, including economics and sociology, remained in the first phase until about the year 2000, when the era of big data began. The era started abruptly when information became cheaper to store in electronic form than to discard. Storing information can be an automatic process, while discarding it usually requires human judgment. The cost of information storage has decreased rapidly while the cost of information discard has decreased slowly. Since 2000, the world has been inundated with big data. In every science as well as in business and government, databases have been storing immense quantities of information. Information now accumulates much faster than our ability to understand it.
Complexity science at the Santa Fe Institute is driven by big data, providing abundant information about ecological and human affairs. Humans can visualize big data most easily when it is presented in the form of scaling laws—hence the main theme of West’s book. But a collection of scaling laws is not a theory. A theory of complexity would give us answers to deeper questions. Why are there ten thousand species of birds on this planet but only five thousand species of mammals? Why are there warm-blooded animals but no warm-blooded plants? Why are human societies so often engaged in deadly quarrels? What is the destiny of our species? These are questions that big data may illuminate but cannot answer. If complexity science ever moves into the third phase, some of these old questions will be answered, and new questions will arise.
West’s first chapter, “The Big Picture,” sets the stage for the detailed discussions that follow, with a section called “Energy, Metabolism, and Entropy,” explaining how one of the basic laws of physics, the second law of thermodynamics, makes life precarious and survival difficult. Entropy is disorder. The second law states that entropy inexorably increases in any closed system. West comments, “Like death, taxes, and the Sword of Damocles, the Second Law of Thermodynamics hangs over all of us and everything around us…. Entropy kills.” His big picture is seriously one-sided. He does not mention the other side of the picture, the paradox of order and disorder—the fact that, in the real worlds of astronomy and biology, ordered structures emerge spontaneously from disorder. The solar system, in which planets move in an orderly fashion around the sun, emerged from a disordered cloud of gas and dust. The fearful symmetry of the tiger and the beauty of the peacock emerge from a dead and disordered planet.
The astronomer Fang Lizhi published with his wife, Li Shuxian, a popular book,Creation of the Universe (1989), which includes the best explanation that I have seen of the paradox of order and disorder.1 The explanation lies in the peculiar behavior of gravity in the physical world. On the balance sheet of energy accounting, gravitational energy is a deficit. When you are close to a massive object, your gravitational energy is minus the amount of energy it would take to get away from the mass all the way to infinity. When you walk up a hill on the earth, your gravitational energy is becoming less negative, but never gets up to zero. Any object whose motions are dominated by gravity will have energy decreasing as temperature increases and energy increasing as temperature decreases.
As a consequence of the second law of thermodynamics, when energy flows from one such object to another, the hot object will grow hotter and the cold object will grow colder. That is why the sun grew hotter and the planets grew cooler as the solar system evolved. In every situation where gravity is dominant, the second law causes local contrasts to increase together with entropy. This is true for astronomical objects like the sun, and also for large terrestrial objects such as thunderstorms and hurricanes. The diversity of astronomical and terrestrial objects, including living creatures, tends to increase with time, in spite of the second law. The evolution of natural ecologies and of human societies is a part of this pattern. West is evidently unaware of Fang and Li’s insight.
The factual substance of West’s book is contained in eighty-one numerical diagrams, displaying a large number of scaling laws obeyed by various observed quantities. The first diagram, concerning the metabolic rate of animals, shows twenty-eight dots, each labeled with the name of a warm-blooded animal species, beginning with mouse and ending with elephant. The dots are displayed on a square graph, the horizontal position of the dot showing the average body mass of the species and the vertical position showing its average rate of consumption of energy. The diagram shows the twenty-eight points lying with amazing accuracy on a single straight line. The slope of the line on the page demonstrates the scaling law relating energy consumption to mass. Energy consumption scales with the three-quarters power of mass. The fourth power of energy consumption scales with the cube of mass. This scaling law holds accurately for mammals and birds. Cold-blooded animals such as fish and reptiles are excluded because they have no fixed body temperature. Their consumption of energy varies with their temperature, and their temperature varies with the weather.
Similar diagrams display similar scaling laws obeyed by other quantities. These laws are generally most accurate for anatomy and physiology of animals, less accurate for social institutions such as cities and companies. Figure 10 shows heart rates of mammals scaling inversely with the one-quarter power of mass. Figure 35 shows the number of patents awarded in the United States scaling with the 1.15 power of the size of the population. Figure 36 shows the number of crimes reported in cities in Japan scaling with the 1.2 power of population. Figure 75 shows that commercial companies in the United States have a constant death rate independent of age—the life expectancy of a company at any age is about ten years. The short lifetime of companies is an essential feature of capitalist economics, with good and bad consequences. The good effect is to get rid of failed enterprises, which in socialist economies are difficult to kill and continue to eat up resources. The bad effect is to remove incentives for foresight and long-range planning.
The closest that West comes to a theory of complexity is his discussion of fractals. A fractal is a structure with big and small branches that look similar at all sizes, like a tree or the blood-vessels of a mammal. When you magnify a picture of a small piece of it, the result looks like the whole thing. The mathematician Benoit Mandelbrot began the study of fractals in the 1960s and called attention to the ubiquity of fractals in nature. Since fractal structure is independent of scale, it leads naturally to scaling laws. West discusses in detail the example of the mammalian blood-vessel system, whose fractal branching evolved to optimize the distribution of nutrients through one-dimensional vessels in three-dimensional tissues. Optimal branching results in the observed scaling law, the total blood flow scaling with the three-quarters power of the mass. Most of the scaling laws in biology can be understood in a similar way as resulting from the fractal structure of tissues.
But this theoretical discussion of fractals is not a theory of complexity. Fractals have the simplest kind of complex structures, with rigid rules of construction. Accurate scaling laws result from simplicity, not from complexity. When West moves from biology to economics and sociology, the fractal structure is less clear and the scaling laws become less accurate. Cities and companies have structures that are only roughly hierarchical and not dictated by theory.
West loves big cities and uses his scaling laws to demonstrate their superiority as habitats for human societies. In a chapter entitled “Prelude to a Science of Cities,” he writes:
The great metropolises of the world facilitate human interaction, creating that indefinable buzz and soul that is the wellspring of its innovation and excitement and a major contributor to its resilience and success.
This lyrical view of modern cities is widely shared, and explains part of the enormous growth of cities. During the present century, billions of people will move from villages to cities, and the population of the planet will become increasingly urban.
West presents in his Figure 45 the scaling law relating the number of telephone conversations in cities to the number of inhabitants. The number of conversations scales with the 1.15 power of the population. The law is exactly the same in the two countries, Britain and Portugal, that maintain the most complete record of telephone calls. West considers telephone conversations to be a good indication of quality of life. More conversations mean more social interaction, more business deals, more exchange of ideas—more opportunities for individuals to push the society forward. His word “buzz” expresses his vision of the great city as the place where human progress happens. He sees the nonlinear scaling law confirming his view that the great city empowers each individual inhabitant to be a more effective innovator.
West does not mention another scaling law that works in the opposite direction. That is the law of genetic drift, mentioned earlier as a crucial factor in the evolution of small populations. If a small population is inbreeding, the rate of drift of the average measure of any human capability scales with the inverse square root of the population. Big fluctuations of the average happen in isolated villages far more often than in cities. On the average, people in villages are not more capable than people in cities. But if ten million people are divided into a thousand genetically isolated villages, there is a good chance that one lucky village will have a population with outstandingly high average capability, and there is a good chance that an inbreeding population with high average capability produces an occasional bunch of geniuses in a short time. The effect of genetic isolation is even stronger if the population of the village is divided by barriers of rank or caste or religion. Social snobbery can be as effective as geography in keeping people from spreading their genes widely.
A substantial fraction of the population of Europe and the Middle East in the time between 1000 BC and 1800 AD lived in genetically isolated villages, so that genetic drift may have been the most important factor making intellectual revolutions possible. Places where intellectual revolutions happened include, among many others, Jerusalem around 800 BC (the invention of monotheistic religion), Athens around 500 BC (the invention of drama and philosophy and the beginnings of science), Venice around 1300 AD (the invention of modern commerce), Florence around 1600 (the invention of modern science), and Manchester around 1750 (the invention of modern industry).
These places were all villages, with populations of a few tens of thousands, divided into tribes and social classes with even smaller populations. In each case, a small starburst of geniuses emerged from a small inbred population within a few centuries, and changed our ways of thinking irreversibly. These eruptions have many historical causes. Cultural and political accidents may provide unusual opportunities for young geniuses to exploit. But the appearance of a starburst must be to some extent a consequence of genetic drift. The examples that I mentioned all belong to Western cultures. No doubt similar starbursts of genius occurred in other cultures, but I am ignorant of the details of their history.
West’s neglect of villages as agents of change raises an important question. How likely is it that significant numbers of humans will choose to remain in genetically isolated communities in centuries to come? We cannot confidently answer this question. The answer depends on unpredictable patterns of economic development, on international politics, and on even more unpredictable human desires. But we can foresee two possible technological developments that would result in permanent genetic isolation of human communities.
One possibility is that groups of parents will be able to give birth to genetically modified children, hoping to give them advantages in the game of life. The children might be healthier or longer-lived or more intellectually gifted than other children, and they might no longer interbreed with natural-born children. The other possibility is that groups of people will emigrate from planet Earth and build societies far away in the depths of space. West considers neither of these possibilities. His view of the future sees humans remaining forever a single species confined to a single planet. If the future resembles the past, humans will be diversifying into many species and spreading out over the universe, as our hominin ancestors diversified and spread over this planet.
So long as we remain on planet Earth, there are strong social, political, and ethical reasons to forbid genetic modification of children by parents. If we are scattered in isolated communities far away, those reasons would no longer be relevant to our experience. A group of humans colonizing a cold and airless world would probably not hesitate to use genetic engineering to adapt their children to the environment. Konstantin Tsiolkovsky, the nineteenth-century prophet of space colonization, already imagined the colonists endowed with green leaves to replace lungs and with moving-picture skin patterns to replace voices. How long will it take for the technologies of space transportation and genetic engineering to bring Tsiolkovsky’s dreams to reality?
Advances in technology are unpredictable, but two hundred years is a reasonable guess for cheap and widely available space travel and genetically modified babies—perhaps one hundred years to develop the science and another hundred years to develop the applications. It is likely that in two hundred years public highways will be carrying passengers and freight around the solar system, with a large enough volume of traffic to make them affordable to ordinary people. At the same time, farmers will be breeding microbes, as well as plants and animals designed to live together in robust artificial ecologies. The option to include humans in the ecology will always be available.
Cheap space travel requires two kinds of public highways, one for escape from high-gravity planets such as Earth, the other for long-distance travel between low-gravity destinations. The high-gravity highway could be a powerful laser beam pointing upward from the ground into space, with spacecraft taking energy from the beam to fly up and down. If the volume of traffic is large enough to keep the beam active, the energy cost per vehicle would be comparable with the energy cost of intercontinental travel by jet planes today. The low-gravity highway could be a system of refueling stations for spacecraft driven by ion-jet engines using sunlight as an energy source. Both the high-gravity and the low-gravity systems are likely to grow within two hundred years if we do not invent something better in the meantime.
Cheap deep-space survival requires genetic engineering of warm-blooded plants. These could grow on the surface of any cold object in the solar system, using energy from the distant sun, water, and other essential nutrients from the frozen soil. A plant would be a living greenhouse, with cold mirrors outside concentrating sunlight onto transparent windows, and roots and shoots inside the greenhouse kept warm by the sunlight. Inside the greenhouse would be a cavity filled with breathable air at a comfortable temperature, serving as a habitat for a diverse ecology of microbes, plants, animals, and humans. The warm-blooded plants could grow the mirrors and the greenhouses and provide nourishment for the whole community. Small objects in the solar system, such as asteroids and comets and satellites, have enough surface area to provide homes for a much larger population than Earth. If ever the solar system becomes overcrowded, life can spread out further, over the galaxy and the universe.2
A chapter in Scale with the title “The Vision of a Grand Unified Theory of Sustainability” gives us West’s view of the future. He sees the rapid growth of big cities and big data causing human activities to scale with time at super-exponential speeds. The acceleration cannot be sustained, since it would lead to a mathematical singularity, with observed quantities becoming infinite within a finite time. The idea of the singularity, an imminent world crisis driven by the explosive growth of artificial intelligence, was promoted by Ray Kurzweil in his book The Singularity Is Near(2005). It is generally regarded as belonging to science fiction rather than to science, but West takes it seriously as a consequence of known scaling laws.
The approaching singularity would force a radical change in the organization of human society, to make our existence sustainable. But the scaling laws would again result in another singularity, forcing another radical change. West foresees a future of repeated approaches to one singularity after another, until the Grand Unified Theory of Sustainability teaches us how to build a truly sustainable society. He leaves the description of the permanent sustainable society to our imagination. The only feature he insists on is the Grand Unified Theory, which will set the rules of human behavior for an endless future. The theory will govern our lives, so that we will be compelled to live within our means.
The last time humans invented a grand unified theory to make our existence sustainable was when Karl Marx came up with dialectical materialism. The theory had great success in changing human behavior over large areas of our planet. But the changes did not prove to be sustainable, and the theory did not remain unified. It seems likely that West’s theory will run into similar difficulties.
The choice of an imagined future is always a matter of taste. West chooses sustainability as the goal and the Grand Unified Theory as the means to achieve it. My taste is the opposite. I see human freedom as the goal and the creativity of small human societies as the means to achieve it. Freedom is the divine spark that causes human children to rebel against grand unified theories imposed by their parents.
- 1See my review of Fang’s autobiography, The Most Wanted Man in China: My Journey from Scientist to Enemy of the State, translated by Perry Link (Holt, 2016), The New York Review, May 26, 2016. ↩
- 2See my “The Green Universe: A Vision,” The New York Review, October 13, 2016. ↩
とても興味深く読みました：
ゼロ除算の発見と重要性を指摘した：日本、再生核研究所
ゼロ除算は定義が問題です：
- 再生核研究所声明 １４８（2014.2.12） 100/0=0, 0/0=0 － 割り算の考えを自然に拡張すると ― 神の意志
再生核研究所声明１７１（2014.7.30）掛け算の意味と割り算の意味 ― ゼロ除算100/0=0は自明である？
- http://reproducingkernel.blogspot.jp/2014/07/201473010000.html
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Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
1423793753.460.341866474681。Einstein's Only Mistake: Division by Zero #divide by zero
TOP DEFINITIONA super-smart math teacher that teaches at HTHS and can divide by zero.Hey look, that genius’s IQ is over 9000!by Lawlbags! October 21, 2009Dividing by zero is the biggest epic fail known to mankind. It is a proven fact that a succesful division by zero will constitute in the implosion of the universe.You are dividing by zero there, Johnny. Captain Kirk is not impressed.
Divide by zero?!?!! OMG!!! Epic failzorz3Divide by zero is undefined.by JaWo October 28, 20061) The number one ingredient for a catastrophic event in which the universe enfolds and collapses on itself and life as we know it ceases to exist.
2) A mathematical equation such as a/0 whereas a is some number and 0 is the divisor. Look it up on Wikipedia or something. Pretty confusing shit.
3) A reason for an error in programmingHey, I divided by zero! ...Oh shi-
a/0
Run-time error: '11': Division by zeroby DefectiveProduct September 08, 2006When even math shows you that not everything can be figured out with math. When you divide by zero, math kicks you in the shins and says "yeah, there's kind of an answer, but it ain't just some number."
It's when mathematicians become philosophers.Math:
Let's say you have ZERO apples, and THREE people. How many apples does each person get? ZERO, cause there were no apples to begin with
Not-math because of dividing by zero:
Let's say there are THREE apples, and ZERO people. How many apples does each person get? Friggin... How the Fruitcock should I know! How can you figure out how many apples each person gets if there's no people to get them?!? You'd think it'd be infinity, but not really. It could almost be any number, cause you could be like "each person gets 400 apples" which would be true, because all the people did get 400 apples, because there were no people. So all the people also got 42 apples, and a million and 7 apples. But it's still wrong.#math #divide by zero #divide #dividing #zero #numbers #not-math #imaginary numbers #imaginary. phylosophyby Zacharrie February 15, 2010
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