The Math Book

by DK

  The title of the lecture was not chosen by Lorenz himself, but by physicist Philip Merilees, the convener of the American Association for the Advancement of Science’s annual meeting in Boston. Lorenz had been late to provide information about his proposed talk, so Merilees had improvised, basing his choice of words on what he knew of Lorenz’s work and an earlier comment that “one flap of a seagull’s wings” could be enough to change the weather forecast.

  A butterfly flaps its wings in the Amazonian jungle, and subsequently a storm ravages half of Europe.

  Terry Pratchett and Neil Gaiman

  British authors

  Chaos theory

  The butterfly effect is a popular introduction to chaos theory, which looks at the way complex systems are highly sensitive to initial conditions and are thus extremely unpredictable. Chaos theory has practical relevance to areas such as population dynamics, chemical engineering, and financial markets, and helps in the development of artificial intelligence.

  Lorenz began investigating climate modeling in the 1950s. By the early 1960s, he was attracting attention for the unexpected results of a toy climate model (“toy” meaning that it was a simplistic model made to demonstrate processes concisely). The model predicted the way the atmosphere would evolve in terms of three data points, such as air pressure, temperature, and wind speed. Lorenz found that the results were chaotic. He compared two sets of results, each starting with near-identical sets of data, noting that the atmospheric conditions developed along near-identical lines at first, but then changed in completely different ways. He also found that while every starting point in his model rendered unique results, they were all confined within certain limits.

  In a Lorenz attractor, small changes in starting conditions result in huge changes to the paths each line takes, yet the lines still fall within the confines of the same shape, providing order within the chaos.

  The amazing thing is that chaotic systems don’t always stay chaotic.

  Connie Willis

  American writer

  Strange attractor

  The computing power available to Lorenz in the early 1960s was unable to plot the modeled atmospheric variables in a three-dimensional space, where the values on the x, y, and z axes represented, for example, air temperature, pressure, and humidity (or triplets of other weather data). In 1963, when it became possible to plot this data, the shape created became known as the Lorenz attractor. Each starting point evolves into a looping line that swings from one quadrant of the space to another—indicating, for example, a change from wet and windy weather to hot, dry conditions, and all states in between. Each starting point leads to a unique evolution, but all the lines, whatever the start point, fall into the same region of the space. After many iterations, run for long periods, that region becomes a beautiful looping surface. The individual lines within the attractor are highly unstable in their trajectories; those that start in the same area often move far apart at a later point, and lines with very different starting points may end up tracking each other closely for long periods. However, the attractor shows that as a whole, the system is stable. There is no possible starting point within the attractor that can lead to a trajectory that escapes from it. This apparent contradiction is at the heart of chaos theory.

  Chaos: when the present determines the future, but the approximate present does not approximately determine the future.

  Edward Lorenz

  Finding the right path

  The roots of chaos theory lie in early attempts to understand and predict motion, especially of heavenly bodies. For example, in the 1600s, Galileo formulated laws about the way pendulums swing and how objects fall; Johannes Kepler showed how planets sweep through space as they orbit the Sun; and Isaac Newton combined this knowledge with physical laws covering gravity and motion. Along with Gottfried Leibniz, Newton is credited with developing calculus, a system of mathematics designed to analyze and predict the behaviors of more complex systems. Using calculus, the relationships between any complex variables can —in theory—be predicted by solving a particular differential equation.

  These physical laws and analytical tools can demonstrate that the Universe is deterministic— if the exact location and condition of an object and all the forces acting upon it are known, it is possible to determine its future location and condition with perfect accuracy.

  The three-body problem

  Nevertheless, Newton found a flaw with this deterministic view of the Universe. He reported difficulties in analyzing the movements of three bodies bound together by gravity—even when those bodies were as seemingly stable as the Earth, Moon, and Sun. Later attempts to analyze the movement of the Moon to improve navigation were plagued by inaccuracies. In 1890, French mathematician Henri Poincaré showed that there was no generalized, predictable way in which three bodies move around each other. In a few cases, where the bodies start in very specific places, the motion is periodic—it repeats the same paths over and over again. Mostly, Poincaré argued, the three bodies do not retrace their paths, and their movement is called aperiodic.

  Mathematicians hoping to solve this “three-body problem” have abstracted it to consider imaginary bodies moving around surfaces and spaces with specific curvature. The curvature of an imaginary body can be a mathematical representation of the forces (such as gravity) acting on it. The path the imaginary body takes in each case is called the geodesic path. In a simple case, such as the movement of a pendulum or the orbit of a planet around a star, this imaginary body oscillates (moves back and forth) around a fixed point on the surface, following a repeating path and creating what is called a limit cycle. In the case of a damped pendulum (one that is losing energy because of friction), the oscillatory motion will diminish until the imaginary body reaches the fixed point—when it stops moving.

  When considering the motion of an imaginary body with respect to several others, the geodesic path becomes very complicated. If it were possible to set the start conditions precisely, it would be possible to create every conceivable path. Some would be periodic, repeating a path of whatever complexity over and over again. Others would be unstable initially but would settle into a limit cycle eventually. A third kind would fly off to infinity—perhaps right away, or perhaps after a period of apparent stability.

  Determinism was equated with predictability before Lorenz. After Lorenz, we came to see that… in the long run, things could be unpredictable.

  Stephen Strogatz

  American mathematician

  Approximations

  Although it has been studied by physicists and mathematicians alike, the three-body problem is largely theoretical. When it comes to a real physical system, there is no way to be absolutely precise about the starting conditions. This is the essence of chaos theory. Even though the system is deterministic, every measurement of that system is an approximation. Therefore, any mathematical model based on those uncertain measurements will very possibly develop in a different way from the real thing. Even a small uncertainty is enough to create chaos.

  The geodesic path of a planet orbiting a star in a predictable way is shown in the left-hand image. The image on the right shows how the presence of three other celestial bodies—perhaps nearby planets or other stars—complicates the planet’s path, making it unpredictable, or chaotic.

  EDWARD LORENZ

  Born in 1917, in West Hartford, Connecticut, Edward Lorenz studied mathematics at Dartford College and Harvard University, gaining a masters degree at Harvard in 1940. After training as a meteorologist, he served with the US Army Air Corps in World War II. After the war, Lorenz studied meteorology at the Massachusetts Institute of Technology and began to develop ways to predict the behavior of the atmosphere. At that time, meteorologists used linear statistical modeling to forecast weather, and they often failed.

  In developing a nonlinear model of the atmosphere, Lorenz stumbled across the area of chaos theory that would later be dubbed the butterfly effect. He showed that even the most po
werful computers could not produce accurate long-term weather forecasts. Lorenz remained physically and mentally active until just before his death in 2008.

  Key work

  1963 Deterministic Nonperiodic Flow

  See also: The problem of maxima • Probability • Calculus • Newton’s laws of motion • Laplace’s demon • Topology • Fractals

  IN CONTEXT

  KEY FIGURE

  Lotfi Zadeh (1921–2017)

  FIELD

  Logic

  BEFORE

  350 BCE Aristotle develops a system of logic that dominates Western scientific reasoning until the 1800s.

  1847 George Boole invents a form of algebra in which variables can have one of only two values (true or false), paving the way for symbolic, mathematical logic.

  1930 Polish logicians Jan Łukasiewiecz and Alfred Tarski define a logic with infinitely many truth values.

  AFTER

  1980s Japanese electronics companies use fuzzy logic control systems in industrial and domestic appliances.

  The binary logic of any computer is clear: given valid inputs, it will provide appropriate outputs. However, binary computer systems are not always well suited for dealing with real-world inputs that are ambiguous or unclear. In the case of handwriting recognition, for example, a binary system would not be sufficiently subtle. A system controlled by fuzzy logic, however, allows for degrees of truth that can better analyze complex phenomena, including human actions and thought processes. Fuzzy logic is an offshoot of the fuzzy set theory developed in 1965 by Lotfi Zadeh, an Iranian–American computer scientist. Zadeh claimed that as a system becomes more complex, precise statements about it become meaningless; the only meaningful statements about it are imprecise. Such situations demand a many-valued (fuzzy) reasoning system.

  Standard set theory allows an element to either belong or not belong to a set, but fuzzy set theory allows degrees of membership or a continuum. Similarly, fuzzy logic allows a range of truth values for a proposition—not just completely true or completely false, the two values of Boolean logic. Fuzzy truth values also require fuzzy logical operators—for example, the fuzzy version of the AND operator of Boolean algebra is the MIN operator, which outputs the minimum of the two inputs.

  The classes of objects encountered in the real physical world do not have precisely defined criteria of membership.

  Lotfi Zadeh

  Creating fuzzy sets

  A basic computer program that mimics the simple human task of soft-boiling an egg might apply a single rule: boil the egg for five minutes. A more sophisticated program would, like a human, take the weight of the egg into account. It might divide eggs into two sets—small eggs of 1.76 oz (50 g) or less, and large ones over 1.76 oz—and boil the former for four minutes, and the latter for six. Fuzzy logicians call these crisp sets: each egg either does or does not belong.

  To achieve a perfectly cooked egg, however, the boiling time must be adjusted to match the weight of the egg. While an algorithm could use traditional logic to divide a set of eggs into precise weight ranges and assign exact cooking times, fuzzy logic achieves this result with a more general approach. The first step is to make the data fuzzy—every egg is regarded as both large and small, belonging to both sets to different degrees. For example, a 1.76 oz egg would have a membership degree of 0.5 for both sets, while an 2.82 oz (80 g) egg would be “large” with degree nearly 1, and also “small” with degree nearly 0. A fuzzy rule is then applied, with large eggs boiled for six minutes and small eggs for four. Through a process called fuzzy inference, the algorithm applies the rule to each egg based on its fuzzy set membership. The system will deduce that an 2.82 oz egg should be boiled for both four and six minutes (with degrees of almost 0 and almost 1 respectively). This output is then defuzzified to give a crisp logical output that can be used by the control system. As a result, the 2.82 oz egg would be assigned a boiling time of nearly 6 minutes.

  Fuzzy logic is now a ubiquitous part of computer-controlled systems. It has many applications, from forecasting weather to trading stocks, and plays a vital role in programming artificial intelligence systems.

  Fuzzy logic recognizes a continuum of truth values instead of the Boolean binary values of “yes” (1) or “no” (0). These fuzzy values resemble probabilities, but are fundamentally quite distinct—they indicate the degree to which a proposition is true, not how likely it is.

  Artificial intelligence

  A humanoid robot using AI works at the front desk of a Henn-na hotel in Tokyo, which claims to be the world’s first hotel with robotic staff.

  Fuzzy control systems can work effectively with uncertainties in the everyday world, and are therefore used in artificial intelligence (AI) systems. The fuzziness of AI helps to give the illusion of a self-directing intelligence, but in reality fuzzy logic processes data to smooth out uncertainty. AI is therefore entirely the product of a pre-programmed set of rules.

  Techniques such as machine learning, in which AIs program themselves by a process of trial and error, and expert systems, in which the AI draws upon a database of knowledge provided by human programmers, have greatly extended the abilities of AI. Nevertheless most AI is “narrow,” in that it is tasked with doing one job very well, generally better than a human can, but it cannot learn to do anything else and is unaware of what it does not know. A general AI that can direct its own learning in the same way as evolved intelligence (such as human intelligence) is the next goal of computer science.

  See also: Syllogistic logic • Binary numbers • Boolean algebra • Venn diagrams • The logic of mathematics • The Turing machine

  IN CONTEXT

  KEY FIGURE

  Robert Langlands (1936–)

  FIELD

  Number theory

  BEFORE

  1796 Carl Gauss proves the quadratic reciprocity theorem, relating the solvability of quadratic equations to prime numbers.

  1880–84 Henri Poincaré develops the concept of automorphic forms—tools that allow us to keep track of complicated groups.

  1927 Austrian mathematician Emil Artin extends the reciprocity theorem to groups.

  AFTER

  1994 Andrew Wiles uses a special case of Langlands’ conjectures to translate Fermat’s last theorem from a problem in number theory to one in geometry, enabling him to solve it.

  In 1967, the young Canadian–American mathematician Robert Langlands suggested a set of profound links between two major and seemingly unconnected areas of mathematics—number theory and harmonic analysis. Number theory is the mathematics of integers, in particular prime numbers. Harmonic analysis (in which Langlands specialized) is the mathematical study of waveforms, exploring how they can be broken down to sine waves. These fields seem fundamentally different: while sine waves are continuous, integers are discrete.

  Langlands’ letter

  In a 17-page handwritten letter to number theorist André Weil in 1967, Langlands offered several conjectures linking number theory and harmonic analysis. Realizing its significance, Weil had the letter typed up and circulated among number theorists through the late 1960s and ’70s. Once they had been made public, Langlands’ conjectures became influential across mathematics, and continue to shape research 50 years later.

  Modular (“clock”) arithmetic involves number systems with finite sets of numbers. On a 12-hour clock, for example, if you count on four hours from 10 o’clock, you get 2 o’clock; 10 + 4 = 2, because the remainder of 14 ÷ 12 is 2. In the Langlands program, numbers are usually manipulated by modular arithmetic.

  Uncovering links

  Langlands’ ideas involve highly technical mathematics. In basic terms, his areas of interest are Galois groups and functions called automorphic forms. Galois groups turn up in number theory and are a generalization of the groups that Évariste Galois used in order to study roots of polynomials.

  Langlands’ conjectures are significant in that they allowed problems from number theory to be reframed in the language
of harmonic analysis. The Langlands Program has been described as a mathematical Rosetta Stone, helping to translate ideas from one area of mathematics into another. Langlands himself has helped to develop the means for working on the Program, including generalizing functoriality—a way of comparing the structures of different groups.

  Langlands’ marriage of harmonic analysis and number theory could lead to a wealth of new tools, just as the 19th-century unification of electricity and magnetism into electromagnetism provided a new understanding of the physical world. By finding new links between mathematical fields that seem profoundly different, the Program has revealed some of the structures at the heart of mathematics. In the 1980s, Ukrainian mathematician Vladimir Drinfel’d expanded the Program’s scope to show that there might be a Langlands-type connection between specific topics within harmonic analysis and others within geometry. In 1994, Andrew Wiles used one of Langlands’ conjectures to help solve Fermat’s last theorem.

  ROBERT LANGLANDS

  Born near Vancouver, Canada, in 1936, Robert Langlands did not plan to go to study at a university until a teacher “took up an hour of class time” to publicly implore him to make use of his talents. He was also a gifted linguist, but at 16, he enrolled at the University of British Columbia, Canada, to study mathematics. He later moved to the US, where he was awarded a doctorate from Yale University in 1960. Langlands taught at Princeton, Berkeley, and Yale before moving to the Institute for Advanced Study (IAS), where he still occupies Einstein’s old office.