Power of Mathematical Thinking: From Newton's Laws to Elections and the Economy

Course No. 1417
Professor Donald G. Saari, Ph.D.
University of California, Irvine
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Course Overview

Scientists studying the universe at all scales often marvel at the seemingly "unreasonable effectiveness" of mathematics—its uncanny ability to reveal the hidden order behind the most complex of nature's phenomena. They are not alone: Economists, sociologists, political scientists, and many other specialists have also experienced the wonder of math's muscle power.

This use of mathematics to solve problems in a wide range of disciplines is called applied mathematics, and it is a far cry from the impression that many people have of math as an abstract field that has no relevance to the real world. Consider the remarkable utility of the following ideas:

  • The n-body problem: Beginning with Isaac Newton, the attempts to predict how a group of objects behave under the influence of gravity have led to unexpected insights into a wide range of mathematical and physical phenomena. One outcome is the new field of chaos theory.
  • Torus: The properties of a donut shape called a torus shed light on everything from the orbits of the planets to the business cycle, and they also explain how the brain reads emotions, how color vision works, and the apportionment scheme in the U.S. Congress.
  • Arrow's impossibility theorem: In an election involving three or more candidates, several crucial criteria for making the vote equitable cannot all be met, implying that no voting rule is fair. This surprising result has had widespread application in the theory of social choice and beyond.
  • Higher dimensions: Whenever multiple variables come into play, a problem may benefit by exploring it in higher dimensions. With a host of applications, higher dimensions are nonetheless difficult to envision—although Salvador Dali came close in some of his paintings.

Math's very abstraction is the secret of its power to strip away inessentials and get at the heart of a problem, giving deep insight into situations that may not even seem like math problems—such as how to present a winning proposal to a committee or to understand the dynamical interactions of street gangs. Given this astonishing versatility, mathematics is truly one of the greatest tools ever developed for unlocking mysteries.

In 24 intensively illustrated half-hour lectures, The Power of Mathematical Thinking: From Newton's Laws to Elections and the Economy gives you vivid lessons in the extraordinary reach of applied mathematics. Your professor is noted mathematician Donald G. Saari of the University of California, Irvine—a member of the prestigious National Academy of Sciences, an award-winning teacher, and an exuberantly curious investigator, legendary among his colleagues for his wide range of mathematical interests.

Inviting you to explore a rich selection of those interests, The Power of Mathematical Thinking is not a traditional course in applied mathematics or problem solving but is instead an opportunity to experience firsthand from a leading practitioner how mathematical thinking can open doors and operate powerfully across multiple fields. Designed to take you down new pathways of reasoning no matter what your background in mathematics, these lectures show you the creative mind of a mathematician at work—zeroing in on a problem, probing it from a mathematical point of view, and often reaching surprising conclusions.

When Elections Go Haywire

Professor Saari is a pioneer in the application of mathematics to problems in astronomy, economics, and other fields, but he is best known to the general public for his influential critique of election rules. In this course, he devotes several lectures to what can go wrong with elections, showing how the least preferred contender in a race with three or more candidates can sometimes end up as the winner—and how this flaw is latent in many apparently fair voting methods. He also shows that similar problems plague other ranking procedures, such as the method of apportioning congressional seats in the U.S. Congress. Among the many cases you explore are these:

  • Suppose your local school ranks students by the number of A's they receive. It sounds like a formula for excellence, but what it means is that the student who gets an A in one course and F's in everything else will be ranked above the student who gets all B's. The same flaw is at the heart of plurality voting.
  • What would you think if a consultant approached your organization and offered to write a fair voting rule that guaranteed whatever outcome you wanted in a vote involving several alternatives? Such consultants may not exist, but their methods do and are in wide use when making paired comparisons.
  • You are on a search committee whose members have voted on four candidates. Before you announce the winner, the lowest vote-getter drops out. Should the committee take a new vote? If it doesn't, the original choice may not represent the true preferences of the members.
  • Your state is entitled to a number of seats in the U.S. House of Representatives proportional to its population. Should you object if the total number of seats in the House is increased? In fact, your state could lose a seat under this scenario, as Alabama did in 1880.

Apart from the fascination of studying such examples, you invariably get the big picture from Professor Saari, as he shows how the power of mathematics comes from reaching beyond, say, a particular election to consider what can possibly happen in any election. And he introduces a set of mathematical ideas that prove remarkably useful at analyzing a wide range of problems at a deep level.

A Mathematical Odyssey

Both entertaining and intellectually exhilarating, this course is based on Professor Saari's own mathematical odyssey—from his early career in celestial mechanics to his discovery that the social sciences are fertile ground for sophisticated applied mathematics. Furthermore, Dr. Saari has delightfully contrarian impulses that make him question why something is true, or, indeed, if it is true at all. In this spirit, you examine Newton's theory of gravitation, Arrow's impossibility theorem, Adam Smith's "invisible hand" concept, and other ideas, pushing beyond the standard interpretations to extract new insights that in many cases represent original contributions by Dr. Saari.

By his enthusiastic example, Professor Saari shows that the abstract nature of mathematics is nothing to fear. Instead, it is something to cherish, nurture, and use with imagination. "In mathematics, we have the ability to transcend our experiences," he says. "We do not want to solve the problems of the past; we want to solve problems that we've never experienced or didn't anticipate." And for that, we need The Power of Mathematical Thinking.

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24 lectures
 |  Average 31 minutes each
  • 1
    The Unreasonable Effectiveness of Mathematics
    Begin your mathematical odyssey across a wide range of topics, exploring the apparently unreasonable effectiveness of mathematics at solving problems in the real world. As an example, Professor Saari introduces Simpson's paradox, which shows that a whole can surprisingly often differ from the sum of its parts. x
  • 2
    Seeing Higher Dimensions and Symmetry
    Many of the examples in this course deal with the geometry of higher-dimension spaces. Learn why this is a natural outcome of situations with several variables and why higher dimensions are easier to understand than you may think. Warm up by analyzing four-dimension cubes and pyramids. x
  • 3
    Understanding Ptolemy's Enduring Achievement
    Although the ancient astronomer Ptolemy was wrong about the sun going around the Earth, his mathematical insights are still applicable to modern problems, such as the shape of the F ring orbiting Saturn. In this lecture, you use Ptolemy's methods to study the motions of Mars and Mercury. x
  • 4
    Kepler's 3 Laws of Planetary Motion
    Delve into Kepler's three laws, which explain the motions of the planets and laid the foundation for Newton's revolution in mathematics, physics, and astronomy. Discover how Kepler used mathematical thinking to make fundamental discoveries, based on the work of observers such as Tycho Brahe. x
  • 5
    Newton's Powerful Law of Gravitation
    Explore Newton's radically different way of thinking in science that makes him a giant among applied mathematicians. By analyzing the mathematical consequences of Kepler's laws, he came up with the unifying principle of the inverse square law, which governs how the force of gravity acts between two bodies. x
  • 6
    Is Newton's Law Precisely Correct?
    According to Newton's inverse square law, the gravitational attraction between two objects changes in inverse proportion to the square of the distance between them. But why isn't it the cube of the distance? In testing this and other alternatives, follow the reasoning that led Newton to his famous law. x
  • 7
    Expansion and Recurrence—Newtonian Chaos!
    While a two-body system is relatively simple to analyze with Newton's laws of motion, the situation with three or more bodies can become chaotically unpredictable. Discover how this n-body problem has led to progressively greater insight into the chaos of "two's company, but three's a crowd." x
  • 8
    Stable Motion and Central Configurations
    When the number of bodies is greater than two, chaos need not rule. Some arrangements—called central configurations—are stable because the forces between the different bodies cancel out. Probe this widespread phenomenon, which occurs with cyclones, asteroids, spacecraft mid-course corrections, and even vortices from a canoe paddle. x
  • 9
    The Evolution of the Expanding Universe
    Use mathematical ideas that you have learned in the course to investigate the evolution of an expanding universe according to Newton's laws. Amazingly, the patterns that emerge from this exercise reflect the observed organization of the cosmos into galaxies and clusters of galaxies. x
  • 10
    The Winner Is ... Determined by Voting Rules
    Focus on the paradoxical results that can occur in plurality voting when three or more candidates are involved. The Borda count, which ranks candidates in order of preference with different points for each level of ranking, is one method for more accurately representing the will of the voters. x
  • 11
    Why Do Voting Paradoxes Occur?
    When voters rank their preferences for different candidates in an election, tallying the results can be tedious and complicated. Learn Professor Saari's ingenious geometric method that makes determining the final rankings as enjoyable as a Sudoku puzzle. x
  • 12
    The Order Matters in Paired Comparisons
    Can you come up with a voting rule that will ensure the election of a candidate that most voters rank near the bottom in a large field of candidates? In fact, there's a method that works, showing that the order in which alternatives are considered can determine the final outcome. x
  • 13
    No Fair Election Rule? Arrow's Theorem
    Explore Arrow's impossibility theorem, which is often summarized as "no voting rule is fair." But is that depiction correct? Dr. Saari shows how the conditions of Arrow's theorem can be modified in small ways to remove paradoxical outcomes and make elections more equitable. x
  • 14
    Multiple Scales—When Divide and Conquer Fails
    Divide and conquer is a tried and true technique for solving complex problems by breaking them into manageable components. But how successful is it? Learn how Arrow's theorem shows that this approach has built-in flaws, much as with voting rules. x
  • 15
    Sen's Theorem—Individual versus Societal Needs
    Expanding on Arrow's theorem, Amartya Sen showed that there is an apparently inevitable restriction on the rights of individuals to make even trivial decisions. But Professor Saari argues that Sen's theorem has a different result—one that helps explain the origins of a dysfunctional society. x
  • 16
    How Majority Improvements Go Wrong
    Use geometry to investigate issues from game theory; namely, how to devise an unbeatable strategy when presenting a proposal to a committee and why too much tinkering can ruin the consensus on a project. Also, see how to produce a stable outcome from a situation involving many choices. x
  • 17
    Elections with More than Three Candidates
    Delve into the problems that can arise when more than three candidates run in a plurality election. For example, with seven candidates, the number of things that can go wrong is 1050—or a one followed by 50 zeros! x
  • 18
    Donuts in Decisions, Emotions, Color Vision
    See how the simple geometry of a donut shape, called a torus, helps unlock an abundance of mysteries, including how to decide where to have a picnic, how the brain reads emotions in faces, and how color vision works. x
  • 19
    Apportionment Problems of the U.S. Congress
    Because a congressional district cannot be represented by a fraction of a representative, a rounding-off procedure is needed. Discover how this explains why there are 435 representatives in the U.S. Congress—and how this mystery is unlocked by using the geometry of a torus. x
  • 20
    The Current Apportionment Method
    Beware of looking at the parts in isolation from the whole—a mathematical lesson illustrated by the subtly flawed current method of apportioning representatives to the U.S. Congress. The problem resides in what happens in the geometry of higher-dimension cubes. x
  • 21
    The Mathematics of Adam Smith's Invisible Hand
    According to Adam Smith's "invisible hand," the unfettered market balances supply and demand to reach an equilibrium price for any commodity. Probe this famous idea with the tools of mathematics to discover that the invisible hand may be shakier than is generally supposed. x
  • 22
    The Unexpected Chaos of Price Dynamics
    The world economy is full of examples in which the invisible hand should have created price stability, but chaos resulted. What went wrong? Discover that many times there isn't enough information to allow the price mechanism to function as Adam Smith envisioned. x
  • 23
    Using Local Information for Global Insights
    Follow Professor Saari into the unknown to see what a simple graph can reveal about a seemingly unpredictable rivalry between street gangs. Then continue your investigation of social interaction by examining how people judge fairness when sharing is in their mutual best interest. x
  • 24
    Toward a General Picture of What Can Occur
    Finish the course by using a concept called the winding number to explain why fairness is judged differently by different cultures. Your analysis captures perfectly the ability of mathematics to make sense of the world through the power of abstraction. x

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Your professor

Donald G. Saari

About Your Professor

Donald G. Saari, Ph.D.
University of California, Irvine
Dr. Donald G. Saari is Distinguished Professor of Economics and Mathematics and the Director of the Institute for Mathematical Behavioral Sciences at the University of California, Irvine. He earned his bachelor's degree from Michigan Technological University and his Ph.D. from Purdue University. Before joining the faculty at UCI, Dr. Saari spent three decades teaching at Northwestern University, where he became the first...
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