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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18 Reviews
22% of reviewers would recommend this product
Course No. 1417
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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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Power of Mathematical Thinking: From Newton's Laws to Elections and the Economy is rated 2.1 out of 5 by 18.
Rated 1 out of 5 by from Lemon-aide Anyone? After viewing over 70 of the Great Courses, I do enjoy taking note of teaching style as well as course content. The course content is boring as opposed to engaging. It seems designed to appeal to a narrow audience. Sharing understanding doesn't seem to be a goal. As to teaching style, this professor is either ignorant of or deliberately avoids the practices of modern andragogy. This course gets my vote as the worst offering of the many Great Courses with which I am familiar.
Date published: 2019-01-21
Rated 1 out of 5 by from Pay Attention to the Ratings Tough to follow, nothing to learn, watched portions of 10 or so lectures after the first just looking for something to spark my interest, no joy. I should have paid attention to the ratings.
Date published: 2017-12-01
Rated 4 out of 5 by from Mathematical value...plus! I bought this even though the low rating and professor comments would make one think twice about doing so. Sometimes the presentation style can be so unusual as to be entertaining and join my Great Courses that I call 'cult classics'. I needed the mathematics value it offered and actually have come to enjoy the presentations of the 'most interesting man in the world'...even if he has to say so himself.
Date published: 2017-09-20
Rated 2 out of 5 by from I simply can't stand it; I almost made it through 17 lectures, but the thought of hearing more of the 'I found', 'I proved', 'I, I, I" was unbearable. The subject of the course has a lot of potential, but the ego of the speaker makes it impossible for him to provide the graphics necessary to follow his train of thought - he appears speaking largely to himself or to a group of mathematicians who are well versed in all the mathematics behind the subject he expounds. .
Date published: 2017-08-15
Rated 4 out of 5 by from A Graduate Seminar The short comings of this professor's presentation are well catalogued by the earlier reviews. However, I believe that this course has more to offer than indicated by these comments. The first point is that this is not a survey course on a particular topic or discipline. Rather it is a set of topics that much more resemble a series of lectures in an open graduate seminar exploring new or advanced topics. These classes remind me of those single afternoon lectures by visiting professors posted on the bulletin board possibly in the lecture halls but more probably in the department offices. Therefore, you need to approach these topics in the spirit of inquiry and curiosity. The lectures do have a loose framework that follows the lecturer's own intellectual journey. The early lectures cover his early work in mathematical astronomy. They introduce the topics on the uses of geometry in the fourth and higher dimensions and Newton's three body problem. The three body problem leads us into the topic of the motion of atoms. This descent into the microscopic transitions the lectures to the professor's later mathematical work in the social sciences in particular decision theory, the Arrow theorem, and efficient voting systems. The sneaky unity of these presentations is illustrated by his solution of problems with holes in them. For example, how do you pick a hospital site for a hospital that will serve three communities on the shores of a large lake. Answer, you consider the lake as the shadow of a donut/torus on the plane and solve the problem on the continuous four dimensional surface of the donut and not the discontinuous surface of the two dimensional plane with the hole in it. In this case apparently independent early topics reappear in the latter lectures. The professor is a controversial figure in the field of voting theory. There is a readable paperback by Poundstone listed in the recommended readings that covers the voting system controversies in political science. if you purchase this course you should read this mass market paperback to get an understanding of the controversies surrounding the political science topics at the end of the lectures. This set of lectures is in part an academic selling his life's work. The ego does show at times and I think this is one source of negative feelings expressed in many of the other reviews. However, the math is good and the controversies are very real and very topical in advanced level debates in economics and political science. I remember from my college days the saying that what you learn in high school is forty years old, what you learn as an undergraduate is twenty years old and if you want to know what we really know/don't know you have to listen to a graduate school lecture. I recommend this class primarily, because it is one of the few places I have found decent coverage of the Arrow theorem on the transitivity of preferences in decision theory applicable to economics and political science. Secondarily, it is interesting example of one man's intellectual journey into cutting edge academic controversies. if you feel adventuresome try it. Take the good with the annoying. There are some gems to pick up along the journey.
Date published: 2016-06-18
Rated 1 out of 5 by from Bellow Expectation There is a quality problem on both the author and the company sides. The author underestimated the audience, and the company may had not given enough guidance to the author as to what good presenters did. In such cases the customer gets a refund or a replacement. By the way, I have purchased 20+ courses, so I am a regular customer.
Date published: 2016-05-02
Rated 1 out of 5 by from So, Kepler formulated a...oh look, a shiny penny! I found this professor's presentation excruciatingly incoherent, disconnected, and amateurish. I read the other reviews and didn't think anyone with his credentials could be that bad, so bought this product anyway (curiosity got the better of me). I now definitely agree with those who say this course is not at all to the standards of any of the other courses I have bought (which are more than 30 and counting). At first I thought he was just nervous or quirky (very quirky). But, as I kept struggling through the lectures (trying my best to ignore his idiosyncratic style), I found that his inability to stick to a train of thought, penchant for arriving at conclusions in mid-sentence, and constant digressions to other concepts, prevented even a basic grasp of what he's talking about. I spent more time on Wikipedia trying to fill in the gaps in details of the topics and conclusions that he raised in order to try and understand what he was trying to say. In fact, that was the only good in this course...getting these concepts mentioned through his "half baked" explanations, at least allowed me to piece together the overall objective of his lectures (I guess) using other resources. I think he may understand what he's trying to teach, but he is not the type of person that should be teaching (if this is how he is in an actual class room environment).
Date published: 2015-08-22
Rated 3 out of 5 by from Saari I didn't enjoy this much... I had some difficulty understanding Dr Saari's presentation, owing to his somewhat eccentric accent, the way he swallows words, raises & lowers his voice, and fails to emphasise the appropriate words. This was a palpable challenge -- a major difference from the high standard of most Great Courses professors. He is also verbose: I was continually waiting eagerly for him to make his points and to get to the meat of his topics. He is a highly-qualified mathematician with excellent teaching experience, but one needs to exercise considerable patience and intense concentration to derive maximum benefit from his lectures. At times I felt he was actually struggling to get his thoughts across. So... I can understand why his course has been rated so low, though I feel three stars overall are called for. The content of the lectures is okay, but some of the mathematical explanations were blurry, and I feel that a LOT more pure mathematics should have been included in the course. It was not clear to me how the power of mathematical thinking could be applied, as promised in the course title. I didn't feel Dr Saari was pompous, rather that he wanted to share his excitement and inspire his audience. I have learned from the talks, though some areas were a struggle. On balance, I'm afraid I cannot recommend this course. It was a chore rather than a pleasure.
Date published: 2014-02-08
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