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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2.1 out of 5
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 Where did you get this guy? Previous comments about condescension and lack of focus are totally correct. Saari is more intent on demonstrating his grasp of specific mathematical ideas that are, clearly in his own mind, well beyond mere mortals. However, I disagree with describing him as arrogant – that word doesn’t begin to touch this aspect of his presentation. In the Timeline, Ptolemy, Copernicus, Newton, Galileo and Euler each earn one mention; Saari’s importance earns him(self) four mentions. And of the approximately 70 items in the bibliography, a full third are those of Donald G. Saari. Fortunately, this course was not my first from the Teaching Company; had it been, it would also have been my last.
Date published: 2013-12-28
Rated 1 out of 5 by from Incoherent I have enjoyed many of teaching company courses but find ideas presented in this course difficult to connect to other ideas in this same course and too the ideas of have from the considerable mathematical knowledge I had before getting this course.
Date published: 2013-01-19
Rated 1 out of 5 by from Falls well short of Teaching Company standards I have purchased and enjoyed many of The Teaching Company’s courses and expect to continue doing so in the future. Had this been the first course I’d tried, however, I might never have considered looking at another. It is, without doubt, the worst I have experienced. I have a strong mathematical and scientific background, but I found his “explanations” poor and often unconvincing. Basic concepts are glossed over so that little real insight is provided for the general public, while insufficient detail for various topics is provided for meaningful understanding by those who are more comfortable with mathematics. It was often difficult to figure out who was his intended audience. The lecturer also appears to be extremely impressed with his own work, and one might conclude that only subjects to which he or his students have made personal contributions are deemed worthy of his attention. At times his lecture style struck me as condescending and arrogant. I was very disappointed with this course and I would not recommend it to anyone.
Date published: 2012-08-26
Rated 4 out of 5 by from Just what I needed I'm surprised at the low ratings by others who viewed this course, and I found this course to be just what I needed. I have several excellent Teaching Co courses that explain the how to do equations, such as algebra, geometry, etc. Those are great for "doing" math. But I've wanted to understand better how math is applied to the real world, how mathematicians think about problems in the first place, then how they go about trying to resolve the problems through math. I found the first 8 lectures on mathematics and astronomy the most coherent and helpful lectures I've seen yet on math and astronomy! I finally get the differences in the early models of the universe, how later scientists revised those models and why, for instance, Kepler's laws were so helpful to Newton, and why Newton's laws made such a big difference to astronomy. The diagrams and explanations are helpful. Yes, by the end ,some of it was lost on me, but I'm confident that as I continue my own education in mathematics, I can later come back to these lectures and get even more out of them. I feel I better understand the importance of math and the mind of the mathematician.
Date published: 2012-04-23
Rated 2 out of 5 by from I Needed More Help This course suffers in comparison with the uniform excellence of the other Teaching Co. math courses I've watched, by Profs. Burger, Starbird, Benjamin, Devadoss, and Zeitz. The material was not explained in such a way that I could follow the reasoning. Granted, some of it was quite difficult, but that was the case in other math courses. "The Shape of Nature" comes to mind: some of the content of that course was beyond me, but Prof. Devadoss took great pains to walk his students through the toughest concepts, and even when I did not completely follow, I felt that I was at least on the right track. Here, I was often at a loss to follow how Prof. Saari got from Point A to Point B. He would state the problem, and then offer a conclusion or solution, but I felt that what came between was lacking. What may have been obvious to him was opaque to me. I have a strong interest and some background (got as far as calculus in college, a long time ago, have read in the field, and have done other math courses), but I was left behind frequently. One example: in the last lectures, he introduced the concept of a "winding number." He gave examples and illustrations, but to me the underlying concept was never made clear. I later found a definition in the guidebook that was somewhat helpful, but unless I dozed off, I don't think the definition was offered in the course itself and the examples didn't leave me feeling that I understood the logic behind the use of this tool. Dr Saari is an enthusiastic and mostly engaging lecturer. He's clearly a brilliant and creative mathematician who has made significant contributions in many areas. (Unlike another reviewer, I didn't mind that he identified his own contributions: they were directly relevant to the course, and why shouldn't he take pride in them? He was scrupulous about identifying colleagues and collaborators who contributed to his results.) I wish, though, that he had pitched this course more toward those like myself who need guidance through the rough patches.
Date published: 2011-07-29
Rated 1 out of 5 by from With respect, a very, very poor course It only takes a glance at Prof. Saari's c.v. to see that he is an extremely accomplished and distinguished mathematician, and to be impressed by the number of teaching awards with which he has been honored. So it is a great disappointment to find that little of his mathematical insight, and none of his teaching ability, are to be found in this course, which may well be the worst the Teaching Company has produced. Prof. Saari is a gifted speaker, whose flowing speech, self-assurance, and obvious conviction of the importance of what he is saying, would be wonderful if the content matched the form. Unfortunately, however, the multiple weaknesses of the content are all too apparent. They are: - Very, very little actual mathematics is discussed or explained. Really. Mostly a problem is briefly presented, a few inadequate equations and diagrams are shown, and Prof. Saari then announces the conclusion. One gets a notion of the problem and its answer, but rarely any substantial idea of how one gets from the former to the latter. - What little mathematics is presented is extremely unclear. This is not because sophisticated background is required for understanding; it is because the explanations provide inadequate background, fuzzy discussions of the mathematical thinking, and poor or no explanations of how one step leads to another. - Prof. Saari, as others have noted, concentrates almost exclusively on areas in which he has made a personal contribution. This would be fine if those areas were of more general interest mathematically. However, the early lectures, on more traditional mathematical topics, provide only rudimentary and haphazard treatment of potentially fascinating topics, while the later lectures, on voting and some other areas of 'behavioral mathematics' were, to me, just boring (immense though their practical importance in the real world might be.) - The actual information content of the lectures is remarkably low. Once one gets past Prof. Saari's excellent delivery, sentences and sometimes entire paragraphs of speech often turn out to be the equivalent of so much hand-waving, whose words add little or nothing to the argument being developed. The frequent repetition of such hyperbolic claims as "we're exploring the power of mathematics to understand almost everything around us," and such reassurances as "you're going to enjoy the next lecture," are also unhelpful. - As one of many possible specific examples of problems, chaos theory - a truly fascinating area - is very poorly explained (lecture 22.) (For a wonderful introduction to chaos try the TC's "Chaos" course by Prof. Steven Strogatz, or Jame's Gleick's outstanding book "Chaos - The Making of a New Science.") Given Prof. Saari's eminence, and the fact that I am not a mathematician, writing this review has been quite uncomfortable - I can only affirm that it represents my honest response to the course, and note that it is obviously at variance with the many groups who have awarded Prof. Saari his many teaching honors. If you do take the course, please review it! More opinions would be greatly appreciated.
Date published: 2011-04-23
Rated 1 out of 5 by from Tedious The content was okay as far as I got, but I couldn't get through it. I found myself unable to continue with the lectures because the professor indulged himself in a series of long, narcissistic asides that had no bearing on anything we were learning. It was unbearably tedious to listen to him go on and on about himself. It wasn't necessary or engaging, and it provided no insight into the material. I think the whole set should be reedited to remove overt self promotion and long drawn out personal stories about his childhood.
Date published: 2010-11-21
Rated 1 out of 5 by from Great Subject; Flawed Content I would have liked to have given Prof. Saari's course a better grade. When I finished it, however, I was inclined to send him a telegram: "Nice try, but no cigar." I imagined I knew how the Israelites might have felt if Moses had pledged to lead them to the Promised Land but, on reaching the last hill in their way, stopped and said, "Sorry, your can't go over the hill and down into the Promised Land, and you can't even go up to the top of the hill to look down into it to see what it looks like." Admittedly, I had a preconception of what the course might be about. When you learn that the square root of 2 is an irrational number in the sixth or seventh grade, the proof is so aesthetically pleasing, you begin to understand the power of mathematics. I expected the course to be more about applied math. As it is, showing the pure logical power of mathematics was not the focus of this course. Prof. Saari wants to show that mathematics is relevant to examining problems in the external world: it is a pre-pre-Mathematical Modeling course. I think there are three flaws and an error in the course. The first flaw is the fact that, while Prof. Saari's intention is to show links between mathematics and analysis of the external world, often his pronouncements fall short of the promise. He doesn't follow through. For example, in Lecture 12, he talks about Complexity ("Complexity is very, very complex!!!") and mentions that problems like Alcoholism are complex. Then he leaves us hanging. How are they complex? I asked. I never got an answer. [Let me recommend that Saari check out course #5181.] Prof. Saari has a tendency to jump around without making the connections he thinks he has established. In Lecture 7, he wraps up his discussion of the N-body problem with a set of nonsequiturs -- or connections I did not comprehend. Next, the level of mathematical competence required to grasp what he is talking about varies extremely. In the early lectures, you can get by without knowing much beyond high school geometry. You must know what a torus is, but he shows you. By lecture 22, you need to be a graduate student to grasp what he's talking about. The conjectures he posits in the last lectures are answerable by analyzing behavior in International Systems (e.g., the Chinese Warring States period, or the Thirty Years War), comparing the conquering styles of Genghis Khan v. Alexander the Great, or studying the meeting of the Mafia that occurred in the late 1950s in the Adirondack Mountains. And it was IBM not Apple that tried to eliminate Microsoft (in the late 1980s). You don't need phony problems. The third flaw is the injection of his own work into the lectures throughout the course. I have never heard a teacher -- and I studied at three universities -- discuss his own work with such unabashed personal promotion. Apparently, he solved the N-body problem for 3 bodies; he solved problems associated with Arrows Theorem, "extending the theorem;" he provides advice on how to guarantee an electoral victory; he solved political apportionment problems; and he thinks math will be useful to understanding general gang behavior. And we found out that he has worked with the most important mathematicians of his generation. By his word, I was thinking that the Swedish Academy should introduce a new Nobel Prize in Mathematics, and give the first one to Saari! Here's the error (of at least omission) he made: in examining a decision analysis problem, he wants to point out the mathematical principles required to proceed. These are: abstraction, continuity, acceptance of unanimity, and reasonableness. His example is two people planning a picnic on a desert island. To cut to the chase, his conclusion is that one of the agents needs to be dominant, if both cannot agree on the spot on the island to have the picnic. However, I was reminded of a European country with a Parliament having an equal number of seats after an election in the 1970s and no desire to share political power. By Saari's reasoning, a coup d'état by one or the other party would have been in order. What in fact happened was that the sides decided to draw lots in instances when voting was tied. By Saari's own assumption of reasonableness or unanimity, the parties can agree to make a random choice -- cast dice, cut cards, or whatever -- eliminating the need for dominance. Even a husband and wife planning a night at the cinema might draw cards from a fair deck for a decision, if they cannot agree on which film to see. This is the problem with mathematical models of social life: how to conceive of and model real options. (And too many Harvard MBAs try to shoe-horn people into mathematically modeled patterns of behavior that contradict more efficient natural tendencies.) There is a growing academic movement challenging the math-based ideal-type, Platonic-Cartesian model of deduction as singularly valid tool for examining social phenomena. Rhetoricians like Perelman and Toulmin are among the leaders. Mathematical models must be connected to evidence, rather than advocating people to adapt their behavior to the conclusions of mathematical models; and in too many instances, one historical case is insufficient to make a general deduction. The search for patterns over time and varying local contexts is something mathematicians can work with, and this is one important point that Saari makes pains to make.
Date published: 2010-10-30
Rated 5 out of 5 by from A Different Look at Math I was quite intrigued by the approach of these lectures, looking at math from a different perspective. If you want 'straight' math, buy the algebra and geometry courses. These lectures will giggle your gray matter, but when the penny eventually droppeth, you'll be a happy mathematical camper.
Date published: 2010-08-17
Rated 3 out of 5 by from Hmmm I was disappointed. I was disappointed in the course content and presentation. I'm very sceptical about some the conclusions drawn from the mathematical results.
Date published: 2010-08-01
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