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

Discover how mathematics can reveal hidden truths about a variety of topics—from the economy to elections to the laws of physics—in this fascinating course that explores the surprising connections between abstract math and physical reality.
Power of Mathematical Thinking: From Newton's Laws to Elections and the Economy is rated 2.5 out of 5 by 21.
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Rated 5 out of 5 by from Take the challenge! Very challenging, but rewarding and doable with the aid of the readings, including the suggested readings; such as, Saari's book, "Disposing Dictators, Demystifying Voting Paradoxes," from the previous year (2009).
Date published: 2024-02-07
Rated 5 out of 5 by from TGC Cult Classic As a helplessly addicted Great Courses junkie (20+ year customer), Donald Saari's "Power of Mathematical Thinking" achieved a legendary status of sorts to me by being perennially listed as the lowest rated course offered. With that type of designation, I just had to purchase it to see how bad a professor must be to be the WOAT. However, like the other 4-5 star reviewers, I ended up being pleasantly surprised by how much the course stretched my brain. And, like many of my fellow satisfied reviewers, I also believe that many/most of the 1 star reviewers were off-base by expecting it to be exactly like a review course of an established topic like Calculus. Dr. Saari's goal is to share the thought process a mathematician goes through while solving a complex problem, for which no obvious mathematical pathway exists. This may not be everyone's cup of tea, but I find the now-ness of this type of learning experience to be a nice change from simply re-hashing what we already know. As for Dr. Saari's accent and minor quirks, they only add to the Cult Classic status to me.
Date published: 2020-12-31
Rated 5 out of 5 by from Inspiring I am amazed that this course has received such negative reviews. I have only watched the lectures on the Universe (lectures 1-9) but these are inspirational. I study mathematics and Donald Saari has done an excellent job at presenting the mathematics of the universe. Sure, there are a few places where I would have liked greater elaboration or felt that a little bit too much hand waving was going on - but, many other mathematics courses that rated better, suffer from this issue as well. The graphics were superb. The style was pleasantly avuncular. The claim that Donald Saari was egocentric is, to my mind, unfounded. He has done some significant work in this field and his references to his work were relevant and not overworked. I will be looking at these lectures repeatedly to cement the ideas further.
Date published: 2020-12-09
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! 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
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One of the secret powers of math is its ability to provide insights into situations that may not even seem like math problems. In 24 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. Join noted mathematician and Professor Donald G. Saari on this dynamic exploration of how math is used to solve important problems in astronomy, economics, politics, and other vital fields.


Donald G. Saari

There is a power of transferability in mathematics—a tremendous transferability of ideas from one discipline to another. This is one of the powerful aspects of mathematical reasoning.


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 Pancoe Professor of Mathematics. Professor Saari's early research focused on dynamical issues such as the evolution of the universe, but his interests later broadened into the social sciences. As a result, he also became a member of Northwestern's Department of Economics, Department of Applied Mathematics and Engineering Science, and the Center for Mathematical Studies in Economics. Professor Saari was twice named Most Influential Professor by students at Northwestern University, among his many other teaching awards. He is also the recipient of honorary doctorates from several prestigious universities, including Purdue. Dr. Saari was Chief Editor of the Bulletin of the American Mathematical Society, and he is a member of the National Academy of Sciences, the Finnish Academy of Science and Letters, and the American Academy of Arts and Sciences. He has also been a Guggenheim Fellow and the chair of the U.S. National Committee of Mathematics.

The Unreasonable Effectiveness of Mathematics

01: 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.

31 min
Seeing Higher Dimensions and Symmetry

02: 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.

33 min
Understanding Ptolemy's Enduring Achievement

03: 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.

32 min
Kepler's 3 Laws of Planetary Motion

04: 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.

31 min
Newton's Powerful Law of Gravitation

05: 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.

31 min
Is Newton's Law Precisely Correct?

06: 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.

30 min
Expansion and Recurrence—Newtonian Chaos!

07: 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."

31 min
Stable Motion and Central Configurations

08: 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.

33 min
The Evolution of the Expanding Universe

09: 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.

33 min
The Winner Is... Determined by Voting Rules

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.

33 min
Why Do Voting Paradoxes Occur?

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.

31 min
The Order Matters in Paired Comparisons

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.

33 min
No Fair Election Rule? Arrow's Theorem

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.

31 min
Multiple Scales—When Divide and Conquer Fails

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.

30 min
Sen's Theorem—Individual versus Societal Needs

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.

31 min
How Majority Improvements Go Wrong

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.

31 min
Elections with More than Three Candidates

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 one followed by 50 zeros!

33 min
Donuts in Decisions, Emotions, Color Vision

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.

31 min
Apportionment Problems of the U.S. Congress

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.

32 min
The Current Apportionment Method

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.

32 min
The Mathematics of Adam Smith's Invisible Hand

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.

32 min
The Unexpected Chaos of Price Dynamics

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.

31 min
Using Local Information for Global Insights

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.

32 min
Toward a General Picture of What Can Occur

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.

32 min