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.
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.
Overview
About
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
