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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.
Course Lecture Titles
1. The Unreasonable Effectiveness of Mathematics
2. Seeing Higher Dimensions and Symmetry
3. Understanding Ptolemy's Enduring Achievement
4. Kepler's 3 Laws of Planetary Motion
5. Newton's Powerful Law of Gravitation
6. Is Newton's Law Precisely Correct?
7. Expansion and Recurrence—Newtonian Chaos!
8. Stable Motion and Central Configurations
9. The Evolution of the Expanding Universe
10. The Winner Is ... Determined by Voting Rules
11. Why Do Voting Paradoxes Occur?
12. The Order Matters in Paired Comparisons
13. No Fair Election Rule? Arrow's Theorem
14. Multiple Scales—When Divide and Conquer Fails
15. Sen's Theorem—Individual versus Societal Needs
16. How Majority Improvements Go Wrong
17. Elections with More than Three Candidates
18. Donuts in Decisions, Emotions, Color Vision
19. Apportionment Problems of the U.S. Congress
20. The Current Apportionment Method
21. The Mathematics of Adam Smith's Invisible Hand
22. The Unexpected Chaos of Price Dynamics
23. Using Local Information for Global Insights
24. Toward a General Picture of What Can Occur