The Power of Mathematical Thinking

The Power of Mathematical Thinking

Salepage

Donald G. Saari, Ph.D.

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.

Institution: University of California, Irvine

Alma mater: Purdue University

Course Overview

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.

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.

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.

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.

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.

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.

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.”

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.

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.

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.

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.

12. Why Do Voting Paradoxes Occur?

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.

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.

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.

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.

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.

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!

18. Donuts in Decisions, Emotions, Color Vision

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.

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.

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.

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.

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.

24. Toward a General Picture of What Can Occur