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

Course Overview

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 by noted mathematician Donald G. Saari. In 24 intensively illustrated half-hour lectures, you will explore the extraordinary reach of applied mathematics and see how it operates across multiple fields.

Professor Saari invites you to experience mathematical thinking as a powerful tool for understanding complex issues. Designed for learners of all backgrounds, this course takes you down new pathways of reasoning, emphasizing how mathematical thought processes can open doors to surprising conclusions.

Video Lectures

1. The Unreasonable Effectiveness of Mathematics – 31 min
   Explore the effectiveness of mathematics in solving real-world problems.
2. Seeing Higher Dimensions and Symmetry – 33 min
   Understand higher-dimensional geometry and its implications.
3. Understanding Ptolemy’s Enduring Achievement – 32 min
   Use Ptolemy’s methods to study planetary motion and their modern applications.
4. Kepler’s 3 Laws of Planetary Motion – 31 min
   Delve into Kepler’s laws and their foundational role in modern mathematics.
5. Newton’s Powerful Law of Gravitation – 31 min
   Examine Newton’s principles and their impact on understanding gravity.
6. Is Newton’s Law Precisely Correct? – 30 min
   Investigate the reasoning behind Newton’s inverse square law of gravitation.
7. Expansion and Recurrence—Newtonian Chaos! – 31 min
   Discover chaos in systems with three or more bodies using Newton’s laws.
8. Stable Motion and Central Configurations – 33 min
   Learn about stable configurations in physical systems and their practical applications.
9. The Evolution of the Expanding Universe – 33 min
   Use mathematical principles to explore cosmic evolution.
10. The Winner Is… Determined by Voting Rules – 33 min
    Analyze how voting methods can lead to paradoxical outcomes.
11. Why Do Voting Paradoxes Occur? – 31 min
    Delve into geometric methods for resolving voting paradoxes.
12. The Order Matters in Paired Comparisons – 33 min
    Examine how the sequence of options can influence decisions.
13. No Fair Election Rule? Arrow’s Theorem – 31 min
    Explore Arrow’s theorem and its implications for equitable voting systems.
14. Multiple Scales—When Divide and Conquer Fails – 30 min
    Understand the limitations of dividing problems for solutions.
15. Sen’s Theorem—Individual versus Societal Needs – 31 min
    Discuss the implications of Sen’s theorem for individual rights.
16. How Majority Improvements Go Wrong – 31 min
    Learn how to devise strategies to ensure fair decision-making.
17. Elections with More than Three Candidates – 33 min
    Investigate complexity arising in plurality elections with multiple candidates.
18. Donuts in Decisions, Emotions, Color Vision – 31 min
    Discover how geometry explains various phenomena in decision-making and perception.
19. Apportionment Problems of the U.S. Congress – 32 min
    Examine how apportionment methods affect congressional representation.
20. The Current Apportionment Method – 32 min
    Analyze the complexities of current congressional apportionment practices.
21. The Mathematics of Adam Smith’s Invisible Hand – 32 min
    Explore the dynamics of free markets through mathematics.
22. The Unexpected Chaos of Price Dynamics – 31 min
    Study why markets can act unpredictably despite the invisible hand.
23. Using Local Information for Global Insights – 32 min
    Learn how local data can reveal broader social patterns.
24. Toward a General Picture of What Can Occur – 32 min
    Conclude by applying mathematical principles to societal issues.