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Great Thinkers, Great Theorems
Course No. 1471
Taught By Professor William Dunham, Ph.D., Ohio State University,
Muhlenberg College

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Mathematics is filled with beautiful theorems that are as breathtaking as the most celebrated works of art, literature, or music. They are the Mona Lisas, Hamlets, and Fifth Symphonys of the field—landmark achievements that repay endless study and that are the work of geniuses as fascinating as Leonardo, Shakespeare, and Beethoven. Here is a sample:

* Pythagorean theorem: Although he didn't discover the Pythagorean theorem about a remarkable property of right triangles, the Greek mathematician Euclid devised an ingenious proof that is a mathematical masterpiece. Plus, it's beautiful to look at!
* Area of a circle: The formula for the area of a circle, A = À r2, was deduced in a marvelous chain of reasoning by the Greek thinker Archimedes. His argument relied on the clever tactic of proof by contradiction not once, but twice.
* Basel problem: The Swiss mathematician Leonhard Euler won his reputation in the early 1700s by evaluating an infinite series that had stumped the best mathematical minds for a generation. The solution was delightfully simple; the path to it, bewilderingly complex.
* Larger infinities: In the late 1800s, the German mathematician Georg Cantor blazed the trail into the "transfinite" by proving that some infinite sets are bigger than others, thereby opening a strange new realm of mathematics.

You can savor these results and many more in Great Thinkers, Great Theorems, 24 half-hour lectures that conduct you through more than 3,000 years of beautiful mathematics, telling the story of the growth of the field through a carefully chosen selection of its most awe-inspiring theorems.

Approaching great theorems the way an art course approaches great works of art, the course opens your mind to new levels of math appreciation. And it requires no more than a grasp of high school mathematics, although it will delight mathematicians of all abilities.

Your guide on this lavishly illustrated tour, which features detailed graphics walking you through every step of every proof, is Professor William Dunham of Muhlenberg College, an award-winning teacher who has developed an artist's eye for conveying the essence of a mathematical idea. Through his enthusiasm for brilliant strategies, novel tactics, and other hallmarks of great theorems, you learn how mathematicians think and what they mean by "beauty" in their work. As added enrichment, the course guidebook has supplementary questions and problems that allow you to go deeper into the ideas behind the theorems.

An Innovative Approach to Mathematics

Professor Dunham has been taking this innovative approach to mathematics for over a quarter-century—in the classroom and in his popular books. With Great Thinkers, Great Theorems you get to watch him bring this subject to life in stimulating lectures that combine history, biography, and, above all, theorems, presented as a series of intellectual adventures that have built mathematics into the powerful tool of analysis and understanding that it is today.

In the arts, a great masterpiece can transform a genre; think of Claude Monet's 1872 canvas Impression, Sunrise, which gave the name to the Impressionist movement and revolutionized painting. The same is true in mathematics, with the difference that the revolution is permanent. Once a theorem has been established, it is true forever; it never goes out of style. Therefore the great theorems of the past are as fresh and impressive today as on the day they were first proved.

What Makes a Theorem Great?

A theorem is a mathematical proposition backed by a rigorous chain of reasoning, called a proof, that shows it is indisputably true. As for greatness, Professor Dunham believes the defining qualities of a great theorem are elegance and surprise, exemplified by these cases:

* Elegance: Euclid has a beautifully simple way of showing that any finite collection of prime numbers can't be complete—that there is always at least one prime number left out, proving that the prime numbers are infinite. Dr. Dunham calls this one of the greatest proofs in all of mathematics.
* Surprise: Another Greek, Heron, devised a formula for triangular area that is so odd that it looks like it must be wrong. "It's my favorite result from geometry just because it's so implausible," says Dr. Dunham, who shows how, 16 centuries later, Isaac Newton used algebra in an equally surprising route to the same result.

Great Thinkers, Great Theorems includes many lectures that are devoted to a single theorem. In these, Professor Dunham breaks the proof into manageable pieces so that you can follow it in detail. When you get to the Q.E.D.—the initials traditionally ending a proof, signaling quod erat demonstrandum (Latin for "that which was to be demonstrated")—you can step back and take in the masterpiece as a whole, just as you would with a painting in a museum.

In other lectures, you focus on the biographies of the mathematicians behind these masterpieces—geniuses who led eventful, eccentric, and sometimes tragic lives. For example:

* Cardano: Perhaps the most bizarre mathematician who ever lived, the 16th-century Italian Gerolamo Cardano was a gambler, astrologer, papal physician, convicted heretic, and the first to publish the solution of cubic and quartic algebraic equations, which he did after a no-holds-barred competition with rival mathematicians.
* Newton and Leibniz: The battle over who invented calculus, the most important mathematical discovery since ancient times, pitted Isaac Newton—mathematician, astronomer, alchemist—against Gottfried Wilhelm Leibniz— mathematician, philosopher, diplomat. Each believed the other was trying to steal the credit.
* Euler: The most inspirational story in the history of mathematics belongs to Leonhard Euler, whose astonishing output barely slowed down after he went blind in 1771. Like Beethoven, who composed some of his greatest music after going deaf, Euler was able to practice his art entirely in his head.
* Cantor: While Vincent van Gogh was painting pioneering works of modern art in France in the late 1800s, Georg Cantor was laying the foundations for modern mathematics next door in Germany. Unappreciated at first, the two rebels even looked alike, and both suffered debilitating bouts of depression.

Describing a common reaction to the theorems produced by these great thinkers, Professor Dunham says his students often want to know where the breakthrough ideas came from: How did the mathematicians do it? The question defies analysis, he says. "It's like asking: ‘Why did Shakespeare put the balcony scene in Romeo and Juliet? What made him think of it?' Well, he was Shakespeare. This is what genius looks like!" And by watching the lectures in Great Thinkers, Great Theorems, you will see what equivalent genius looks like in mathematics.

About Your Professor

Dr. William Dunham is the Truman Koehler Professor of Mathematics at Muhlenberg College in Allentown, Pennsylvania. He earned his undergraduate degree from the University of Pittsburgh and his M.S. and Ph.D. in Mathematics from The Ohio State University.

Before his current appointment at Muhlenberg, Dr. Dunham taught at Hanover College in Indiana, receiving teaching awards from both institutions as well as the Award for Distinguished College or University Teaching from the Eastern Pennsylvania and Delaware Section of the Mathematical Association of America. He was a Visiting Professor at Ohio State and at Harvard University, where he was invited to teach an undergraduate course on the work of Leonhard Euler and to deliver the Clay Public Lecture in 2008.

Dr. Dunham's "great theorems" approach to teaching mathematics was fostered by a 1983 summer grant from the Lilly Endowment, which also led to his first book, Journey Through Genius: The Great Theorems of Mathematics—a Book-of-the-Month Club selection that has been translated into five languages. Other books followed in addition to articles on mathematics and its history, earning him numerous awards from the Mathematical Association of America and other organizations. He has presented popular talks on mathematics throughout the United States and has appeared on the BBC and NPR's Talk of the Nation: Science Friday.

Available Exclusively on DVD

This richly illustrated course comes filled with 3-D animations, step-by-step visual walkthroughs of major theorems, an animated timeline, photographs and illustrations, and other visual elements to enhance your learning experience.

Course Lecture Titles

24 Lectures
30 minutes / lecture

1. Theorems as Masterpieces
Certain theorems stand out as great masterpieces of mathematics that can be appreciated as great works of art. After hearing Professor Dunham explain this approach, discover the two ways of proving a theorem: direct proof and indirect proof. Also, meet some of the great thinkers whose ideas you will be studying.

2. Mathematics before Euclid
Investigate three non-Greek civilizations that had robust traditions in mathematics. Then encounter a pair of Greek mathematicians who predated Euclid, but who left very deep footprints: Thales and Pythagoras—the latter renowned for the theorem that bears his name.

3. The Greatest Mathematics Book of All
Begin your exploration of the work widely considered the greatest mathematical text of all time: Euclid's Elements. Discover why these 13 succinct books have been so influential for so long as you delve into the ground-laying definitions, postulates, common notions, and theorems from book I.

4. Euclid's Elements—Triangles and Polygons
Continuing your journey through Euclid, work your way toward his most famous result: his proof of the Pythagorean theorem—a demonstration of remarkable visual and intellectual beauty. Also, cover some of the techniques from book IV for constructing regular polygons.

5. Number Theory in Euclid
In addition to being a geometer, Euclid was a pioneering number theorist, a subject he took up in books VII, VIII, and IX of the Elements. Focus on his proof that there are infinitely many prime numbers, which Professor Dunham considers one of the greatest proofs in all of mathematics.

6. The Life and Works of Archimedes
Even more distinguished than Euclid was Archimedes, whose brilliant ideas took centuries to fully absorb. Probe the life and famous death of this absent-minded thinker, who once ran unclothed through the streets, shouting "Eureka!" ("I have found it!") on solving a problem in his bath.

7. Archimedes' Determination of Circular Area
See Archimedes in action by following his solution to the problem of determining circular area—a question that seems trivial today but only because he solved it so simply and decisively. His unusual strategy relied on a pair of indirect proofs.

8. Heron's Formula for Triangular Area
Heron of Alexandria (also called Hero) is known as the inventor of a proto-steam engine many centuries before the Industrial Revolution. Discover that he was also a great mathematician who devised a curious method for determining the area of a triangle from the lengths of its three sides.

9. Al-Khwarizmi and Islamic Mathematics
With the decline of classical civilization in the West, the focus of mathematical activity shifted to the Islamic world. Investigate the proofs of the mathematician whose name gives us our term "algorithm": al-Khwarizmi. His great book on equation solving also led to the term "algebra."

10. A Horatio Algebra Story
Visit the ruthless world of 16th-century Italian universities, where mathematicians kept their discoveries to themselves so they could win public competitions against their rivals. Meet one of the most colorful of these figures: Gerolamo Cardano, who solved several key problems. In secret, of course.

11. To the Cubic and Beyond
Trace Cardano's path to his greatest triumph: the solution to the cubic equation, widely considered impossible at the time. His protégé, Ludovico Ferrari, then solved the quartic equation. Norwegian mathematician Niels Abel later showed that no general solutions are possible for fifth- or higher-degree equations.

12. The Heroic Century
The 17th century saw the pace of mathematical innovations accelerate, not least in the introduction of more streamlined notation. Survey the revolutionary thinkers of this period, including John Napier, Henry Briggs, René Descartes, Blaise Pascal, and Pierre de Fermat, whose famous "last theorem" would not be proved until 1995.

13. The Legacy of Newton
Explore the eventful life of Isaac Newton, one of the greatest geniuses of all time. Obsessive in his search for answers to questions from optics to alchemy to theology, he made his biggest mark in mathematics and science, inventing calculus and discovering the law of universal gravitation.

14. Newton's Infinite Series
Start with the binomial expansion, then turn to Newton's innovation of using fractional and negative exponents to calculate roots—an example of his creative use of infinite series. Also see how infinite series allowed Newton to approximate sine values with extraordinary accuracy.

15. Newton's Proof of Heron's Formula
Return to Heron's ancient formula for determining the area of a triangle to consider Newton's proof using algebraic techniques—an approach he also applied to other geometry problems. The steps are circuitous, but the result bears Newton's stamp of genius.

16. The Legacy of Leibniz
Probe the career of Newton's great rival, Gottfried Wilhelm Leibniz, who came relatively late to mathematics, plunging in during a diplomatic assignment to Paris. In short order, he discovered the "Leibniz series" to represent ?, and within a few years he invented calculus independently of Newton.

17. The Bernoullis and the Calculus Wars
Follow the bitter dispute between Newton and Leibniz over priority in the development of calculus. Also encounter the Swiss brothers Jakob and Johann Bernoulli, enthusiastic supporters of Leibniz. Their fierce sibling rivalry extended to their competition to outdo each other in mathematical discoveries.

18. Euler, the Master
Meet history's most prolific mathematician, Leonhard Euler, who went blind in his sixties but kept turning out brilliant papers. A sampling of his achievements: the number e, crucial in calculus; Euler's identity, responsible for the most beautiful theorem ever; Euler's polyhedral formula; and Euler's path.

19. Euler's Extraordinary Sum
Euler won his spurs as a great mathematician by finding the value of a converging infinite series that had stumped the Bernoulli brothers and everyone else who tried it. Pursue Euler's analysis through the twists and turns that led to a brilliantly simple answer.

20. Euler and the Partitioning of Numbers
Investigate Euler's contribution to number theory by first warming up with the concept of amicable numbers—a truly rare breed of integers until Euler vastly increased the supply. Then move on to Euler's daring proof of a partitioning property of whole numbers.

21. Gauss—the Prince of Mathematicians
Dubbed the Prince of Mathematicians by the end of his career, Carl Friedrich Gauss was already making major contributions by his teen years. Survey his many achievements in mathematics and other fields, focusing on his proof that a regular 17-sided polygon can be constructed with compass and straightedge alone.

22. The 19th Century—Rigor and Liberation
Delve into some of the important trends of 19th-century mathematics: a quest for rigor in securing the foundations of calculus; the liberation from the physical sciences, embodied by non-Euclidean geometry; and the first significant steps toward opening the field to women.

23. Cantor and the Infinite
Another turning point of 19th-century mathematics was an increasing level of abstraction, notably in the approach to the infinite taken by Georg Cantor. Explore the paradoxes of the "completed" infinite, and how Cantor resolved this mystery with transfinite numbers, exemplified by the transfinite cardinal aleph-naught.

24. Beyond the Infinite
See how it's possible to build an infinite set that's bigger than the set of all whole numbers, which is itself infinite. Conclude the course with Cantor's theorem that the transcendental numbers greatly outnumber the seemingly more abundant algebraic numbers—a final example of the elegance, economy, and surprise of a mathematical masterpiece.

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