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Great Thinkers, Great Theorems

Delve into the mechanics of some of math's greatest and most awe-inspiring achievements.
Great Thinkers, Great Theorems is rated 4.8 out of 5 by 125.
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Rated 5 out of 5 by from Improve Your Mind This was actually a beautiful and fun course with profound content and a delightful teacher, William Dunham. If you like math at all, take it or if you don’t like math very much, take it. Trying to follow the development of such intellectual achievements has to be good for ones brain. It would be helpful also to beginning calculus students. As I often say, I wish I had had this earlier.
Date published: 2024-06-03
Rated 3 out of 5 by from Some good insights I learned many good insights about mathematical topics. However, majority of topics were high school mathematics and very basic to discuss. Being an Engineer myself, I already knew them well, but the course did exhibit some novel ways to see things differently. If you are already at an intermediate or advanced level in mathematics, perhaps this course is not for you. Regardless you will still somewhat enjoy watching it if you need to understand your already known topics with further depth.
Date published: 2024-04-09
Rated 5 out of 5 by from Save Money: Read Dunham’s “Dividing Line" This 2010 course’s Professor Biography tells us that the National Endowment for the Humanities has honored Dunham’s portrayal of great theorems as a work of art. Dunham’s method [Lecture 1 = (L1)]: “…we will judge (theorems) by certain characteristics: elegance, or economy, and an element of unexpectedness, or surprise.” For example, Euclid did not have the use of equations with exponents for his “windmill" proof of the Pythagorean theorem (L4). One of the course rewards therefore becomes being able to understand math in unfamiliar, marvelously creative ways. Archimedes’ proof that the area of a circle (L7) is equal to that of a triangle with a side equal to the radius and another equal to the circumference involved some math. The basic concept started with the area of a hexagon being summed up as the areas of triangles (each triangle defined by the apothem center base point and the base length of each hex). Though Archimedes did not have Algebra, Dunham’s formula is: ½ * (apothem) * (perimeter). It then becomes immediately obvious that Archimedes will use “exhaustion” to approximate the proof. Marvelous, but if this 3-sentence summary is difficult, the lecture is not. For whatever reason, it has always bothered me that I had nothing but tables to find the square root of large numbers (where guessing is impractical). Heron of Alexandria’s method in the opening section of L8 provides a method: If x is the approximation at any stage of finding the square root of N, the next approximation will be x/2 + N/2x. Why? Watch Dunham’s straightforward proof. Heron’s formula for a non-right triangular [Given the semi perimeter “s” (equal to 1/2 of the perimeter of a triangle with sides a, b, & c) then A = the square root of s(s-a)(s-b)(s-c).] Dunham himself found this fantastic - when he was in 9th grade!. Though he says the proof is complex, he shows how it implies the Pythagorean theorem. As the lectures progress, the math gets more difficult – often exceeding many non-mathematicians' “need to know”. As a physician, I did not need to retain Cardano’s solutions (L11) for cubic or quartic equations (though the method was interesting). However, Dunham's historical lecturing (especially L13 on Newton’s life and L16 on Liebnitz) make the course interesting for those less mathematically inclined. From L17 the lectures also get more "esoteric". Though well ordered, they are less likely to be useful for those who don't heavily use math in their careers. CONCLUSION: SINCE WASTING TIME IN COLLEGE is so expensive, high school students may find that L5 is worth the price of the course. There Dunham states that a DIVIDING LINE separating mathematicians from those destined for other fields is a HEARTFELT APPRECIATION for Euclid’s theorem 20 of Book IX: ““Prime numbers are more than any assigned multitude of prime numbers”. See what you think and perhaps save $22K plus a year on the wrong college major…this test alone makes the course rate a 5!
Date published: 2023-12-19
Rated 5 out of 5 by from Outstanding Man, what a course. So much enjoyable and informative.
Date published: 2023-11-21
Rated 5 out of 5 by from Great Thinkers, Great Theorems These are lectures that everyone should study. I am very happy with this course and encourage The Great Courses to offer similar courses. The subject of these lectures is not easy to explain and to understand. Dr. Dunham does an exceptional work explaining this difficult subject.
Date published: 2023-09-01
Rated 5 out of 5 by from Great Professor, Great Lectures This was the first Great Course I started and the first one I have finished. Professor Dunham is absolutely brilliant. He has a very charming personality and has an incredible passion for mathematics. He presents the subject matter with true enthusiasm and with almost childlike wonder. He is truly a joy to watch and listen to, and he presents the information in such a way as to leave the viewer anxious for the next lecture. And it’s about math! The series is clearly intended for a high school graduate audience and he does a fantastic job of presenting the content in a digestible, yet still dense way. It’s no wonder he’s such a celebrated teacher. The content itself spans the history of Western mathematics from the ancient Greeks to the modern day, and he gives an excellent survey of the major players and issues. I wish I were able to thank Professor Dunham personally for presiding over such a beautifully presented course, though few things are as beautiful as the wonderful works of art which are mathematics’ great theorems. My previously passive interest in mathematics has turned into a a real desire to read the great works of mathematics, and to study further beyond what I have learned in school. I highly recommend this course not only to the mathematically inclined, but everyone. It will certainly incite greater understanding, spark an interest in the subject, and elicit an appreciation for the natural beauty of the internal logic of the universe: mathematics.
Date published: 2023-06-18
Rated 5 out of 5 by from Excellent This course is amazing. I have a bachelor's degree in mathematics and I had briefly read and discussed these great mathematics when their theorems would appear. To go through history with professor Dunham and see the mathematics evolve from Geometry and the Greeks to Calculus in European 17th century is eye opening. He allows you to consider how absurd the genius of these great men were that laid the these theorems with only their predecessors to build on. It is enough to make one want to live in another time with the ridiculous idea that one could contribute.
Date published: 2023-04-16
Rated 5 out of 5 by from A truly enjoyable learning experience! It's been more than 40 years since I took my last college math class during my studies in electrical engineering. Through a dozen or so courses I was exposed to many of the names discussed by Prof. Dunham and even a few of the background stories, but beyond that I simply learned to use the tools first discovered by these great thinkers. It's been a true pleasure to see how some of the proofs were constructed while gaining an appreciation for how far above my skills these geniuses lived. Thanks!
Date published: 2023-03-14
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Overview

Explore the most awe-inspiring theorems in the 3,000-year history of mathematics with the 24 lectures of Great Thinkers, Great Theorems. Professor William Dunham, an award-winning teacher with a talent for conveying the essence of mathematical ideas, reveals how great minds like Pythagoras, Newton, and Euler crafted theorems that would revolutionize our understanding of the world. Approaching great theorems the way an art course approaches great art, Professor Dunham will open your mind to new levels of mathematics appreciation.

About

William Dunham

Mathematics is perhaps the most useful of subjects, but my interests are not utilitarian.  Rather, I look to the great landmarks from the history of mathematics and present them as creative achievements that are a joy to behold.

INSTITUTION

Muhlenberg College

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.

By This Professor

Theorems as Masterpieces

01: 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.

32 min
Mathematics before Euclid

02: 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.

31 min
The Greatest Mathematics Book of All

03: 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.

29 min
Euclid's Elements-Triangles and Polygons

04: 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.

32 min
Number Theory in Euclid

05: 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.

29 min
The Life and Works of Archimedes

06: 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.

29 min
Archimedes' Determination of Circular Area

07: 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.

32 min
Heron's Formula for Triangular Area

08: 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.

31 min
Al-Khwarizmi and Islamic Mathematics

09: 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."

30 min
A Horatio Algebra Story

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.

29 min
To the Cubic and Beyond

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.

32 min
The Heroic Century

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.

31 min
The Legacy of Newton

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.

30 min
Newton's Infinite Series

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.

31 min
Newton's Proof of Heron's Formula

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.

32 min
The Legacy of Leibniz

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 π, and within a few years he invented calculus independently of Newton.

31 min
The Bernoullis and the Calculus Wars

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.

31 min
Euler, the Master

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.

30 min
Euler's Extraordinary Sum

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.

31 min
Euler and the Partitioning of Numbers

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.

31 min
Gauss-the Prince of Mathematicians

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.

30 min
The 19th Century-Rigor and Liberation

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.

31 min
Cantor and the Infinite

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.

29 min
Beyond the Infinite

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.

31 min