Mathematics, Philosophy, and the "Real World"

Course No. 1440
Professor Judith V. Grabiner, Ph.D.
Pitzer College
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Course No. 1440
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Course Overview

Mathematics has spread its influence far beyond the realm of numbers. The concepts and methods of mathematics are crucially important to all of culture and affect the way countless people in all spheres of life look at the world. Consider these cases:

  • When Leonardo da Vinci planned his mural The Last Supper in the 1490s, he employed geometric perspective to create a uniquely striking composition, centered on the head of Jesus.
  • When Thomas Jefferson sat down to write the Declaration of Independence in 1776, he composed it on the model of a geometric proof, which is what gives it much of its power as a defense of liberty.
  • When Albert Einstein developed his theory of general relativity in the early 20th century, he used non-Euclidean geometry to prove that the path of a ray of light, in the presence of a gravitational field, is not straight but curved.

Intriguing examples like these reflect the important dialogue between mathematics and philosophy that has flourished throughout history. Indeed, mathematics has consistently helped determine the course of Western philosophical thought. Views about human nature, religion, truth, space and time, and much more have been shaped and honed by the ideas and practices of this vital scientific field.

Award-winning Professor Judith V. Grabiner shows you how mathematics has shaped human thought in profound and exciting ways in Mathematics, Philosophy, and the "Real World," a 36-lecture series that explores mathematical concepts and practices that can be applied to a fascinating range of areas and experiences.

Believing that mathematics should be accessible to any intellectually aware individual, Professor Grabiner has designed a course that is lively and wide-ranging, with no prerequisites beyond high school math. For those with an interest in mathematics, this course is essential to understanding its invaluable impact on the history of philosophical ideas; for those with an interest in philosophy, Professor Grabiner's course reveals just how indebted the field is to the mathematical world.

Math Meets Philosophy

In a presentation that is clear, delightful, and filled with fascinating case histories, Professor Grabiner focuses on two areas of mathematics that are easily followed by the nonspecialist: probability and statistics, and geometry. These play a pivotal role in the lives of ordinary citizens today, when statistical information is everywhere, from medical data to opinion polls to newspaper graphs; and when the logical rules of a geometric proof are a good approach to making any important decision.

Mathematics, Philosophy, and the "Real World" introduces enough elementary probability and statistics so that you understand the subtleties of the all-important bell curve. Then you are immersed in key theorems of Euclid's Elements of Geometry, the 2,200-year-old work that set the standard for logical argument. Throughout the course, Professor Grabiner shows how these fundamental ideas have had an enormous impact in other fields. Notably, mathematics helped stimulate the development of Western philosophy and it has guided philosophical thought ever since, a role that you investigate through thinkers such as these:

  • Plato: Flourishing in the 4th century B.C.E., Plato was inspired by geometry to argue that reality resides in a perfect world of Forms accessible only to the intellect—just like the ideal circles, triangles, and other shapes that seem to exist only in the mind.
  • Descartes: Writing in the 17th century, René Descartes used geometric reasoning in a systematic search for all possible truths. In a famous exercise, he doubted everything until he arrived at an irrefutable fact: "I think, therefore I am."
  • Kant: A century after Descartes, Immanuel Kant argued that metaphysics was possible by showing its kinship with mathematics. The perfection of Euclidean geometry led him to take for granted that space has to be Euclidean.
  • Einstein: Working in the early 20th century with a concept of "straight lines" that was different from Euclid's, Albert Einstein showed that gravity is a geometric property of non-Euclidean space, which is an essential idea of his general theory of relativity.

Non-Euclidean Geometry Explained

The discovery of non-Euclidean geometry influenced fields beyond mathematics, laying the foundation for new scientific and philosophical theories and also inspiring works by artists such as the Cubists, the Surrealists, and their successors leading up to today.

Non-Euclidean geometry was a stunning intellectual breakthrough in the 19th century, and you study how three mathematicians, working independently, overthrew the belief that Euclid's geometry was the only possible consistent system for dealing with points, lines, surfaces, and solids. Einstein's theory of relativity was just one of the many ideas to draw on the non-Euclidean insight that parallel lines need not be the way Euclid imagined them.

Professor Grabiner prepares the ground for your exploration of non-Euclidean geometry by going carefully over several of Euclid's proofs so that you understand Euclid's theory of parallel lines at a fundamental level. You even venture into the visually rich world of art and architecture to see how Renaissance masters used Euclidean geometry to map three-dimensional space onto flat surfaces and to design buildings embodying geometrical balance and symmetry. The Euclidean picture of space became internalized to a remarkable extent during and after the Renaissance, with a far-reaching effect on the development of philosophy and science.

Change the Way You Think

Mathematics has not only changed the way specialists think about the world, it has given the rest of us an easily understandable set of concepts for analyzing and understanding our surroundings. Professor Grabiner provides a checklist of questions to ask about any statistical or probabilistic data that you may encounter. Her intriguing observations include the following:

  • Statistics: Biologist and author Stephen Jay Gould, who developed abdominal cancer, was told his disease had an eight-month median survival time after diagnosis. The diagnosis sounded hopeless, but his understanding of the characteristics of the median (as opposed to the mean or mode) gave him a strategy for survival.
  • Bad graphs: There are many ways to make a bad graph; some deliberately misleading, others merely badly conceived. Beware of a graph that starts at a number higher than zero, since comparisons between different data points on the graph will be exaggerated.
  • Polls: The Literary Digest poll before the 1936 U.S. presidential election was the largest ever conducted and predicted a landslide win for Alf Landon over Franklin Roosevelt. Yet the result was exactly the opposite due to an unrecognized systematic bias in the polling sample.
  • Probability: Intuition can lead one astray when one is judging probabilities. You investigate the case of an eyewitness to an accident who has done well on tests of identifying the type of vehicle involved. But a simple calculation shows that she is more likely wrong than not.

The Power of Mathematical Thinking

Mathematics, Philosophy, and the "Real World" focuses on mathematics and its influence on culture in the West. But for an alternative view, Professor Grabiner devotes a lecture to mathematics in classical China, where geometers discovered some of the same results as the ancient Greeks but with a very different approach. One major difference is that the Chinese didn't use indirect proof, a technique that proves a proposition true because the assumption that it is false leads to a contradiction.

In another lecture, Professor Grabiner gives time to the critics of mathematics—philosophers, scientists, poets, and writers who have argued against the misuse of mathematics. Charles Dickens speaks for many in his memorable novel Hard Times, which depicts the human misery brought by Victorian England's obsession with statistics and efficiency.

But even more memorable are the cases in which mathematics turns up where it is least expected. "We hold these truths to be self evident ..." So wrote Thomas Jefferson in the second sentence of the Declaration of Independence. He had originally started, "We hold these truths to be sacred and undeniable ... " The change to "self-evident" was probably made at the suggestion of Benjamin Franklin, a great scientist as well as a statesman, who saw the power of appealing to scientific thinking. A Euclidean proof begins with axioms (self-evident truths) and then moves through a series of logical steps to a conclusion.

With her consummate skill as a teacher, Professor Grabiner shows how Jefferson laid out America's case against Great Britain with all of the rigor he learned in Euclid's Elements, working up to a single, irrefutable conclusion: "That these United Colonies are, and of Right ought to be Free and Independent States."

There is arguably no greater demonstration of the power of mathematics to transform the real world—and it's just one of the fascinating insights you'll find in Mathematics, Philosophy, and the "Real World."

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36 lectures
 |  Average 30 minutes each
  • 1
    What's It All About?
    Professor Grabiner introduces you to the approach of the course, which deals not only with mathematical ideas but with their impact on the history of thought. This lecture previews the two areas of mathematics that are the focus of the course: probability and statistics, and geometry. x
  • 2
    You Bet Your Life—Statistics and Medicine
    At age 40, the noted biologist Stephen Jay Gould learned he had a type of cancer whose median survival time after diagnosis was eight months. Discover why his knowledge of statistics gave him reason for hope, which proved well founded when he lived another 20 years. x
  • 3
    You Bet Your Life—Cost-Benefit Analysis
    A mainstay of today's economics, cost-benefit analysis has its origins in an argument justifying belief in God, proposed by the 17th-century philosopher Blaise Pascal. Examine his reasoning and the modern application of cost-benefit analysis to a disastrous decision in the automotive industry. x
  • 4
    Popular Statistics—Averages and Base Rates
    In the first of three lectures on the popular use of statistics, investigate three ways of calculating averages: the mean, median, and mode. The preferred method depends on the nature of the data and the purpose of the analysis, which you test with examples. x
  • 5
    Popular Statistics—Graphs
    Learn how to separate good graphs from bad by examining cases of each and reviewing questions to ask of any graphically presented information. The best graphs promote fruitful thinking, while the worst represent poor statistical reasoning or even a deliberate attempt to deceive. x
  • 6
    Popular Statistics—Polling and Sampling
    Concluding your survey of popular statistics, you look at public opinion polling and the sampling process that makes it possible. Professor Grabiner uses a bowl of M&Ms as a realistic model of sampling, and she discusses important questions to ask about the results of any poll. x
  • 7
    The Birth of Social Statistics
    Geometry has been around for more than 2,000 years, but social statistics is a relatively new field, developed in part by Adolphe Quetelet in the 19th century. Investigate what inspired Quetelet to apply mathematics to the study of society and how the bell curve led him to the concept of the "average man." x
  • 8
    Probability, Multiplication, and Permutations
    Probing deeper into the origin of the bell curve, focus on the definition of probability, the multiplication principle, and the three basic laws of probability. Also study real-world examples, with an eye on the broader historical and philosophical implications. x
  • 9
    Combinations and Probability Graphs
    Adding the concept of combinations to the material from the previous lecture, Professor Grabiner shows why a bell curve results from coin flips, height measurements, and other random phenomena. Many situations are mathematically like flipping coins, which raises the question of whether randomness is a property of the real world. x
  • 10
    Probability, Determinism, and Free Will
    Explore two approaches to free will. Pierre-Simon Laplace believed that probabilistic reasoning only serves to mask ignorance of what, in principle, can be predicted with certainty. Influenced by the kinetic theory of gases, James Clerk Maxwell countered that nothing is absolutely determined and free will is possible. x
  • 11
    Probability Problems for Fun and Profit
    This lecture conducts you through a wide range of interesting problems in probability, including one that may save you from burglars. Conclude by examining the distribution of large numbers of samples and their relations to the bell curve and the concept of sampling error. x
  • 12
    Probability and Modern Science
    Turning to the sciences, Professor Grabiner shows how probability underlies Gregor Mendel's pioneering work in genetics. In the social sciences, she examines the debate over race and IQ scores, emphasizing that the individual, not the averages, is what's real. x
  • 13
    From Probability to Certainty
    This lecture introduces the second part of the course, which examines geometry and its interactions with philosophy. Begin by comparing probabilistic and statistical reasoning on the one hand, with exact and logical reasoning on the other. What sorts of questions are suited to each? x
  • 14
    Appearance and Reality—Plato's Divided Line
    Plato's philosophy is deeply grounded in mathematical ideas, especially those from ancient Greek geometry. In this lecture and the next, you focus on Plato's Republic. Its central image of the Divided Line is a geometric metaphor about the nature of reality, being, and knowledge. x
  • 15
    Plato's Cave—The Nature of Learning
    In his famous Myth of the Cave, Plato depicts a search for truth that extends beyond everyday appearances. Professor Grabiner shows how Plato was inspired by mathematics, which he saw as the paradigm for order in the universe—a view that had immense impact on later scientists such as Kepler and Newton. x
  • 16
    Euclid's Elements—Background and Structure
    Written around 300 B.C.E., Euclid's Elements of Geometry is the most successful textbook in history. Sample its riches by studying the underpinnings of Euclid's approach and looking closely at his proof that an equilateral triangle can be constructed with a given line as its side. x
  • 17
    Euclid's Elements—A Model of Reasoning
    This lecture focuses on the logical structure of Euclid's Elements as a model for scientific reasoning. You also examine what Aristotle said about the nature of definitions, axioms, and postulates and the circumstances under which logic can reveal truth. x
  • 18
    Logic and Logical Fallacies—Why They Matter
    Addressing the nature of logical reasoning, this lecture examines the forms of argument used by Euclid, including modus ponens, modus tollens, and proof by contradiction, as well as such logical fallacies as affirming the consequent and denying the antecedent. x
  • 19
    Plato's Meno—How Learning Is Possible
    The first of two lectures on Plato's Meno shows his surprising use of geometry to discover whether learning is possible and whether virtue can be taught. Professor Grabiner poses the question: Is Plato's account of how learning takes place philosophically or psychologically plausible? x
  • 20
    Plato's Meno—Reasoning and Knowledge
    Continuing your investigation of Meno, look at Plato's use of hypothetical reasoning and geometry to discover the nature of virtue. Conclude by going beyond Plato to consider the implications of his ideas for the teaching of mathematics today. x
  • 21
    More Euclidean Proofs, Direct and Indirect
    This lecture returns to Euclid's geometry, with the eventual goal of showing the key theorems he needs to establish his logically elegant and philosophically important theory of parallels. Working your way through a series of proofs, learn how Euclid employs his basic assumptions, or postulates. x
  • 22
    Descartes—Method and Mathematics
    Widely considered the founder of modern philosophy, René Descartes followed a Euclidean model in developing his revolutionary ideas. Probe his famous "I think, therefore I am" argument along with some of his theological and scientific views, focusing on what his method owes to mathematics. x
  • 23
    Spinoza and Jefferson
    This lecture profiles two heirs of the methods of demonstrative science as described by Aristotle, exemplified by Euclid, and reaffirmed by Descartes. Spinoza used geometric rigor to construct his philosophical system, while Jefferson gave the Declaration of Independence the form of a Euclidean proof. x
  • 24
    Consensus and Optimism in the 18th Century
    Mathematics, says Professor Grabiner, underlies much of 18th-century Western thought. See how Voltaire, Adam Smith, and others applied the power of mathematical precision to philosophy, a trend that helped shape the Enlightenment idea of progress. x
  • 25
    Euclid—Parallels, Without Postulate 5
    Having covered the triumphal march of Euclidean geometry into the Age of Enlightenment, you begin the third part of the course, which charts the stunning reversal of the semireligious worship of Euclid. This lecture lays the groundwork by focusing on Euclid's theory of parallel lines. x
  • 26
    Euclid—Parallels, Needing Postulate 5
    Euclid's fifth postulate, on which three of his propositions of parallels hinge, seems far from self-evident, unlike its modern restatement used in geometry textbooks. Work through several proofs that rely on Postulate Five, examining why it is necessary to Euclid's system and why it was so controversial. x
  • 27
    Kant, Causality, and Metaphysics
    The first of two lectures on Immanuel Kant examines Kant's question of whether metaphysics is possible. Study Kant's classification scheme, which confines metaphysical statements such as "every effect has a cause" to a category called the synthetic a priori. x
  • 28
    Kant's Theory of Space and Time
    Learn how geometry provides paradigmatic examples of synthetic a priori judgments, required by Kant's view of metaphysics. Kant's picture of the universe takes for granted that space is Euclidean, an idea that went unquestioned by the greatest thinkers of the 18th century. x
  • 29
    Euclidean Space, Perspective, and Art
    Art and Euclid have gone hand in hand since the Renaissance. Investigate how painters and architects, including Piero della Francesca, Leonardo da Vinci, Albrecht Dürer, Michelangelo, and Raphael, used Euclidean geometry to map three-dimensional space onto flat surfaces and to design buildings embodying geometric balance. x
  • 30
    Non-Euclidean Geometry—History and Examples
    This lecture introduces one of the most important discoveries in modern mathematics: non-Euclidean geometry, a new domain that developed by assuming Euclid's fifth postulate is false. Three 19th-century mathematicians—Gauss, Lobachevsky, and Bolyai—independently discovered the self-consistent geometry that emerges from this daring assumption. x
  • 31
    Non-Euclidean Geometries and Relativity
    Delve deeper into non-Euclidean geometry, distinguishing between three types of surfaces: Euclidean and flat, Lobachevskian and negatively curved, and Riemannian and positively curved. Einstein discovered that a non-Euclidean geometry of the Riemannian type had the properties he needed for his general theory of relativity. x
  • 32
    Non-Euclidean Geometry and Philosophy
    Philosophers had long valued Euclidean geometry for giving a self-evidently true account of the world. But how did they react to the possibility that we live in a non-Euclidean space? Explore the quest to understand the geometric nature of reality. x
  • 33
    Art, Philosophy, and Non-Euclidean Geometry
    This lecture charts the creative responses to non-Euclidean geometry and to Einstein's theory of relativity. Examine works by artists such as Picasso, Georges Braque, Marcel Duchamp, René Magritte, Salvador Dal', Max Ernst, and architects such as Frank Gehry. x
  • 34
    Culture and Mathematics in Classical China
    Other cultures developed complex mathematics independently of the West. Investigate China as a fascinating example, where geometry long flourished at a sophisticated level, employing methods very different from those in Europe and in a context much less influenced by philosophy. x
  • 35
    The Voice of the Critics
    Survey some of the thinkers who have criticized the influence of mathematics on culture throughout history, ranging from Pascal and Malthus to Dickens and Wordsworth. A sample of their objections: Mathematical reasoning gives a false sense of precision, and mathematical thinking breeds inhumanity. x
  • 36
    Mathematics and the Modern World
    After reviewing the major conclusions of the course, Professor Grabiner ends with four modern interactions between mathematics and philosophy: entropy and why time doesn't run backward; chaos theory; Kurt Gödel's demonstration that the consistency of mathematics can't be proven; and the questions raised by the computer revolution. x

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Your professor

Judith V. Grabiner

About Your Professor

Judith V. Grabiner, Ph.D.
Pitzer College
Dr. Judith V. Grabiner is the Flora Sanborn Pitzer Professor of Mathematics at Pitzer College, one of the Claremont Colleges in California, where she has taught since 1985. She earned her B.S. in Mathematics, with General Honors, from the University of Chicago. She went on to earn her Ph.D. in the History of Science from Harvard University. Professor Grabiner has numerous achievements and honors in her field. In 2012 she...
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Mathematics, Philosophy, and the "Real World" is rated 4.8 out of 5 by 50.
Rated 5 out of 5 by from Mathematics and Beyond I bought this DVD course several years ago and, thanks to having more time at home due to the Pandemic of 2020, I finally had time to watch it. I am so sorry I did not do so sooner! I have "taken" nearly seventy TGC offerings, and this was one of my favorites. Professor Grabiner really knows her stuff, but even better, she is a gifted instructor, who makes her rather novel topic fascinating, even for an intermediate student like me. While my mathematics background extended to a year of engineering calculus in college, where I also took a few ancient philosophy courses (all in pursuit of an Architecture degree), I was fascinated to hear her lessons discussing how mathematics ties into some most practical real-life applications (like surveys and polling). Then, she moved on to discuss how mathematical approaches informed (and still inform) how we conceive of our universe and of ourselves. She touched on how and why young people study -- or should study -- what math they do in high school. This course took math out of that very restricted place of simply being a subject we study to being something that helps us understand how to process and relate to data and concepts in both the concrete and abstract worlds. The professor's presentations were clear, with plenty of helpful visual examples. Her style was extremely engaging. Despite not feeling that I am really a "math" person, I was, nonetheless, able to follow her lessons easily. I looked forward to each lecture. The title of this course intrigued me, as it implied there would be a novel approach to relating seemingly unrelated fields. I am glad I explored this with Professor Grabiner.
Date published: 2020-10-19
Rated 5 out of 5 by from An Excellent Presentation Especially for Teachers This course integrates mathematics and philosophy and it does so without scaring a non-mathematician. I'm a retired high school math teacher who finally learned how philosophy connects to the mathematics and science I taught. I wish I'd known these connections when I was teaching. Practicing math teachers should take this course.
Date published: 2020-08-02
Rated 5 out of 5 by from Excellent teacher!! Well done and real interesting
Date published: 2020-06-03
Rated 4 out of 5 by from Math review Instructor provided great insights and real world examples. I wish The Great Courses was offered as an app under Amazon prime (only Great Courses Plus is an app and I do not want that)
Date published: 2020-05-11
Rated 5 out of 5 by from Engaging and Rewarding Lecture Series Excellent in every respect: She is a truly talented lecturer.
Date published: 2019-09-05
Rated 5 out of 5 by from Great background material I have a background in applied mathematics including statistical analysis of instrument error. Professor Judith Grabiner provides the historical context to the math behind the flood of everyday information clothed in statistical packets. I especially liked the integration of statistics to Western Civilization in regards philosophical thinking and history. Thank you Judith for a most satisfactory learning experience.
Date published: 2019-03-18
Rated 5 out of 5 by from Don’t Let Another Two Years Go By I must assume that the two-year gap between reviews is not reflective of the number of people who have taken this course over that period. If so, many have missed out, for this is a course that is both marvelous and unique. Professor Grabiner has melded philosophy, math and the real world in a manner so seamless that I often forgot when she had made the transition from a math concept to philosophy or some part of the world. To be sure her math is confined to basic statistics and geometry, but in the later case I suspect that many taking this course may not be at all conversant with non-Euclidian geometry. As with many TC courses, the first few lectures may seem a bit slow and simplistic, at least from the math perspective. During the second lecture I was wondering when (and if) we were going to get to something a bit past really basic statistics. This lecture was enlivened by her telling the interesting and compelling story of biologist Steven Jay Gould’s approach to his diagnosis of cancer, using a statistical method. The statistics may have been elementary, but the context was not. Here as she is throughout the course, her presentation is superb, not appearing to talk down to her (supposed) non-math audience, while at the same time not shying away from a difficult concept, either in math or philosophy. In any case, don’t be put off should you wish more rigor in your math, as Dr. Grabiner rarely goes past easily understandable proofs. Perhaps not enough to satisfy a math major, but that is not the point of her course. It is in going from math to philosophy that is the real meat of the course. For example, early on we are presented to the most understandable depiction of Plato’s “Divided Line”, that I have seen, heard or read. In my youth, I really struggled with this concept for a long time. I only wish I’d come across her example much earlier, as I found it crystal clear. Not an easy feat. Other reviewers have mentioned appreciating the linking of Jefferson and Euclid and other math/philosophy examples. As for the real world, she brings in art, architecture and more as the course goes from a mathematical concept to easily understandable, existing examples existing in the world. For me, the best lectures were the ones on non-Euclidean geometry. Her bringing in not only philosophy to this topic (easily imaginable) but art as well made me wish that a lecture on quantum mechanics and modern art could be developed. My only wish for this course was that she had included music in her “real world” examples. It seems like a natural.
Date published: 2019-02-09
Rated 5 out of 5 by from Great course I listened to one of the last lectures by Professor Grabiner on non-Euclidean Geometry. I have found most of the Great Courses extremely interesting, mostly in areas in which I had very little previous knowledge, and have been amazed at the extensive knowledge of the several lecturers. I enjoyed Professor Grabiner’s from the beginning, partly because they discussed mathematics and physical sciences, which were close to the disciplines that I had studied and with which I had some previous understanding, I thought. I further appreciated them because Professor Grabiner tied those disciplines in with “philosophies” with which I had only, through the Great Courses lectures, a beginner’s rudimentary knowledge. Now, near the end of the series she had discussed in several lectures, the non-Euclidean space theories, and more then theories, those which appear to be fact as our human reach of observation extends farther. I cannot imagine a human mind so small as to not find her lectures fascinating, nor one so great as to find them trivial. They make me recognize even more the profound meaning of Jesus’ answer to Pilot’s question, “What is truth?”
Date published: 2017-03-20
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