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Mathematics, Philosophy, and the "Real World"

Mathematics, Philosophy, and the "Real World"

Course No.  1440
Course No.  1440
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Course Overview

About This Course

36 lectures  |  30 minutes per lecture

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.
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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
  • 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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Judith V. Grabiner
Ph.D. Judith V. Grabiner
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 was named a Fellow of the American Mathematical Society. In 2003 she won the Mathematical Association of America's (MAA's) Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching, one of the most prestigious mathematics awards in the country. She also won the Distinguished Teaching Award of the Southern California Section of the MAA, and the Outstanding Professor Award from California State University, Dominguez Hills. In addition, she is a four-time winner of the MAA's Lester R. Ford Award, given for excellence in scholarship. Professor Grabiner has published widely in her field, the history of mathematics, and she has long taught courses to non-mathematicians with the goal of helping them see that mathematics is fun, fascinating, and useful. In turn, her students have taught her much, directing her to mathematical applications in their own specialties-from Leibniz's philosophy and forensic science to quilting and baseball.
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Reviews

Rated 4.8 out of 5 by 30 reviewers.
Rated 5 out of 5 by Three Courses in One! If I’ve done the math correctly, there are three courses combined in this thirty-six-lecture program: (1) mathematics with a focus on probability, statistics, and the history of geometry + (2) Western philosophy, as encapsulated in a 2,500-year survey + (3) the role that mathematics plays in our daily lives = The sum total: A Great Course called “Mathematics, Philosophy, and the ‘Real World’”! In an all-encompassing liberal arts format, the series begins with the mathematical theories of the ancient Greeks (especially Euclid), then traces the relationship of mathematics and philosophy through the ages. The result is a humanities-based course filled with interesting sidebars and personal reflections of the instructor. Professor Judith V. Grabiner is such a good lecturer that she could read aloud from an electronics manual and find a way to make the information both intelligible and interesting. There is also a refreshing down-to-earth quality in her lecturing. The course is packed with lively personal anecdotes, demonstrations, and the wisdom that comes from a master teacher. As part of the inspiration for this course, Professor Grabiner cites Plato’s famous dictum that was inscribed above his Academy in Athens: “Let no one ignorant of geometry enter here.” Starting with Plato and his teacher Socrates, the lecturer demonstrates how through the ages, philosophers’ knowledge of the world has been shaped by mathematics. The discussions of two seventeenth-century French mathematician-philosophers, Pascal and Descartes, reveal how philosophy, science, and religion were intertwined. And in the section on eighteenth-century rationalism, there is a fascinating analysis of how Thomas Jefferson was thinking mathematically when he drafted the Declaration of Independence. The lectures also touch on art history, revealing how mathematics was influential in painting and architecture. There was a detailed explanation of the uses of linear perspective in Renaissance art, wherein the painters drew upon mathematical principles to evoke three-dimensional space on the canvas. While the Renaissance painters were fundamentally Euclidean in their artistic principles, modern abstract artists like Picasso, Dali, and Duchamp were non-Euclidean in their styles of the collage, cubism, and surrealism, as apparent in a sampling of their famous works of art. In this highly eclectic course, there were many memorable and surprising moments. Of all Professor Grabiner’s engaging stories, the most meaningful one addressed the life of the evolutionary scientist Stephen J. Gould. The lecturer described how when he was diagnosed with cancer, Gould applied his knowledge of mathematics and statistics to assess his chances of survival by charting numbers on a graph. He was ultimately successful in overcoming the cancer. As recounted by the professor in a moving and unforgettable way, this story combines all three of the course topics of mathematics, philosophy, and the real world. That story alone made this lecture series an enlightening educational experience. COURSE GRADE: A January 24, 2014
Rated 5 out of 5 by Math, Philosophy and the 'Real World" This course was pure joy for me. I especially relished the tie-in of Euclid and the thinking of Immanuel Kant, whom I first read some 70 years ago. Now I totally get it; my Eureka moment came during this wonderful course. Thank you, Professor Grabiner! July 12, 2014
Rated 5 out of 5 by Combine with Great Theorems I wish this course and Great Thinkers Great Theorems were the basis of teaching math in primary and secondary school rather than the current vacuum. Presentation is very good, material is excellent. In almost every lecture I have an aha moment. May 20, 2014
Rated 5 out of 5 by Another wonderful course with an elegantly articulate teacher whose enthusiasm for her subject propels the topic forward in a very nonintimidating and accessible way. Although I had mathematically based courses in probability and a lot of math in my college work, it's been a long time so I fully appreciated the review of the subjects. I was struck by this lecturer's down to earth delivery of information without a smidgen of arrogance or superiority. I also had a courses in philosophy, art and the history of science in my undergraduate years. All of those broadened my appreciation and understanding of my undergraduate BS degree in chemistry. This professor furthers that experience. I have always felt that the beautiful world of mathematics is frequently contaminated by an arrogance in the instruction of the subject. This course confirms the value of connecting the subject of mathematics with art and philosophy without the "snakiness" of superiority that contaminates the presentation of mathematics and vice versa. Thank you Professor Grabiner. March 17, 2014
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