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Mathematics, Philosophy, and the “Real World”

Discover how mathematics helped stimulate the development of Western philosophy and shaped how these fundamental ideas and practices can be applied to a fascinating range of areas and experiences.
Mathematics, Philosophy, and the "Real World" is rated 4.8 out of 5 by 59.
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Rated 5 out of 5 by from Awesome Course; one of my Favorites! Professor Grabiner is an excellent presenter and she organized this course to offer intellectual stimulation. She brought many concepts together and inspired me to pursue additional study concerning philosophy through the lens of mathematics. I highly recommend this course.
Date published: 2024-03-30
Rated 5 out of 5 by from Enlightening. Living twenty years after being diagnosed with cancer that was supposed to kill you in 8 mos., Stephen Jay Gould wrote an essay: "The Median Is Not the Message" The main point was: "It's not the statistical abstraction, the average, or the median that is real; it's all the different individuals that are real." This lecture series is one of the best, as should be apparent from the near perfect review rating from so many people. It is engaging with an overflow of enlightening information about things I never imagined could be at all, from the shocking use of statistics that resulted in untoward and even fatal consequences, to Pascal and his use of it in religious faith, I would have to admit my amazement and even disgust with humanity. There are times, when you learn of some things that make it easy to understand why Bertrand Russell stated: "I hate the world and almost all the people in it. I hate the Labour Congress and the journalists who send men to be slaughtered, and the fathers who feel a smug pride when their sons are killed, and even the pacifists who keep saying human nature is essentially good, in spite of all the daily proofs to the contrary. I hate the planet and the human race – I am ashamed to belong to such a species." Letter to Colette, December 28, 1916 There has not been any other thing in all of anything that I have learned in these last few years that I have connected with more truly and deeply than Bertrand Russell's words here and it has not left me - not for even one single day - since I became aware of the sick things and that there were people who believed and even supported these sick things that have continued unending for all these years. Human nature is not anything to be proud of ever - not ever! There has been no greater disappointment and there will be no greater relief than when it all ends permanently. I can relate FULLY to the monk, the priest, the nun and others who cannot stay with the secular masses of people and their perversion. Believe without doubt, there are things that are far worse than death and the masses of people are unstoppable once they believe in a sick thing and the best thing that could happen; as in all those unfortunate women accused of being witches - is to die, hopefully very quickly, OH LORD PLEASE come quickly, I beg you, please let your mercy come now - even today! I wish I were not such a coward - I could bear it - the death and pain for terrible - horrific moments - even if it is to burn like that poor monk that set himself on fire to protest the sickness of humanity and the masses that allow and even participate in the sickness - and then worship THEIR GOD - who has given them permission and blessed them and made them mighty and powerful. Job laid it all at God, all of it - and so did Jesus. It really is a sick thing, beyond sick words to express, beyond sick pictures of the starving and brutalized, beyond it all and everything... there could be nothing better than the extinction of such a "humanity" - such a "mob" mentality that would allow such things and then make it all flowery and justifiable in their sick, pathetic minds!
Date published: 2024-02-17
Rated 5 out of 5 by from Ties together at broad range of ideas Presenter Grabiner has an impressive breadth of knowledge and uses it to trace the intellectual history of mathematical and geometric thinking through a variety of subjects. Very interesting!
Date published: 2024-01-24
Rated 5 out of 5 by from One of the Best courses I have taken I have taken well over 200 Great Courses courses and this is one the very best. A retired physicist and amateur philosopher, I have delved into the subjects covered in this course within academia and on my own. This course pulled together things I had never considered before. Anyone who is interested in understanding mathematics and its relevance to modern living would benefit from this course.
Date published: 2024-01-10
Rated 5 out of 5 by from An excellent course even for engineers. When I was in undergraduate engineering college I took at least ten math courses. I think I learn more about how to use math to solve problems in the engineering college than I did at the math college. However what I didn’t learn at either was how math and philosophy affected each other. Between this course and the course Mathematics: Queen of Science, I now have a greater understanding and appreciation of the development of mathematics over time and the people behind that development. Thank you.
Date published: 2024-01-05
Rated 5 out of 5 by from Real depth in a college-level course This is probably the best course I have bought from the Great Courses -- and I've bought a lot of them. The instructor clearly had thought in depth about her subject and brought in perspectives from a variety of viewpoints. Well done!
Date published: 2022-04-27
Rated 5 out of 5 by from Excellent course! This is an amazing, eye-opening course. It has inspired me to continue to learn more about the subject and individuals discussed. Dr. Grabiner is absolutely outstanding: Her lectures are clear, enthusiastic, and down-to-earth yet the material remains intellectually challenging, fun, and stimulating. A prior understanding of math and philosophy is not necessary to benefit from the course. Well worth the money and time. I wish there were more Teaching Company courses by Dr. Grabiner!
Date published: 2021-12-08
Rated 5 out of 5 by from Fantastic Course! The course has given me a good background in the history of math and it has inspired me to learn more about this topic. Dr. Grabiner was a great instructor. She is really engaging and you can tell she really loves her subject matter. Many interesting real life examples were used.
Date published: 2021-10-21
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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.


Judith V. Grabiner

Mathematics is a unique human creation.  It proves its claims by logic and reason, and it applies to all subjects.  I hope to help listeners appreciate its beauty and to use it to become more empowered citizens in a free society.


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.

By This Professor

What's It All About?

01: 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.

33 min
You Bet Your Life—Statistics and Medicine

02: 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.

32 min
You Bet Your Life—Cost-Benefit Analysis

03: 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.

31 min
Popular Statistics—Averages and Base Rates

04: 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.

30 min
Popular Statistics—Graphs

05: 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.

32 min
Popular Statistics—Polling and Sampling

06: 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.

31 min
The Birth of Social Statistics

07: 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."

31 min
Probability, Multiplication, and Permutations

08: 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.

31 min
Combinations and Probability Graphs

09: 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.

31 min
Probability, Determinism, and Free Will

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.

31 min
Probability Problems for Fun and Profit

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.

30 min
Probability and Modern Science

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.

32 min
From Probability to Certainty

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?

31 min
Appearance and Reality—Plato's Divided Line

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.

30 min
Plato's Cave—The Nature of Learning

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.

30 min

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.

30 min

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.

31 min
Logic and Logical Fallacies—Why They Matter

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.

30 min

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?

29 min

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.

30 min
More Euclidean Proofs, Direct and Indirect

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.

29 min
Descartes—Method and Mathematics

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.

31 min
Spinoza and Jefferson

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.

31 min
Consensus and Optimism in the 18th Century

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.

31 min
Euclid—Parallels, Without Postulate 5

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.

31 min
Euclid—Parallels, Needing Postulate 5

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.

30 min
Kant, Causality, and Metaphysics

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.

32 min
Kant's Theory of Space and Time

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.

30 min
Euclidean Space, Perspective, and Art

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.

31 min
Non-Euclidean Geometry—History and Examples

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.

31 min
Non-Euclidean Geometries and Relativity

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.

32 min
Non-Euclidean Geometry and Philosophy

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.

31 min
Art, Philosophy, and Non-Euclidean Geometry

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.

30 min
Culture and Mathematics in Classical China

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.

31 min
The Voice of the Critics

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.

30 min
Mathematics and the Modern World

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.

32 min