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.
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.
Rated 5 out of
5
by
kuzaleks from
One of the best courses on philosophy
I would like to say many thanks to Professor Grabiner for this course. I enjoyed every hour of watching the lectures.
Despite of being interested both in Math and Philosophy for a long time, I still found many new exciting ideas both about particular concepts in math (e.g. how algebraic and logical symbolic expressions appeared from the generalisation of some particular examples), and about influence of the mathematics to culture.
I'd also like to note the exceptional pedagogical skills of Professor Grabiner with which she managed to explained so many complex concepts in such clean and clear way.
Date published: 2024-08-25
Rated 5 out of
5
by
FloridaPat 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
Tom Miron 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
MarkMel 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
Waxhaw 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
RavWayLady 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
Wood75 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
Kath17 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
Overview
About
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
