Mathematics, Philosophy, and the “Real World”
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.
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.
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.
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.
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.
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.
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."
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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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: 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: 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.
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.
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.
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.
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.