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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Reviews

Mathematics, Philosophy, and the "Real World" is rated 4.8 out of 5 by 46.
Rated 5 out of 5 by from I liked the influence of Math on modern arts It is amazing how Professor Judith Grabiner shows the great impact of two topics of mathematics: Probability-Statistics and Geometry (Euclidean and Non-Euclidean) on the way we try to understand reality. I liked very much the lectures on non-Euclidean geometry whit its conceptions of new spaces and its influence on architecture and modern art. This course is very original and useful for expanding my perspectives to look at the world.
Date published: 2016-05-21
Rated 5 out of 5 by from Impossible to exaggerate how good this course is I just finished this course a few days ago, and I'm still excited and amazed by all that I've learned. When I bought this course, I thought (from the title and from other reviews) that I would be learning about the connection between math, philosophy, and their connection to the real world. Well, I got that, of course, but I got so, so much more! Indeed, what I got was a tour de force that examined the profound implications that two broad mathematics ways of thinking--the "probabilistic" and the "geometric"--have had (and continue to have) on philosophers, mathematicians, scientists, artists, corporations, policy makers, etc. I read a lot of philosophy, and have often read about how Euclid's geometry influenced certain thinkers, but never before was the extent of that influence so clearly (and powerfully) laid out than by Professor Grabiner in this course. In the first part of the course, Professor Grabiner provides the viewer with the basic foundations of statistics and probability, discussing such things as the (somewhat surprising, for me, anyway) philosophical and religious roots of cost-benefit analysis and expected value (lecture 3), standard deviation (lecture 4), the history of the graph (lecture 5), and polling and sampling techniques (lecture 7). A lot of this information provided a nice review of my college probability and statistics classes, but was not too earth shattering ... until lecture 10. Here, Professor Grabiner connected the knowledge provided in the previous lectures to a discussion of determinism v. free will, effortlessly discussing the effect that probability theory had on leading mathematicians, scientists, and philosophers, and the effect that figures in each field had on each other. For instance, did you know that Bohr's theory of the atom was influenced by Kierkegaard's leap of faith? Or that Maxwell was influenced by Lucretius' "swerve"? Or even that the debates that were played out among religious figures centuries ago is now being mirrored, in a very specific way, in the debates between the two competing theories of the universe, quantum mechanics and relativity? It is here, in drawing connections between different fields of knowledge, where the course paid major dividends, and where Professor Grabiner is at her best. Similarly, in the second part of the course, where Euclidean thinking is emphasized and explained, Professor Grabiner links up the philosophical story of Plato's cave to Kepler's laws of planetary motion and Newton's law of universal gravitation (lecture 15). What?!! I had no idea there was a connection here, yet now, I am convinced. I also became aware of the enormous debt to Euclid that was owed by such monumental thinkers as Descartes (lecture 22), Spinoza and Jefferson (lecture 23), and Kant (lecture 27). Simply put, without Euclidean geometry, what these thinkers said and wrote would not have been possible. But, even more than this, I now feel that, by understanding the geometric way of thinking that underlay the philosophy of these great thinkers, I now understand what they were trying to do, and the strengths and weaknesses of their approaches, in a way that I never have before. In fact, more than this, I am now convinced that I really never understood these thinkers before, and Professor Grabiner's course made me skeptical that anyone else really can either without understanding the extent to which, and in what way, these minds were directly influenced by Euclid. I now realize that I should have taken the sign that once hung above Plato's academy, which read "Let no one ignorant of geometry enter", much more seriously than I have in the past, and that it might profitably be attached as a warning label to those who would read the thinkers just mentioned. But the reverse is probably also true. We might have a sign attached above the great physics labs across the world saying something like "Let no one ignorant of philosophy enter," for, at the most fundamental level, the debate between modern physicists over their particular world views are, at their core, philosophical. "And", to quote Vonnegut, "so it goes". Other aspects of this course that I really enjoyed learning about was geometry's impact on art (lecture 29 and 33), the tremendous philosophical, mathematical, and scientific impact of non-Euclidean forms of thinking (lectures 30-32), and the importance of Euclidean and non-Euclidean ways of thinking in the modern world (lectures 34-36). This course was a real eye-opener for me, and I can honestly say that I will not look at the world the same way again. Grade: A+
Date published: 2016-02-14
Rated 5 out of 5 by from A Tour de Force This is a spectacular presentation of mathematics for those with a background in liberal arts. Dr. Grabiner is so plain spoken and straightforward that it takes a bit to realize what a deep understanding of classical philosophy and intellectual history she has. It is simply wonderful.
Date published: 2016-01-20
Rated 5 out of 5 by from Fabulous! Brilliant course, an eye-opener Who said mathematics could not be fun? This is a course everyone should have: it's interesting, informative, at times very surprising, a real education, but it is also an entertaining series of lectures. Professor Grabiner is a rather straightforward teacher, but never boring or dull. The many illustrations she uses make her points very clear -- and memorable. Here's a course that is indeed a keeper. I recommend it to everyone over the age of 16.
Date published: 2015-04-01
Rated 5 out of 5 by from Profound! Video Review: Dr. Judith Grabiner presents a unique way to teach the value of mathematics to the Liberal Arts student. Her course shows how mathematics influenced philosophy and intellectual history (primarily) in Western Civilization. Though she briefly introduces some newer mathematics (e.g. fractals) late in the course, the mathematics is focused on probability/statistics and geometry. Her unique approach is to show how the mathematical relationships (e.g. formulas) are developed from first principles and then show how these basic mathematical principles influenced philosophy, intellectual thought, and science. The "real world' comes into play in not only science, but everyday examples such as use (and misuse) of graphs to make a point and use of geometry in art to give perspective. For anyone who can recall laboring through "proofs" in math classes, Dr. Grabiner shows the importance of these in the Euclidean geometry section, for establishing truths not only in mathematics but in the development of logical thinking as evidenced by the enlightenment. Tracing the developments in mathematics from Plato's "Republic" and "Meno" to Euclid's "Elements of Geometry" to enlightenment thinkers Descartes, Spinoza, Voltaire and Newton, she tells the fascinating story of how mathematics was a core element of philosophical thinking. One especially fascinating learning was how Thomas Jefferson applied the basic "if p, then q" logic of mathematics to the Declaration of Independence; which was subsequently used as the basis for other such declarations by countries all over the world. Dr.Grabiner's approach of teaching the core mathematics through the laborious approach of counting actual events to determine probabilities and working through exhaustive proofs in geometry, may seem somewhat elementary. But by using this approach she clearly connects the foundations of mathematics to the development of philosophy and demonstrates how mathematics is evident in the "real world" all around us. The beginning section focuses on how probability and statistics affect real world examples of everyday life. She shows examples of how polls can be conducted to give a valid statistical sample and how they can be biased. She demonstrates how "correlation does not equal causation" with several real world examples (her example of the student who suffered hangovers on three consecutive mornings after drinking bourbon one night, then gin the next, and rum the next, all having been mixed with ginger ale, and concludes that he has to stop drinking ginger ale, is a classic!). She shows the value of proper (and detriment of improper) use of cost-benefit analyses. The example of the Ford Pinto is a poignant one of improper use. A personal example of probability she uses which is quite profound is Stephen Jay Gould's choice for cancer treatment: given two treatment options, one that is 95% effective and one that is 99% effective, he realizes, through the math of probability, that the relative difference is significant and chooses the later approach. The section on geometry really provides the bulk of the mathematics influence on philosophy and science. Dr. Grabiner demonstrates how Euclid's "Elements" influenced Aristotle to extend Euclid's postulates to investigating the universe, i.e. science. Descartes expands on this in the development of analysis and Newton extends this further in his "Principia" to establish his basic laws of nature. Kant goes further to establish cause and effect. Art (e.g DaVinci's "The Last Supper") gains perspective from Euclidean views of 3 dimensional space. While Euclidean geometry carries the day for centuries, the 19th century brings to light non-Euclidean views by Bolyai, Lobachevsky, Riemann, and Gauss. Non-Euclidean spaces on the surfaces of spheres and "saddles" are influences on Einstein in his development of both the Special and General theories of Relativity. In the last few lectures, Dr. Grabiner touches on Chinese developments in mathematics where it appears that visual art influenced geometrical proofs, whereas in the Western world geometry influenced art. She also briefly introduces, at a conceptual level, newer math ideas, such as statistics and entropy, group theory, Boolean logic and the digital computer, and concepts of the infinite. Dr. Grabiner is an excellent presenter. Though she is clearly reading from a teleprompter she does so with inflection, body language and enthusiasm. Appropriate illustrations and other visual aides are used throughout the course, leading me to recommend the video version of the course. The accompanying course guide is excellent as it contains crisp outlines to summarize lectures, a timeline of events, biographical notes, and an annotated bibliography. In summary, while the actual mathematics taught in this course is relatively elementary, the philosophy, intellectual history, and applications to science and society are quite mind opening. Though this course was designed for the humanities or fine arts major (and really hits the mark for this audience), I would also recommend it to any scientist, engineer, social scientist, or business person who is interested in understanding the origin and philosophy behind the mathematics that is deeply embedded in your professions and our everyday world.
Date published: 2015-02-26
Rated 5 out of 5 by from 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!
Date published: 2014-07-12
Rated 5 out of 5 by from 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.
Date published: 2014-05-20
Rated 5 out of 5 by from 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.
Date published: 2014-03-18
Rated 4 out of 5 by from Philosophical Concepts make the course If you are well-versed in mathematics as I am, you will be able to skip the first 12 lectures on statistical fundamentals. The real concepts of the course are the relationships shown between famous (and not so famous) philosophers and the mathematical knowledge of their times. Starting with the ancient Greeks and working up through Einstein, the professor presents much interesting subject matter that will keep even technically savvy students interested. It is nearly a symposium on the human thought process seen through the eyes of logical (i.e., mathematically inclined) philosophers. Overall, a fine job of presenting much new material to this retired engineer.
Date published: 2014-02-10
Rated 5 out of 5 by from 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
Date published: 2014-01-24
Rated 5 out of 5 by from Charming I enjoyed the course, mostly lectures 13-36. Intricacies of axiomatics of Euclidian geometry, its interplay with philosophical concepts and basics of non-Euclidian geometries – very thought provoking. Charming 1970s style presentation manner with lots of energy and passion for the subject. Great course.
Date published: 2013-09-17
Rated 5 out of 5 by from Food for thought I gave 5 stars to this course and 4 stars to the professor presentation. The content of the course is excellent. It's a very interesting historical approach to maths and philosophy. I just finished watching the course today and it's real food for thought. For sure I'll have to go through it again in the future, because there are lots of concepts and references that need to be "digested". If you like history, philosophy and you think maths play an important role in our lives, this course is an excellent buy.
Date published: 2013-06-26
Rated 4 out of 5 by from About Average I wasn'i too happy with this course. It seems to really drag in spots. That may just be the way math stuff is. When you hear people say that math is exciting, you have to wonder what exactly they are comparing it to.
Date published: 2012-06-05
Rated 5 out of 5 by from Inspiring As a life-long student of all things scientific and mathematical, for pleasure as well as intellectual improvement, the professor gave an outstanding overview of the subject with pertinent historical and biographical data bringing us from antiquity to the present day. It has opened up new avenues for personal reading and study in the philosophical sphere. An outstanding lecturer. I wished the series would have continued past the 36 lectures.
Date published: 2012-04-03
Rated 4 out of 5 by from Often Fascinating and Important, If Very Basic Prof. Grabiner provides a remarkably comprehensible look at fascinating points at the intersections of mathematics with philosophy, art, and politics, a.k.a. "the real world." While her choice of topics is necessarily limited and somewhat idiosyncratic, many truly remarkable and important areas are covered. I was particularly impressed by her ability to clearly and succinctly explain some difficult philosophical ideas (including the fundamental's of Kant's metaphysics and how math and philosophy developed from and beyond his insights), and the extraordinary concepts of non-Euclidean geometry. Much else is included, from Plato and Euclid to Gödel and Einstein, all equally well explained. And all of what she covers, in my sincerely humble opinion, will be of interest to everyone desiring a broad knowledge of the achievements of human thought and culture. Prof. Grabiner is herself extraordinary, a superb professor (easily in the company of TGC's other great mathematics lecturers, Michael Starbird and Arthur Benjamin.) She speaks articulately, clearly, and beautifully, in a down-to earth manner about very not-down-to-earth topics, with an infectious enthusiasm and a wonderful assurance while ranging over subjects that could fill a university catalog. So why not 5 stars? - For me (and this will obviously be a very individual response), much of the course was simply too basic. The lectures on the fundamentals of statistics, probability theory, and Euclidean geometry, essential as they are for what comes next, will be of little benefit to anyone with even an elementary familiarity with these areas. In retrospect, I wish I had skipped them, and I recommend - if you find you are learning little new here - that you do so, and get to the engrossing discussions of their applications to so many other areas. I nevertheless recommend the course wholeheartedly for anyone not already quite knowledgeable in the great variety of subjects discussed. Many of these ideas are among the highest achievements of the human mind, fascinating in and of themselves, and of crucial importance to the history of human ideas and civilization.
Date published: 2012-01-11
Rated 4 out of 5 by from Awesome and a bit of a mess I’m conflicted. MATHEMATICS, PHILOSOPHY, AND THE “REAL WORLD” is both awesome and a bit of a mess. Dr Grabiner sets out with three goals: 1) Describe how mathematics influenced the evolution of Western thought (philosophy especially, but also the arts), 2) Introduce a specialized field, the philosophy of mathematics, and 3) Explain basic probability theory and non-Euclidean geometry, two pillars of modern science (hard and soft). Her basic thesis is that mathematics at the most creative level may only attract a tiny minority, but its success in helping us manipulate nature and society has magnified its influence to such an extent — morally, politically and culturally — that we should understand its strength and limitations better. So far so good. She weaves these three goals in rather startling ways. Let’s look at them one-by-one. 1) She does an excellent job of explaining the role of mathematics in Plato through Aristotle and then Euclid. The Greek vision of true knowledge as something eternal, necessary, and universal owed much to geometry. Unfortunately, before dealing with the Greeks, she devotes 13 lessons to explaining basic probability, something entirely foreign to ancient thought. 2) As she proceeds through the Renaissance to the 18th Century and Kant, certain philosophical issues implicit in Euclid pop up with increasing frequency: Are mathematical forms and our notions of space innate to the human mind or are they derived from sense experience? Is deductive logic sufficient to develop mathematics or is empirical confirmation necessary? In what sense are numbers “real”? All this pertains to philosophy of mathematics, a field too rarefied to influence the wider culture in my opinion. 3) Logically, probability theory should have come in the second half with Pascal in the 17th Century. This would leave less space for the bell curve and more to its growing use in the physical and social sciences. As for, non-Euclidean geometry, it evolved from Euclidean constrains when mathematicians started modifying Euclid’s “Fifth Postulate” dealing with parallel lines. The crucial point here was that formal consistency (Is this completely logical?) was more clearly distinguished from empirical confirmation (Is it real?). And once it was created, it turned out to be very useful after all in relativity theory. All of the above may appear ludicrous. She is the expert, not me. My only point is that her approach negates part of her objective. Her decision to spend so much time on elementary probability mathematics leaves very little space for the ethical ramifications of probability theory in 19th Century economics, biology or the nascent social sciences. Quoting Dickens to show outrage at Victorian slums is fine, but the connection with mathematical thought remains obscure. Then she quotes Hannah Arendt on the Nazi death camps and their reliance on ethically-challenged technocrats. All true, but does mathematics really count compared to ideologies such as nationalism and ethnic cleansing? Now for the positive. She is a very able lecturer in a demanding, fairly abstract field. This lecture series is work, not play. At least it was for me. She held my attention. And I know I will listen to her again. The course guidebook was also excellent. Recommended if mathematics really intrigues you.
Date published: 2011-11-18
Rated 5 out of 5 by from Absolutely phenomenal! This course was outstanding! It is a mandatory foundation course for anyone interested in philosophy (or even art!). The teaching is brilliant and content spell binding! It traces the concept of space and time from Kant to today with exceptional clarity . It demonstrates the power of deductive logic (other TC philosophy courses lead one to think that with the introduction of induction, deduction was relegated to oblivion!). It covers and clarifies the mathematical discoveries that destroyed our concepts of truth and certainty. After a somewhat prolonged initial 12 lectures on probabilty (which is done with brilliance, but can be skipped if you know probabilty), the course takes off on a mind-blowing journey, never to look back. I cannot recommend this course highly enough. If you are interested in philosophy, you cannot affort not to take this course. The rewards are the clarification of many other philosophy courses.
Date published: 2011-10-20
Rated 5 out of 5 by from Opens your eyes and expands your mind! This is a great course that gives you the many connections between great thinkers throughout history not just in Math and Philosophy but in art and other sciences. The Professor is smart, witty, and provided a framework to consider the many ways we as humans have discovered and discussed who and what we are. I have a philosophy background and my husband a math background and we both learned new ideas and new ways of looking at the world. A great course to open your eyes and expands your mind!
Date published: 2011-03-24
Rated 5 out of 5 by from A tough job done well Being a retired math professor, I know Dr. Grabiner has taken on a tough job, and she's done well with it. You can't just listen; you have to think. My wife, a non-mathematician, found the course interesting and understandable (except for the formulas). We both highly recommend this course. This course does indeed have a lot of pertinence to the "Real World."
Date published: 2011-03-18
Rated 4 out of 5 by from Good start-and stronger finish! I'm a fan of interdisciplinary courses, but realize they are hard to produce. I really enjoyed and learned from this course. But at times this seems like a math course, at other like a light philosophy course, and only rarely like an integrated math and philosophy course. But the information is still fascinating. I highly recommend this course. In terms of the three Teach Co. math courses I've completed, this is a more substantial than "The Joy of Thinking" but less so than the "Art of Mathematical Problem Solving." But this course is also much more easily digested than "Mathematical Problem Solving." I definitely recommend this course.
Date published: 2011-03-11
Rated 5 out of 5 by from Excellent history and mathematics Especially fine presentation of Euclid, Kant, and non-Euclidean geometry. Highly recommended.
Date published: 2011-02-10
Rated 5 out of 5 by from Excellent Substance and Outstanding Teaching I usually avoid math (certainly geometry and statistics)—it’s hard for me to understand and it does not hold my attention. That often remained true for this course also, but I am nevertheless happy to recommend this course for its excellence, substantiveness, and even its interest. Not that it always held my attention! Professor Grabiner’s teaching was always excellent, but even she often could not keep me interested in most of what I call the “details” (e.g., postulates and proofs, etc.), which apparently are necessary to a proper study of mathematics. Thanks to Grabiner’s expertise, I did learn some proofs and other details. But more important to me was that, when I couldn’t completely follow the details, I still learned and enjoyed main points and “the big picture,” which includes a lot of interesting, sometimes fascinating, material. I did not ENJOY this course nearly as much as I did Brettell’s beautiful course on Impressionism—I was easily and completely absorbed in Impressionism and THOROUGHLY enjoyed it. Comparatively, Grabiner’s course was work; it required a lot more effort, though I now feel well repaid: I got to see the history of how mathematics has shaped and continues to undergird a variety of prominent disciplines. And that has been quite satisfying to me. With excellent analogies and demonstrations, masterful command of the subject, and her easy, engaging, conversational style, Professor Grabiner has done outstanding work here. It’s the most interesting math course I’ve ever encountered, and I recommend it--even for my fellow mathphobes.
Date published: 2011-01-19
Rated 5 out of 5 by from Fascinating course taught by a master teacher Of the 100+ courses I have purchased, this course ranks in the top echelon. The subject is incredibly fascinating, and I learned a tremendous amount. Dr. Grabiner is a truly gifted teacher. Her explanations are so crystal clear that even I can follow them with ease! It is obvious that she loves to teach. I found her use of everyday examples to clarify her explanations very helpful. Now I know why Plato insisted that his students know geometry. I only hope that Professor Grabiner does another Teaching Company course!
Date published: 2010-12-30
Rated 5 out of 5 by from This is the best course This is an extraordinary course. I am very happy that I followed the general comments and bought this set. I have so much enjoyed each lecture of Dr. Grabiner and would really like to recommend the course to anyone who might be interested in math and science. Thanks, Dr. Grabiner!
Date published: 2010-10-07
Rated 4 out of 5 by from Great Teacher and Great Message! This is a good course from TTC. Here are my positives and one area for the course to improve on: Positives: + Prof Judith Grabiner is very clear and passionate - on par with other top professors from TTC. She can make tough topics v easy to understand - mark of a great teacher! + She gave good real-life examples for her arguments/ideas (e.g., Ford Pinto case (in lecture 3) and correlation examples (lecture 4)) + I love her summaries at the end of some lectures (she provided key questions/checklists as well as summary bullet points) + I love it when answers to some numerical / technical questions are provided at the end of some lectures (other TTC courses should aim to do this as well) + I love it how she explained key mathematics from first principles (e.g., permutation and combination) - i.e., providing the WHY rather than just giving the HOW (or giving the formulae) + Good use of diagrams, slides and examples + I particularly love her simple and clear description of non-Euclidian geometry (including Einstein's application of it in General Theory of Relativity) and its impact on philosophy, culture and art + I also particularly liked her message that the application of mathematics should be balanced between certainty/precision and probabilistic/statistical(individual-by-individual) thinking. Particularly, I like her message that, at the end of the day, humans are "humans" (i.e., need to consider the moral, emotional and ethical implications of life and mathematics - can't be completely rational) Area to improve on: - I feel that some of the lectures are too slow / too easy. I recommend packing more insights per lecture (e.g., some of the Euclidian lectures could perhaps be combined into one or two lectures; Plato's Republic and Meno messages that are relevant to the course could have been combined into one or two lectures, etc.) Overall, it's a good course presented by a great teacher.
Date published: 2010-07-20
Rated 4 out of 5 by from This is one of those select few, must-see courses This is a much-needed course to fill in some gaps of the Teaching Company repertoire. With so many of their courses already placing an emphasis on philosophy, and then their sudden flurry of mathematics courses in the past few years, this is the one course that will bridge the gap between them. Judith Grabiner seem so very appropriate to teach this material, so it's also got that a a big plus going for it. She is very practical in her approach, using plenty of real world examples in statistics to start off the first 12 lectures, followed by another 14 on Greek philosophy and mathematics, then the big finale on modernity and how non-Euclidean geometry has shaped our world. These are unestimably important concepts that everyone should know, and only now has Judith laid it all out for us in an accessible manner. Everyone should know how philosophy and mathematics are fields of study so very close, and of course the history of their evolution is one of the most fascinating and intriguing stories one can tell. There are a select set of Teaching Company courses which are simply "musts" in the sense that their subject matter is so essentially fundamental, that regardless of the presentation, which in this case is excellent, one should have these concepts under their belt in order to have a truly well-rounded education. I believe that is one of the key goals of the Teaching Company's mission, and this course fits so very well into that sort of intention.
Date published: 2010-04-13
Rated 5 out of 5 by from A real enlightenment Professor Grabiner is one of the few scholars who know the history and inherent connection between philosophy and math. I'm an engineer and interested in religion and philosophy, and now I know why. Journey through this course was truly an eye opening experience and real enlightenment for me.
Date published: 2010-02-27
Rated 5 out of 5 by from Excellent links from Math to Philosophy Prof Gabiner has developed a thorough and coherent course. She explains concepts and ideas clearly and provides excellent examples from life. Initially she spends a lot of time on Euclid's proofs, probably a little too much time. However, she is developing the ground work to show how these proofs link into so many other ideas. Similar to previous reviewers, I was amazed at the direct links from Euclid's proofs to Thomas Jefferson, Emanuel Kant and even Martyn Luther King’s famous “I had a Dream” speech. She demystified some of Emanuel Kant’s writings for me by using Euclid’s proofs. I thought this link was excellent and very helpful. She further developed the ideas of Non Euclidean geometry, that I had seen in other Teaching Company lectures (Maths from the Visual World and Alex Fillipenko’s wonderful Astronomy series), into Art and Philosophy. I enjoyed the links to the Artistic Styles of Cubism and Surrealism. Finally she links Kurt Gödel’s “Incompleteness Theorem” to modern Philosophy and Mathematics. This theorem seems to affect so many areas: Maths, Physics, Philosophy, Computer Science, Linguistics, Logic, Thinking and Consciousness. Given Kurt Gödel’s, prominence in a number of Teaching Company Lectures, I would like one of the Professors to explain his “Incompleteness Theorem” in more detail. Overall, an excellent course for those wanting to see the links between Mathematics and Philosophy.
Date published: 2009-12-04
Rated 5 out of 5 by from top notch I was a mathematics major as an undergraduate and so I was very familiar with most of the math concepts that she covered. What I was not familiar with was the depth of the connections to philosophy. I enjoyed this course immensely.
Date published: 2009-09-06
Rated 5 out of 5 by from Mathematics for the artist in us This course really started opening up for me when we got to Euclid's geometry and it's application to argument formation. A striking example was when Prof. Grabiner showed how Thomas Jefferson's Declaration of Independence followed the style of Euclid's geometry proofs. Euclid's geometry is a ubiquitous and powerful tool in our everyday world and we can benefit greatly by learning to harness its power. Then we learn about non-Euclidean geometry and things get really exciting. This opens up the world of creativity in art, architecture, space, and beyond. From there on out we were on a wonderful ride through philosophy and real world anecdotes from everyday life. Professor Grabiner shines the light of understanding and order in the dark corners of our world and brings the satisfaction we get when we reach an "Ah-ha" moment when things just seem to fall into place. The world is less of a mystery and becomes more comfortable when we gain the understanding of laws and principles which govern it, and this course brings us closer to that understanding. A truly delightful course.
Date published: 2009-08-16
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