Kung D. Mind-Bending Math. Riddles and Paradoxes

The Great Courses, 2015. — 207 p.The course begins with surprising puzzles that arise from the most basic mathematics: logic and arithmetic. If some sentences—for example, “This sentence is false”—are neither true nor false, then what are they ? From barbers cutting hair to medieval characters who either always tell the truth or always lie, you will learn the importance of self-referential “strange loops” that reappear throughout the 24 lectures. With more complicated topics come more head-scratching conundrums. How can knowing the sex of one child affect the sex of the other ? How is it possible that the statistically preferred treatment for a kidney stone could change based on a test result, no matter what the result is ? Probability and statistics are fruitful grounds for puzzling results—and for honing your thinking ! Certain paradoxes have a long and storied history, such as the Greek philosopher Zeno’s argument that motion itself is impossible. The key concept of infinity underlies many of these puzzles, and it wasn’t until around 1900 that Georg Cantor used some ingeniously twisted logic to tame this feared beast. The oddities of infinity, including the infinitely many sizes of infinity, surprised even Cantor, who wrote about one particularly surprising result: “I see it, but I don’t believe it.” You will be able to both see and believe his amazing work, with the help of carefully crafted explanations and illuminating visual aids. The course culminates with a set of geometrical and topological paradoxes, bending space like the lectures will have bent your mind, leading up to a truly unbelievable result. The Banach-Tarski paradox proves that, at least mathematically, one can cut up a ball and reassemble the pieces into two balls, each the same size as the original ! Usually a result reserved for high-level mathematics courses, the main ideas of this astonishing claim are presented in an elementary way, with no background knowledge required.
By fighting your way through the thickets of mathematical mind benders, you will discover where our everyday “common sense” is accurate—and where it leads us astray. Tapping into the natural human curiosity for solving puzzles, you will expand your mind and, in the process, become a better thinker.