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$7.75The Story
An introduction to aweinspiring ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, and computability theory.This awardwinning book introduces the reader to aweinspiring issues at the intersection of philosophy and mathematics. It explores ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, computability theory, the Grandfather Paradox, Newcombs Problem, the Principle of Countable Additivity. The goal is to present some exceptionally beautiful ideas in enough detail to enable readers to understand the ideas themselves (rather than watereddown approximations), but without supplying so much detail that they abandon the effort. The philosophical content requires a mind attuned to subtlety; the most demanding of the mathematical ideas require familiarity with collegelevel mathematics or mathematical proof.The book covers Cantors revolutionary thinking about infinity,which leads to the result that some infinities are bigger than others; time travel and free will, decision theory, probability, and the BanachTarski Theorem, which states that it is possible to decompose a ball into a finite number of pieces and reassemble the pieces so as to get two balls that are each the same size as the original. Its investigation of computability theory leads to a proof of Gdels Incompleteness Theorem, which yields the amazing result that arithmetic is so complex that no computer could be programmed to output every arithmetical truth and no falsehood. Each chapter is followed by an appendix with answers to exercises. A list of recommended reading points readers to more advanced discussions. The book is based on a popular course (and MOOC) taught by the author at MIT.
Description
An introduction to aweinspiring ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, and computability theory.This awardwinning book introduces the reader to aweinspiring issues at the intersection of philosophy and mathematics. It explores ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, computability theory, the Grandfather Paradox, Newcombs Problem, the Principle of Countable Additivity. The goal is to present some exceptionally beautiful ideas in enough detail to enable readers to understand the ideas themselves (rather than watereddown approximations), but without supplying so much detail that they abandon the effort. The philosophical content requires a mind attuned to subtlety; the most demanding of the mathematical ideas require familiarity with collegelevel mathematics or mathematical proof.The book covers Cantors revolutionary thinking about infinity,which leads to the result that some infinities are bigger than others; time travel and free will, decision theory, probability, and the BanachTarski Theorem, which states that it is possible to decompose a ball into a finite number of pieces and reassemble the pieces so as to get two balls that are each the same size as the original. Its investigation of computability theory leads to a proof of Gdels Incompleteness Theorem, which yields the amazing result that arithmetic is so complex that no computer could be programmed to output every arithmetical truth and no falsehood. Each chapter is followed by an appendix with answers to exercises. A list of recommended reading points readers to more advanced discussions. The book is based on a popular course (and MOOC) taught by the author at MIT.
