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On the Brink of Paradox: Highlights from the Intersection of Philosophy and Mathematics
An introduction to awe-inspiring ideas at the brink of infinities of different sizes, time travel, probability and measure theory, and computability theory.
This award-winning book introduces the reader to awe-inspiring issues at the intersection of philosophy and mathematics. It explores ideas at the brink of infinities of different sizes, time travel, probability and measure theory, computability theory, the Grandfather Paradox, Newcomb's 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 watered-down 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 college-level mathematics or mathematical proof.
The book covers Cantor's 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 Banach-Tarski 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 Gödel's 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.
This award-winning book introduces the reader to awe-inspiring issues at the intersection of philosophy and mathematics. It explores ideas at the brink of infinities of different sizes, time travel, probability and measure theory, computability theory, the Grandfather Paradox, Newcomb's 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 watered-down 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 college-level mathematics or mathematical proof.
The book covers Cantor's 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 Banach-Tarski 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 Gödel's 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.
320 pages, Hardcover
First published March 8, 2019
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Displaying 1 - 4 of 4 reviews
August 13, 2020
Great materials for great course on EdX.
January 31, 2021
"The Banach-Tarski Theorem is an astonishing result. We have decomposed a ball into finitely many pieces, moved around the pieces without changing their size or shape, and then reassembled them into two balls of the same size as the original.
I think the theorem teaches us something important about the notion of volume. As noted earlier, it is an immediate consequence of the theorem that some of the Banach-Tarski pieces must lack definite volumes and, therefore, that not every subset of the unit ball can have a well-defined volume. A little more precisely, the theorem teaches us that there is no way of assigning volumes to the Banach-Tarski pieces while preserving three-dimensional versions of the principles we called Uniformity and (finite) Additivity in chapter 7.
(Proof: Suppose that each of the (finitely many) Banach-Tarski pieces has a definite finite volume. Since the pieces are disjoint, and since their union is the original ball, Additivity entails that the sum of the volumes of the pieces must equal the volume of the original ball. But Uniformity ensures that the volume of each piece is unchanged as we move it around. Since the reassembled pieces are disjoint, and since their union is two balls, Additivity entails that the sum of their volumes must be twice the volume of the original ball. But since the volume of the original ball is finite and greater than zero, it is impossible for the sum of the pieces to equal both the volume of the original ball and twice the volume of the original ball.)
If I were to assign the Banach-Tarski Theorem a paradoxicality grade of the kind we used in chapter 3, I would assign it an 8. The theorem teaches us that although the notion of volume is well-behaved when we focus on ordinary objects, there are limits to how far it can be extended when we consider certain extraordinary objects - objects that can only be shown to exist by assuming the Axiom of Choice."
I think the theorem teaches us something important about the notion of volume. As noted earlier, it is an immediate consequence of the theorem that some of the Banach-Tarski pieces must lack definite volumes and, therefore, that not every subset of the unit ball can have a well-defined volume. A little more precisely, the theorem teaches us that there is no way of assigning volumes to the Banach-Tarski pieces while preserving three-dimensional versions of the principles we called Uniformity and (finite) Additivity in chapter 7.
(Proof: Suppose that each of the (finitely many) Banach-Tarski pieces has a definite finite volume. Since the pieces are disjoint, and since their union is the original ball, Additivity entails that the sum of the volumes of the pieces must equal the volume of the original ball. But Uniformity ensures that the volume of each piece is unchanged as we move it around. Since the reassembled pieces are disjoint, and since their union is two balls, Additivity entails that the sum of their volumes must be twice the volume of the original ball. But since the volume of the original ball is finite and greater than zero, it is impossible for the sum of the pieces to equal both the volume of the original ball and twice the volume of the original ball.)
If I were to assign the Banach-Tarski Theorem a paradoxicality grade of the kind we used in chapter 3, I would assign it an 8. The theorem teaches us that although the notion of volume is well-behaved when we focus on ordinary objects, there are limits to how far it can be extended when we consider certain extraordinary objects - objects that can only be shown to exist by assuming the Axiom of Choice."
January 2, 2024
This book is a mind-blowing intellectual journey covering selected topics in mathematics from a philosophical point of view, related to infinity, paradoxes, time-travel, and other counter-intuitive topics. The book is written for intellectually curious people, not only mathematicians and philosophers. The topics are well motivated by simple cases/examples that spark the readers interest. Basic mathematical concepts are explained from scratch and simple logic is used to walk the reader to more advanced results, without the need of complicated proofs. A vast amount of exercises with solutions are included. The book is generally easy to read, although the general level of abstraction is high.
May 2, 2020
An incredible exposition of interesting topics on math. A good introduction for the amateur.
Displaying 1 - 4 of 4 reviews