An introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs.This book offers an introduction to the art and craft of proof-writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory, and real analysis. The goal is to show students and aspiring mathematicians how to write proofs with elegance and precision.

A rich, thorough exploration of infinity that combines mathematical precision with playful, accessible coverage of a fascinating variety of paradoxes and conundrums.In this rich, thorough account of all things infinite, Joel David Hamkins combines playful thought experiments with mathematically precise exposition to explore the vast, fascinating spectrum of paradoxes and conundrums of infinity. Written with equal parts wonder and rigor, the book moves from classical paradoxes (Zeno, Hilbert’s hotel, Aristotle’s wheel) to Cantor’s transfinite arithmetic, the continuum hypothesis, surreal numbers, infinitesimals, epistemic logic puzzles, and contemporary debates about potential vs. actual infinity. Twenty-five lively, self-contained chapters are enriched by extensive illustrations, diagrams, figures, and Fermi-style estimation puzzles. Highly readable and pedagogically rich, The Book of Infinity is ideal for undergraduate philosophy and math students and enthusiasts of logical and mathematical curiosities alike.Approaches a wide range of ideas with both rigor and delightRenders deep mathematical ideas accessible to beginnersEmphasizes intuition and deep conceptual understandingIntegrates historical perspectives and contemporary issuesSuits anyone interested in paradoxes, large numbers, and conceptual puzzles

How to write mathematical proofs, shown in fully-worked out examples.This is a companion volume Joel Hamkins's Proof and the Art of Mathematics, providing fully worked-out solutions to all of the odd-numbered exercises as well as a few of the even-numbered exercises. In many cases, the solutions go beyond the exercise question itself to the natural extensions of the ideas, helping readers learn how to approach a mathematical investigation. As Hamkins asks, "Once you have solved a problem, why not push the ideas harder to see what further you can prove with them?" These solutions offer readers examples of how to write a mathematical proofs.     The mathematical development of this text follows the main book, with the same chapter topics in the same order, and all theorem and exercise numbers in this text refer to the corresponding statements of the main text.

An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice.In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.

An introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs.This book offers an introduction to the art and craft of proof-writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory, and real analysis. The goal is to show students and aspiring mathematicians how to write proofs with elegance and precision.

A rich, thorough exploration of infinity that combines mathematical precision with playful, accessible coverage of a fascinating variety of paradoxes and conundrums.In this rich, thorough account of all things infinite, Joel David Hamkins combines playful thought experiments with mathematically precise exposition to explore the vast, fascinating spectrum of paradoxes and conundrums of infinity. Written with equal parts wonder and rigor, the book moves from classical paradoxes (Zeno, Hilbert’s hotel, Aristotle’s wheel) to Cantor’s transfinite arithmetic, the continuum hypothesis, surreal numbers, infinitesimals, epistemic logic puzzles, and contemporary debates about potential vs. actual infinity. Twenty-five lively, self-contained chapters are enriched by extensive illustrations, diagrams, figures, and Fermi-style estimation puzzles. Highly readable and pedagogically rich, The Book of Infinity is ideal for undergraduate philosophy and math students and enthusiasts of logical and mathematical curiosities alike.Approaches a wide range of ideas with both rigor and delightRenders deep mathematical ideas accessible to beginnersEmphasizes intuition and deep conceptual understandingIntegrates historical perspectives and contemporary issuesSuits anyone interested in paradoxes, large numbers, and conceptual puzzles

How to write mathematical proofs, shown in fully-worked out examples.This is a companion volume Joel Hamkins's Proof and the Art of Mathematics, providing fully worked-out solutions to all of the odd-numbered exercises as well as a few of the even-numbered exercises. In many cases, the solutions go beyond the exercise question itself to the natural extensions of the ideas, helping readers learn how to approach a mathematical investigation. As Hamkins asks, "Once you have solved a problem, why not push the ideas harder to see what further you can prove with them?" These solutions offer readers examples of how to write a mathematical proofs.     The mathematical development of this text follows the main book, with the same chapter topics in the same order, and all theorem and exercise numbers in this text refer to the corresponding statements of the main text.

An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice.In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.