# Joel David Hamkins

Joel David Hamkins is Professor of Logic at Oxford University and Sir Peter Strawson Fellow in Philosophy at University College, Oxford. He has published widely in refereed research journals in mathematical logic and set theory and is the creator of the popular blog Mathematics and Philosophy of the Infinite. He is a prominent contributor to MathOverflow, where he has posted more than 1,000 mathematical arguments.

• ### Proof and the Art of Mathematics

Examples and Extensions

How to write mathematical proofs, shown in fully worked-out examples.

This companion volume to Joel Hamkins's Proof and the Art of Mathematics provides 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.

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• ### Lectures on the Philosophy of Mathematics

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.

Hamkins shows, for example, how number systems set the stage for discussions of such philosophical issues as platonism, logicism, and the nature of abstraction. Consideration of the rise of rigor in the calculus leads to a discussion of whether the indispensability of mathematics in science offers grounds for mathematical truth. Sophisticated technical developments in set theory give rise to a necessary engagement with deep philosophical concerns, including the criteria for new mathematical axioms. Throughout, Hamkins offers a clear and engaging exposition that is both accessible and sophisticated, intended for readers whose mathematical backgrounds range from novice to expert.

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• ### Proof and the Art of Mathematics

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.

The book is organized around mathematically rich topics (rather than methods of proof), allowing students to learn to write proofs with material that is itself intrinsically interesting. Students will find the early chapters the easiest. Chapter 4 explains the method of mathematical induction, which is used in many arguments throughout the book. Later chapters offer chapter-length developments of major theorems, and the final chapters are more abstract. The book is generously illustrated; an extended chapter on proofs-without-words shows the power of figures and diagrams to communicate mathematical ideas—but also acknowledges the dangers of such an approach. Each chapter includes exercises, and sample answers are provided at the end of the book.

• Paperback \$30.00