Paradoxes: Guiding Forces in Mathematical Exploration

Description

A Riveting Account of Paradoxes and their Impact on Mathematics Does .999...=1? Can a sphere be reassembled into two identically sized spheres? Is the consistency of mathematical systems unprovable? Surprisingly, the answer to all of these questions is yes! And at the heart of each question, there lies paradox.  For millennia, paradoxes have shaped mathematics and guided mathematical progress. From the ancient paradoxes of Zeno to the modern paradoxes of Russell, paradoxes remind us of the constant need to revamp our mathematical understanding. It is for this reason that paradoxes are so important.  Paradoxes: Guiding Forces in Mathematical Exploration provides a survey of mathematical paradoxes spanning a wide variety of topics; delving into each paradox mathematically, philosophically, and historically. Readers will gain a full picture of how paradoxes have contributed to and guided the progress of mathematics in many ways. Paradoxes also provides the reader with a robust argument for the inclusion of paradoxes in education. This is all presented in a way that is accessible to anyone with a high school background in mathematics! Entertaining and educational, this book will appeal to any reader looking for a mathematical and philosophical challenge. Author's Note Introduction Why Mathematical Paradoxes Matter A Shift in Mathematics Education? The Paradox as an Illusion The Paradox as a Magic Trick New Mathematics A Guide on How to Use This Book Paradoxes in Science Foundational Paradoxes Zeno's Paradoxes The Three Paradoxes The Dichotomy Argument The Arrow in Flight Achilles and the Tortoise Some Resolutions A Purely Mathematical Attempt Legacy Russell's Paradox: The Necessity of Rules Naive Set Theory Russell and his Paradox The Tragic Story of a Genius Resolution Grelling's Paradox Gödel's Incompleteness Theorems Math wars Logicism Formalism Intuitionism Gödel's Surprise Background The Toppling of a Regime Hilbert's Dream: Post-Gödel Era Infinity Does .999...=1? An Algebraic Argument The Construction of the Real Numbers Geometric Series Alternate Number Systems Ultrafinitism Infinitesimals and Hyperreal Numbers Hilbert's Hotel Paradox What Do Our Methods Tell Us? Rethinking the Problem Finite Guests Infinitely Many Guests Infinitely Many Buses With Infinite Guests The New Mathematics Are Some Infinities Bigger Than Others? Some Simple Observations Diagonalization In Popular Culture Transfinite Numbers The Continuum Hypothesis Even Larger Infinities? Grandi's Series History Grandi Leibniz Euler Other Approaches Resolution Legacy Thomson's Lamp 1+2+3+4+... Some Approaches Ramanujan's Approaches Zeta Function Regularization Applications in Physics Geometry That Defies Intuition Gabriel's Horn History Derivation Insight The Banach-Tarski Paradox A Useful Analogy A Little Background History How is This Theorem Unique? Hyper-webster Measure A Simplified Explanation Real World Examples? Mind-bending Statistics Berkson's Paradox Medical Data Are Attractive People Jerks? More Examples Does Hollywood Ruin Film Adaptations? Simpson's Paradox College Admission Scandal? Visualization is Key A Send-off Appendix A Brief Introduction to Calculus A Brief Introduction to Set theory Notation Guide Acknowledgements References and Further Reading Index