Philosophy of mathematics Books

Book SynopsisThe Quadrivium consists of the four Liberal Arts of Number, Geometry, Music, and Cosmology, studied from antiquity to the Renaissance as a way of glimpsing the nature of reality. They synthesize number, space, and time. Geometry is number in space, music is number in time, and the cosmos expresses number in space and time. Number, music, and geometry are metaphysical truths, good and beautiful everywhere at all times. Life across the universe investigates them. They foreshadow the physical sciences. This is the first volume to bring together the Quadrivium for many hundreds of years.Trade Review"Music to the eye" - The New York Times. "Fascinating" - Financial Times. "Excellent" - New Scientist. "Genuinely mind-expanding" - Fortean Times. "Engaging and accessible" - Seattle Times. "Beautiful" - London Review of Books. "Rich and artful" - The Lancet.

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Book SynopsisA theoretical physicist takes readers on an awe-inspiring journey—found in 'no other book' (Science)—to discover how the universe generates everything from nothing at all: 'If you want to know what's really going on in the realms of relativity and particle physics, read this book' (Sean Carroll, author of The Biggest Ideas in the Universe). In Waves in an Impossible Sea, physicist Matt Strassler tells a startling tale of elementary particles, human experience, and empty space. He begins with a simple mystery of motion. When we drive at highway speeds with the windows down, the wind beats against our faces. Yet our planet hurtles through the cosmos at 150 miles per second, and we feel nothing of it. How can our voyage be so tranquil when, as Einstein discovered, matter warps space, and space deflects matter?   The answer, Strassler reveals, is that empty space is a sea, albeit a paradoxically strange one.

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Book SynopsisTrade ReviewOne of Choice's Outstanding Academic Titles for 2013 Shortlisted for the 2013 BSHM Neumann Book Prize, British Society for the History of Mathematics "Once a mainstay of mathematics, spherical trigonometry no longer appears on school curricula. Here, Glen Van Brummelen reasserts the field's importance, sharing in illuminating detail how it figured in astronomy, cartography and our understanding of Earth's rotation."--Rosalind Metcalfe, Nature "The present book is very well written; it leaves a clear impression that the author intended to endear--not merely present and teach--spherical trigonometry to the reader. Although not a history book, there are separate chapters shedding light on the approaches to the subject in the ancient, medieval, and modern times. There are also chapters on spherical geometry, polyhedra, stereographic projection and the art of navigation. The book is thoroughly illustrated and is a pleasant read. Chapters end with exercises; the appendices contain a long list of available and not so available textbooks and recommendations for further reading organized by individual chapters. The book made a valuable addition to my library. I freely recommend it to math teachers and curious high schoolers."--Alexander Bogomolny, CTK Insights "A no-nonsense introduction to spherical trigonometry."--Book News, Inc. "A beautiful popular book."--ThatsMaths.com "Full of academic, textbook content, the book is a delight to math students. So if you are game for a journey into the world of spherical trigonometry, pick up the book. Van Brummelen gives exercises at the end of the chapters that can be fun."--R. Balashankar, Organiser "Heavenly Mathematicsis a truly enjoyable description of the somewhat forgotten science of spherical trigonometry... As readers discover this discipline, they will also appreciate the beauty inherent in the topic."--Choice "Heavenly Mathematics proves the value of bringing a fascinating piece of mathematical history within the grasp of the general reader."--Florin Diacu, Literary Review of Canada "Van Brummelen has written a wonderful introduction ... that draws on the history of [spherical trigonometry] to illuminate the mathematics itself and at the same time gives readers a real sense of what research in the history of early mathematics is all about."--Metascience "[Heavenly Mathematics] is an excellent survey of spherical trigonometry... Simply an appreciation of a beautiful lost subject, with historical overtones... [D]istinguishable for its appealingly fresh style."--Mathematical Reviews "[Heavenly Mathematics] is a lovely book to read... [A] wonderful introduction for anyone who wishes to learn more about this subject... I am in full agreement with the author that spherical trigonometry ought to be brought to a wider audience, and I believe that this is the book to do it."--Mathematics Today "Engaging, clear and not overly technical; you can safely lend this book to your friends in the history department... [Heavenly Mathematics] is excellent."--Zentralblatt MATH "Heavenly Mathematics will be of interest to mathematically inclined historians of science and also to students of mathematics and engineering. Because spherical trigonometry is relevant in applications of modern science, this elegant book may even contribute to a renaissance of the subject."--Jan P. Hogendijk, Isis "This book could serve as an excellent textbook for any secondary school mathematics classroom at or above the level of geometry and certainly trigonometry; as the basis for a high school honors class; or as a textbook and seminar topic for college students."--Teresa Floyd, Mathematics Teacher "Any reader of this book (and there should be many) will see how present day mathematics may be viewed through the kaleidoscope of its historical origins... Glen Van Brummelen has written a beautifully produced book that includes fascinating biographical detail at every stage of his narrative."--P.N. Ruane, Mathematical Gazette "An engaging read that will appeal to historians of science, mathematicians, trigonometry teachers, and anyone interested in the history of mathematics."--Elizabeth Hamm, Aestimatio Critical Reviews in the History of ScienceTable of ContentsPreface vii 1 Heavenly Mathematics 1 2 Exploring the Sphere 23 3 The Ancient Approach 42 4 The Medieval Approach 59 5 The Modern Approach: Right- Angled Triangles 73 6 The Modern Approach: Oblique Triangles 94 7 Areas, Angles, and Polyhedra 110 8 Stereographic Projection 129 9 Navigating by the Stars 151 Appendix A. Ptolemy's Determination of the Sun's Position 173 Appendix B. Textbooks 179 Appendix C. Further Reading 182 Index 189

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Book Synopsis A remarkable account of Kurt Gödel, weaving together creative genius, mental illness, political corruption, and idealism in the face of the turmoil of war and upheaval. At age 24, a brilliant Austrian-born mathematician published a mathematical result that shook the world. Nearly a hundred years after Kurt Gödel''s famous 1931 paper On Formally Undecidable Propositions appeared, his proof that every mathematical system must contain propositions that are true - yet never provable within that system - continues to pose profound questions for mathematics, philosophy, computer science, and artificial intelligence. His close friend Albert Einstein, with whom he would walk home every day from Princeton''s famous Institute for Advanced Study, called him the greatest logician since Aristotle. He was also a man who felt profoundly out of place in his time, rejecting the entire current of 20th century philosophical thought in his belief that mathematical truths existed independent of theTrade ReviewBudiansky opens up the history of a time where great progress was achieved in foundations of mathematics against the backdrop of the Second World War. This is an easily accessible account for those who did not have the chance to meet Kurt Gödel in person...For the generations that possibly enjoy the t-shirt version of Gödel's work, I would hope that the variety of his work would spark more diversified Gödel merchandise * Annika Kanckos, Metascience *It would be easy to fall into the trap of repeating somewhat exaggerated anecdotes and to ridicule the leading character of this biography. Therefore, it is a pleasure to read a book on the life of Gödel that does all but that. The book offers a serious and unapologetic account of Gödel's life ... The new take on the topics is refreshing and brings the past to life through a coherent narrative. * Annika Kanckos, Metascience *Selected as a 2021 Book of the Year in the Times Literary Supplementwonderfully engrossing * Adam Gopnik, The New Yorker *Budiansky, for all his tremendous efforts and exhaustive interrogations of Gödel's times and places, acquaintances and offices, can only leave us, at the end, with an immeasurably enriched version of Gödel the wise child. It's an undeniably distracting and reductive picture. But - and this is the trouble - it's not wrong. * Simon Ings, Spectator *Journey to the Edge of Reason covers [Gödel's life and work] engagingly and clearly, which is quite a feat given the difficulty of the material. The author... also manages successfully to convey Gödel's naivety, eccentricity and paranoia as well as his genius. * Nick Spencer, Financial Times *In this excellent new biography, Stephen Budiansky introduces the reader to Gödel's stunning achievements in logic, illuminates his devastating mental illness and considers how the two might be related. * Cheryl Misak, Times Literary Supplement *An engaging read, both on a personal and professional level. * David Lorimer, Paradigm Explorer *One of the great geniuses of the 20th century, barely known outside the academy today, receives a much-needed expert biographical treatment ... An outstanding biography of a man of incomprehensible brilliance. * Kirkus reviews *Journey to the Edge of Reason is an intimate and haunting portrait of one of the most elusive gods on Princeton's Mt. Olympus. A triumph of research and a wonderful read. * Sylvia Nasar, author of A Beautiful Mind *Kurt Gödel's mathematical results on incompleteness and undecidable propositions leave it up to us, as individuals, to choose whether to mourn these limits to the power of formal systems, or celebrate his proof that even the most rigid numerical bureaucracy contains the tools by which higher truth will always be able to effect an escape. Stephen Budianksy's Journey to the Edge of Reason expertly and humanely frames these results between Gödel's childhood under the dark shadow of the Austrian and Nazi bureaucracies, his escape to America, his descent into physical and mental illness, and his achievement of a reconciliation between spiritual faith and scientific proof. * George Dyson, author of Analogia and Turing's Cathedral *A painstakingly researched and lucidly presented biography—a close-up of one of the most influential and enigmatic thinkers of the twentieth century—full of vivid detail and sharp historical insight. * Karl Sigmund, professor of mathematics, University of Vienna, and author of Exact Thinking in Demented Times *A brilliant biography of one of the most original thinkers of all time, Journey to the Edge of Reason is as deep and precise as the genius it describes. In a paradox befitting Gödel himself, it takes a tale of logic and its limits and finds, at its heart, something strangely soulful and sympathetic. * Steven Strogatz, professor of mathematics, Cornell University, and author ofInfinite Powers *Prepare yourself for a great adventure and fascinating book that can be picked up at ease and read at pace..Budiansky provides a gripping and interesting dialogue fitting the great story of this fascinating figure...an excellent book. * Kenny Green, Mathematics Today *

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Book Synopsis"Let no one ignorant of geometry enter here," Plato warned would-be philosophers. Mathematician Karl Sigmund agrees.In The Waltz of Reason, he shows how mathematics and philosophy together have shaped our understanding of space, chance, logic, cooperation, voting, and the social contract. Sigmund shows how game theory is integral to moral philosophy, how statistics shaped the meaning of reason, and how the search for a logical basis for math leads to deep questions about the nature of truth itself. But this is no dry tome: Sigmund's wit and humour shine as brightly as his erudition.The Waltz of Reason is an engrossing history of ideas as vibrant as a ballroom full of dancers, one that empowers as it entertains, following the complex and occasionally dizzying steps of the thinkers who have moulded our thought and founded our world.

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Book SynopsisUncover the language of our universe - numbers - in this wide-ranging whistle-stop tour of the history and majesty of mathematics.Our world simply wouldn't function if we didn't have numbers. But where do they come from? Why do we cut cake the wrong way? How can there be different sizes of infinity?All these questions and more are answered in this engaging romp through the history of numbers by acclaimed science writer, Colin Stuart. From the mathematicians who have (and haven't) shouted 'Eureka!' to the theories that affect and inform our everyday lives; Numbers shows us that maths was never boring - we were just being taught it in the wrong way.Consisting of ten bite-sized essays, there's no better guide to this fundamental science.

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Book SynopsisTrade Review"This book could well serve as a history of mathematics. … [Stillwell] has done an amazing job of collecting and categorizing many of the most important ideas in this area."---Jim Stein, New Books in Mathematics"Stillwell’s [The Story of Proof] joins his two other Princeton University Press books in having my highest recommendation. I just wish they had been around when I was a student."---George Hacken, Computing Reviews"I hugely enjoyed this book."---Jonathan Shock, Mathemafrica"This book would be perfect for any keen undergraduate, keen amateur, or indeed a teacher of mathematics, who wants a book to dip into to use for the classroom."---Jonathan Shock, Mathemafrica"A well-crafted, thought-provoking meditation on the concept of proof in mathematics. . . .It is a substantive book that deserves to be read and reflected upon."---Tommy Murphy, Irish Mathematical Society Bulletin"This is a work that mathematicians, historians, and philosophers will find especially engaging, as will anyone with a serious interest in mathematics and the limits of certainty that it is constantly probing."---J.W. Dauben, Choice

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Book SynopsisThe weird and wonderful quest for unfathomably large numbersTrade Review‘A wonderful new book… if you love journeying into imagined mathematical worlds and simply exploring, then [this book] is pure, unadulterated escapism… brilliant.’ -- New Scientist‘We are taken on an amazing adventure… [with] witty humour and fascinating facts… a comprehensive read that I would struggle to find fault in and for anyone with a passion for maths, or a knack for numbers, I couldn’t recommend it enough!’ -- Astronomy Ireland‘The brilliant combination of an accomplished science writer and a young mathematical prodigy has resulted in page after page that oozes enthusiasm, clarity and intrigue.’ -- Bobby Seagull, on Weirder MathsTable of ContentsIntroduction 1 Of Sand and Stars 2 At the Limits of Reality 3 Maths Unbound 4 Up, Up and Away 5 G Whizz 6 Conway’s Chains 7 Ackermann and the Power of Recursion 8 Figure This – If You Can 9 Infinite Matters 10 Growing Fast 11 Does Not Compute! 12 The Strange World of the Googologist 13 Bridge to Beyond 14 The Biggest Number of All Acknowledgements Bibliography Useful websites and webpages References

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Book SynopsisTrade ReviewHonorable Mention for the 2015 PROSE Award in Popular Science & Popular Mathematics, Association of American Publishers "A book that stimulates the mind as well as the eye."--Scientific American "The combination of art and exposition was quite effective. The writing is accessible to most reasonably well-educated laypeople, and I imagine that many such people would derive considerable pleasure dipping into this attractive and interesting book."--Mark Hunacek, MAA Reviews "Eli Maor's lively writing benefits in equal parts from the geometry of ancient Greece and the eye-popping images conjured by artist Eugen Jost."--Bill Cannon, Scientist's Bookshelf "Graphic illustrations serve as both beautiful abstract art and helpful explanations in this overview of geometric theorems and patterns."--Science News "[Beautiful Geometry] achieves its aim to demonstrate that there is visual beauty in Mathematics. I heartily recommend it."--LSE Review of Books "The explanations are clear, and cover the background to the paintings in a manner that will be appreciated by readers whatever their level of mathematical knowledge... Anyone with any interest in visual mathematics will love this book."--Times Higher Education "A good-looking, large-format book suitable for the coffee table, but with lots of mathematical ideas packed in among the colorful illustrations... [A] handsome book for browsing and for some deep thought, and would be a superb gift for anyone (especially a young person) who has interest in mathematics."--Rob Hardy, Columbus Dispatch "It is a handsome book for browsing and for some deep thought, and would be a superb gift for anyone (especially a young person) who has interest in mathematics."--Rob Hardy, Dispatch "The book by Maor and Jost should be given to everyone--young or old--embarking on the study of mathematics or anyone teaching mathematics. The book will act as a source of inspiration and as a reminder of why it is that mathematics has fascinated the human race for millennia."--Henrik Jeldtoft Jensen, LMS Newsletter "The content is accessible to anyone with even a high school course in geometry. The writing is very clear."--Choice "Clear and lively... The mathematics in this book is first-rate, but the real surprise is how well the art reflects and illuminates the topic at hand... All of it is lovely to look at... [Beautiful Geometry] rises to the level of a coffee-table art book, only with a lot more depth."--Mathematical Reviews "[E]erily captivating book... Maor's style of writing is conversational, and the writing is engaging."--Annalisa Crannell, Journal of Mathematics and the Arts "At a very reasonable price, this is a book which would grace the coffee-table of any mathematics department, and many of the ideas in it will stimulate valuable discussions in the classroom."--Gerry Leversha, Mathematical Gazette "It presents as a coffee-table book for mathematicians and would be a good addition to a classroom library, available for students of all ages to explore."--Susan Mielechowsky, Mathematics Teaching in the Middle School "Visually stunning... [Beautiful Geometry] raises fundamental questions, answered thousands of years later and evidencing the progress made... This is an engaging book of broad appeal and a colourful approach to the history of geometry."--Mathematics TodayTable of ContentsPrefaces ix 1.Thales of Miletus 1 2.Triangles of Equal Area 3 3.Quadrilaterals 6 4.Perfect Numbers and Triangular Numbers 9 5.The Pythagorean Theorem I

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Book SynopsisTHE INTERNATIONAL BESTSELLER''An entertaining tour that will change how you see the world'' Sean Carroll, author of Something Deeply HiddenIs there a secret formula for improving your life? For making something a viral hit? For deciding how long to stick with your current job, Netflix series, or even relationship?This book is all about the equations that make our world go round. Ten of them, in fact. They are integral to everything from investment banking to betting companies and social media giants. And they can help you to increase your chance of success, guard against financial loss, live more healthily and see through scaremongering. They are known only by mathematicians - until now.With wit and clarity, mathematician David Sumpter shows that it isn''t the technical details which make these formulas so successful. It is the way they allow mathematicians to view problems from a different angle - a way of seeing the world that anyone cTrade ReviewSometimes books about numbers come along and we're so ecstatic that we just pop with delight. One such book is The Ten Equations that Rule The World -- Tim Harford * More or Less BBC4 *Hugely entertaining, erudite and at times genuinely witty . . . it's nice to be spoken to in grown-up language by a genius. You will come away from Sumpter's book with a much clearer idea of why the world is less messy than it appears * E&T Magazine *These aren't the equations of Newton or Einstein -- crisp relations describing the evolution of a clockwork universe. These are the equations of randomness, expectation, and imperfect information. The equations, in other words, of the real world. David Sumpter provides an entertaining tour that will change how you see the world -- Sean Carroll author of Something Deeply HiddenSumpter writes fascinatingly about his experiences as a consulting mathematician. . . I will encourage my mathematics undergraduates to read this book since it will inspire them by showing the relevance of mathematics to today's world and make them think about the moral issues they will face as mathematicians * Times Higher Education *

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Book SynopsisA mathematical sightseeing tour of the natural world from the author of THE MAGICAL MAZEWhy do many flowers have five or eight petals, but very few six or seven? Why do snowflakes have sixfold symmetry? Why do tigers have stripes but leopards have spots?Mathematics is to nature as Sherlock Holmes is to evidence. Mathematics can look at a single snowflake and deduce the atomic geometry of its crystals; it can start with a violin string and uncover the existence of radio waves. And mathematics still has the power to open our eyes to new and unsuspected regularities - the secret structure of a cloud or the hidden rhythms of the weather. There are patterns in the world we are now seeing for the first time - patterns at the frontier of science, yet patterns so simple that anybody can see them once they know where to look.

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Book SynopsisThis Element addresses the categoricity arguments that have been more effective in historical cases that reflect philosophically on internal mathematical matters than in recent questions of pre-theoretic metaphysics.Table of Contents1. Introduction; 2. Dedekind in 'Was sind und was sollen die Zahlen?' (1888); 3. Dedekind in 'Was sind und was sollen die Zahlen?' (1888); 4. Kreisel in 'Informal rigor and incompleteness proofs' (1967) and 'Two notes on the foundations of set theory'(1969); 5. Parsons in 'The uniqueness of the natural numbers' (1990) and 'Mathematical induction' (2008); 6. Parsons in 'The uniqueness of the natural numbers' (1990) and 'Mathematical induction' (2008); 7. Conclusion; References.

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Book Synopsis'Who knew numbers could be so charming? ... Suri takes us on a light-hearted journey all the way from nothing (zero) to infinity' Karen Joy Fowler, New York Times bestselling author of We Are All Completely Beside Ourselves Our universe has multiple origin stories, from religious creation myths to the Big Bang of scientists. But if we leave those behind and start from nothing – no matter, no cosmos, not even empty space – could we create a universe using only maths? In this new mathematical origin story, mathematician and award-winning novelist Manil Suri creates a natural progression of ideas needed to design our world, starting with numbers and continuing through geometry, algebra, and beyond. With evocative and engaging examples ranging from multidimensional crochet to the Mona Lisa’s asymmetrical smile, as well as ingenious storytelling that helps illuminate complex concepts like infinity and relativity, The Big Bang of Numbers charts a playful, inventive course to existence. Distilled from almost four decades of teaching experience, and offering both striking new perspectives for maths aficionados and an accessible introduction for enthusiastic novices, The Big Bang of Numbers proves that we can all fall in love with maths.Trade ReviewThe fundamental[s] of maths presented like a delightful fairy-tale. Hugely enjoyed it * Dara O'Brien *A beautifully written meditation on mathematics: whimsical, thought-provoking and deep * Alex Bellos, author of Alex's Adventures in Numberland *Who knew numbers could be so charming? So industrious? Suri takes us on a light-hearted journey all the way from nothing (zero) to infinity. Math has rarely been so readable. * Karen Joy Fowler, New York Times bestselling author of We Are All Completely Beside Ourselves *In The Big Bang of Numbers, Manil Suri invites the reader to create a universe made of mathematical ideas, sparking a thrill that may catch you off guard—an exhilarating sensation of playfulness, power, and insight. * Steven Strogatz, New York Times bestselling author of Infinite Powers and The Joy of X *Numerophobic? This infinitely fascinating book will cure you ... Manil Suri’s march through maths is brimming with entertaining and yet discombobulating thoughts ... Suri has a knack for clarity and a welcome habit of grounding tricky concepts in the tangible * The Times *A most unusual, creative, and fascinating account of mathematics that relies not on equations or formulas, but on metaphors, paradoxes, and lovely vignettes. * John Allen Paulos, author of Innumeracy and A Mathematician Reads the Newspaper *A delightful ride of a book. Before I knew it, I was rooting for primes, doubting the wisdom of dividing by zero, and holding my breath as the universe starts to emerge from triangles and cones and planes. The book was so enjoyable and understandable, it almost made me want to take another stab at calculus. Almost. Proof that when a smart person who writes well and honestly explores their passion, that passion is contagious. * Ken Krimstein, author of When I Grow Up *An excellent new book that could make anyone fall in love with math * Washingtonian *

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Book SynopsisWe all learned that the ratio of the circumference of a circle to its diameter is called pi and that the value of this algebraic symbol is roughly 3.14. What we weren't told, though, is that behind this seemingly mundane fact is a world of mystery, which has fascinated mathematicians from ancient times to the present. Simply put, pi is weird. Mathematicians call it a "transcendental number" because its value cannot be calculated by any combination of addition, subtraction, multiplication, division, and square root extraction. In this delightful layperson's introduction to one of math's most interesting phenomena, Drs. Posamentier and Lehmann review pi's history from prebiblical times to the 21st century, the many amusing and mind-boggling ways of estimating pi over the centuries, quirky examples of obsessing about pi (including an attempt to legislate its exact value), and useful applications of pi in everyday life, including statistics.This enlightening and stimulating approach to mathematics will entertain lay readers while improving their mathematical literacy.Trade Review""There is something for everyone in this book and everyone should read this book because it will be for some, a revelation that mathematics can be fun and beautiful, something they may not have realized during earlier encounters. Math teachers will find a host of ideas to enrich their instruction since Pi, as you know, comes up everywhere. This book is highly recommended and should provide a major step toward increasing the popularity of mathematics.”-Education Update “A joyful exploration…written in a conversational style reminiscent of children's science books. The writing is clear and crisp and draws the reader into the author's enthusiasm…I highly recommend [this book] to high school and college students and teachers of both. The book captures the excitement and fascination of pi and can serve as a starting point for more detailed discussion.”-Mathematics Teacher“I enjoyed reading the book…for its many applications, curiosities, and anecdotes.”-Science “Readers curious about pi could start here…Recommended.” -Choice

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Book SynopsisUncertainty is everywhere. It lurks in every consideration of the future - the weather, the economy, the sex of an unborn child - even quantities we think that we know such as populations or the transit of the planets contain the possibility of error. It's no wonder that, throughout that history, we have attempted to produce rigidly defined areas of uncertainty - we prefer the surprise party to the surprise asteroid. We began our quest to make certain an uncertain world by reading omens in livers, tea leaves, and the stars. However, over the centuries, driven by curiosity, competition, and a desire be better gamblers, pioneering mathematicians and scientists began to reduce wild uncertainties to tame distributions of probability and statistical inferences. But, even as unknown unknowns became known unknowns, our pessimism made us believe that some problems were unsolvable and our intuition misled us. Worse, as we realized how omnipresent and varied uncertainty is, we encountered chaos, quantum mechanics, and the limitations of our predictive power. Bestselling author Professor Ian Stewart explores the history and mathematics of uncertainty. Touching on gambling, probability, statistics, financial and weather forecasts, censuses, medical studies, chaos, quantum physics, and climate, he makes one thing clear: a reasonable probability is the only certainty.Trade ReviewIntriguing ... a challenging but rewarding trip through a quantum world of uncertainties. * Publisher's Weekly *Praise for Ian Stewart: Stewart is Britain's most brilliant and prolific populariser of maths -- Alex BelosThis is not pure maths. It is maths contaminated with wit, wisdom, and wonder ... He guides us on a mind-boggling journey from the ultra trivial to the profound. Thoroughly entertaining * New Scientist *Humbling and inspiring. Stewart shows with his typical clarity how the power of pure thought has shaped our world for over two millennia. -- Jim Al-Khalili, FRSThis is a superb Cabinet of Mathematical Curiosities that deserves a place with the classics of the genre * Mathematics Today *'With captivating stories and his signature clarity, Ian Stewart shows us how math makes the world - and the rest of the universe - go round. -- Steven Strogatz, Professor of Mathematics, Cornell University, and author of The Joy of XStewart has served up the instructive equivalent of a Michelin-starred tasting menu, or perhaps a smorgasbord of appetisers. And of course, appetisers are designed to give you an appetite for more. -- Tim Radford * Guardian *

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Book SynopsisTrade Review"Nahin includes gems from all over mathematics, ranging from engineering applications to beautiful pure-mathematical identities... It would be good to have more books like this."--Timothy Gowers, Nature "Nahin's tale of the formula e[pi] i+1=0, which links five of the most important numbers in mathematics, is remarkable. With a plethora of historical and anecdotal material and a knack for linking events and facts, he gives the reader a strong sense of what drove mathematicians like Euler."--Matthew Killeya, New Scientist "It is very difficult to sum up the greatness of Euler... This excellent book goes a long way to explaining the kind of mathematician he really was."--Steve Humble, Mathematics Today "What a treasure of a book this is! This is the fourth enthusiastic, informative, and delightful book Paul Nahin has written about the beauties of various areas of mathematics... This book is a marvelous tribute to Euler's genius and those who built upon it and would make a great present for students of mathematics, physics, and engineering and their professors."--Henry Ricardo, MAA Reviews "The heart and soul of the book are the final three chapters on Fourier series, Fourier integrals, and related engineering. One can recommend them to all applied math students for their historical development and sensible content."--Robert E. O'Malley, Jr., SIAM Review "This is a book for mathematicians who enjoy historically motivated mathematical explanations on a high mathematical level."--Eberhard Knobloch, Mathematical Reviews "It is a 'popular' book, written for a general reader with some mathematical background equivalent to a first-year undergraduate course in the UK."--Robin Wilson, London Mathematical Society NewsletterTable of Contents*FrontMatter, pg. i*Contents, pg. ix*Preface to the Paperback Edition, pg. xiii*Preface, pg. xxix*Introduction, pg. 1*Chapter 1. Complex Numbers, pg. 13*Chapter 2. Vector Trips, pg. 68*Chapter 3. The Irrationality of pi2, pg. 92*Chapter 4. Fourier Series, pg. 114*Chapter 5. Fourier Integrals, pg. 188*Chapter 6. Electronics and -1, pg. 275*Euler: The Man and the Mathematical Physicist, pg. 324*Notes, pg. 347*Acknowledgments, pg. 375*Index, pg. 377

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Book SynopsisAn inclusive vision of mathematics—its beauty, its humanity, and its power to build virtues that help us all flourishTrade Review“Beautifully written, contains well-chosen and interesting mathematical puzzles, and offers an important viewpoint for mathematicians to consider. . . . The book is aimed at a broader audience and is also a call to being more inclusive, to recognising that there are many paths to success.”—Pamela Gorkin, Mathematical IntelligencerAwarded Book of the Year by Aleo ReviewWinner of the Euler Book Prize, sponsored by the Mathematical Association of AmericaSelected for the 2021 Phi Beta Kappa Award for Science Short List“The ancient Greeks argued that the best life was filled with beauty, truth, justice, play and love. The mathematician Francis Su knows just where to find them.”—Kevin Hartnett, Quanta Magazine “Please read this beautiful, compelling, galvanizing book if you care about mathematics, social justice, or humanity, which I hope is everyone.”—Eugenia Cheng, author of The Art of Logic in an Illogical World “The world desperately needs this all‑embracing and deeply human perspective on what mathematics is and why it matters. The key qualities developed by mathematical thinking are characteristics that we should all value and long for.”—Eddie Woo, author of It’s a Numberful World “I was mesmerized by this unusual, sublime book. Original insights and engaging puzzles made me feel young again, discovering a way to Zen and the Art of Mathematics.”—Nalini Joshi, University of Sydney “Francis Su believes that math can make us better humans—and he leads by example. Every page is a work of generosity and compassion. Plus, the puzzles will haunt you for weeks.”—Ben Orlin, author of Math with Bad Drawings “A celebration of mathematics and the human spirit. Learning mathematics enriches our lives, and Su wants everyone to have a seat at the banquet.”—Edward Scheinerman, author ofvThe Mathematics Lover’s Companion “A delightful mixture of philosophy, mathematical illustrations, and compassion.”—John Cook, Singular Value Consulting “Francis Su has written a lyrical meditation on the beauty of mathematics and how it connects to our common humanity.”—John Urschel, author of Mind and Matter: A Life in Math and Football “Su elegantly uncovers the beauty and power of mathematics as they relate to our desires to be loved, trusted, and accepted. A powerful narrative of mathematical beauty, this book is the antidote for a mathematically fixed mindset.”—Talithia Williams, author of Power in Numbers: The Rebel Women of Mathematics “This is perhaps the most important mathematics book of our time. Francis Su shows mathematics is an experience of the mind and, most important, of the heart.”—James Tanton, Global Math Project“The ancient Greeks argued that the best life was filled with beauty, truth, justice, play and love. The mathematician Francis Su knows just where to find them.”—Kevin Hartnett, Quanta Magazine

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Book SynopsisThis is a cultural history of mathematics and art, from antiquity to the present. Mathematicians and artists have long been on a quest to understand the physical world they see before them and the abstract objects they know by thought alone. Taking readers on a tour of the practice of mathematics and the philosophical ideas that drive the disciplinTrade Review"This is a marvelous coffee table book ... very well researched and documented. It touches upon so many fundamental questions that philosophers, scientists, mathematicians and artists have asked since antiquity. But yet it guides the reader smoothly through all these competing visions and theories without becoming dull or getting lost in abstraction. This is the history of Western civilization with particular interest in art and mathematics, illuminating and instructive, and all wrapped up in a rich, colorful, and fancy book."--Adhemar Bultheel, European Mathematical Society "This is the beauty and power of this book: [Mathematics and Art] is an intellectual tour de force of art history and its interaction with mathematics that will draw most readers, including me, back for further reading and study."--Frank Swetz, MAA Reviews "Excellent new book... Overall this is a comprehensive, valuable and detailed book. It is written in an accessible style, with enough mathematics to interest the technical reader without overwhelming one with an arts background... Its rich anthology is particularly relevant today, given the explosion of interest in the digital arts and the need for digital artists to use maths creatively. I will definitely be keeping it close at hand."--William Latham, New Scientist "The author does an artful job in creating a wide-ranging and beautifully illustrated survey that mathematicians and art historians will enjoy."--John Barrow, The Art Newspaper "This sumptuously illustrated book chronicles the history of mathematics through its intersection with the development of visual art... Gamwell articulates the compelling, far-reaching connections within these fields in a way that is rewarding for scholars yet accessible to non-specialists."--Choice "Beautiful books that display the beauty of art are fine additions to many coffee tables; beautiful books that display the beauty of mathematics are fine additions to few coffee tables. Gamwell's impressive work integrates the beauty of these two disciplines to create a work larger than their sum... A book for all ages and of all ages: truly a brilliant 'millennial' composition!"--Sandra L. Arlinghaus, Mathematical Reviews "This splendidly produced volume will appeal to everybody interested in mathematics and art and offers room for agreement and disagreement with the author... This volume stands out by its richness in contents, its wealth of colour reproductions and, last but not least, its very affordable price."--Dirk Werner, Zentralblatt MATH "This wonderful book gives a very thorough overview of the impact of mathematics (and science) of the visual arts (and architecture) over the centuries."--Eos "An interesting read, filled with paradigm-shifting history and art, the book still posits a linear perspective of the relationship of art and mathematics, specifically recounting the ways math has influenced art."--Karie Brown, Mathematics Teacher "A monumental volume... Excellently illustrated by 523 images... Many highlighted quotations from writings of outstanding personalities of the sciences and the arts make the volume more colourful."--Gyorgy Darvas, Symmetry "Mathematics and Art is an enjoyable read accessible to anyone interested in the history of mathematics and art."--Andre Michael Hahn, British Journal for the History of ScienceTable of ContentsFOREWORD by Neil deGrasse Tyson IX PREFACE XI 1 Arithmetic and Geometry 1 2 Proportion 73 3 Infinity 109 4 Formalism 151 5 Logic 197 6 Intuitionism 225 7 Symmetry 249 8 Utopian Visions after World War I 277 9 The Incompleteness of Mathematics 321 10 Computation 355 1 1 Geometric Abstraction after World War II 385 12 Computers in Mathematics and Art 455 13 Platonism in the Postmodern Era 499 NOTES 512 ACKNOWLEDGMENTS 547 CREDITS 548 INDEX 549

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Book SynopsisIn 1983 Gerry Kennedy set off on a tour through Russia, China, Japan and the USA to visit others involved in the global anti-war movement. Only dimly aware of his Victorian ancestors: George Boole, forefather of the digital revolution and James Hinton, eccentric philosopher and advocate of polygamy, he had directly followed in the footsteps of two dynasties of radical thinkers and doers.Their notable achievements, in which the women were particularly prominent, involved many spheres. Boole's wife, Mary Everest, niece of George Everest, surveyor of the eponymous mountain, was an early advocate of hands-on education. Of the five talented Boole daughters, Ethel Voynich, wife of the discoverer of the enigmatic, still unexplained Voynich Manuscript, campaigned with Russian anarchists to overthrow the Tsar. Her 1897 novel The Gadfly, filmed later with music by Shostakovich, sold in millions behind the Iron Curtain. She was rumoured to have had an affair with the notorious 'Ace of Spies', Sidney Reilly. One of Ethel's sisters married Charles Howard Hinton: a leading exponent of the esoteric realm of the fourth dimension and inventor of the gunpowder baseball-pitcher.Of their descendants, Carmelita Hinton also pioneered progressive education in the USA at her school in Putney, Vermont. Her children dedicated their lives to Mao's China. Appalled by the dropping on Japan of the atomic bomb that she had helped design, Joan Hinton defected to China and actively engaged in the Cultural Revolution. William Hinton wrote the influential documentary Fanshen based on his experience in 1948 of revolutionary change in a Shanxi village. Other members of the clan became renowned in their fields of physics, entomology and botany. Their combined legacy of independent and constructive thinking is perhaps typified by the invention of the Jungle Gym: the climbing-frame now used by children the world over. In The Booles and the Hintons the author embarks on a quest to reveal the stories behind their remarkable lives.Table of ContentsContentsAcknowledgements viiiForeword ixWho's Who xii 1 GRAVY OVER A TABLECLOTH 1The author's own background in West London. 'Discovery' of the Booles, brief outline of the Boole/Hinton major characters.2 THE UGLY DUCKLING 12The Voynich Manuscript in brief: its history and research to date.3 BRINGING STARRY WISDOM DOWN 29The author's visit to Lincoln. George Boole's life and background.4 THE MISSUS 59Mary Everest Boole: her life with George Boole and her own views on life and education.5 TADPOLES INTO FROGS 83The five Boole daughters' early lives.6 THE WIZARD 89James Hinton: his life and philosophy.7 TRAPDOORS AND VELVETEEN 104Tsarist Russia in the late nineteenth century. Russian anarchists: Kropotkin and Stepniak. Charlotte Wilson and Wyldes Farm.Radical politics in England in the 1880s.8 CITIZENS' DIPLOMACY 119The author's peace politics, Moscow dissidents and journey on the Trans-Siberian railway, 1983. Ethel Boole's stay in St Petersburg, 1887-89.9 ROUND AND ROUND THE GARDEN 141James Hinton's comeuppance. Olive Schreiner and Havelock Ellis. Caroline Haddon. The Men and Women's Club.10 THE RIFF-RAFF OF RASCALDOM 153The Society of Friends of Russian Freedom. Russian anarchists and related novels of the period. London arrival of Wilfrid Voynich.11 COMMUNING WITH SPACE 170Charles Howard Hinton and the fourth dimension. His and Mary Ellen Boole's exile to Japan and the USA in 1887. Hinton'swritings and influence in Europe. Peter Ouspensky in Russia.12 THE CITADEL 197The author's visit to Warsaw. Proletariat: early Polish socialists. History of Wilfrid's escape from Siberia.13 DEAD AS MUTTON? 213Ethel Voynich's novel The Gadfly. Wilfrid Voynich establishes his book trade. Jack Raymond, another novel by Ethel.14 THE ENGLISH AUNTS 227The five Boole sisters: Mary Ellen, Margaret, Alice, Lucy and Ethel's later lives. Ethel and Ivor Gurney.15 BRINGING HOME THE BACON 247The Voynich Manuscript again. Ethel and Wilfrid in the USA.16 THE BIGGLES OF COMMUNISM 259Ethel in New York; her last novel. Ethel 're-discovered'. Scandal with the spy, Sidney Reilly.17 OLIVE AND OVOD 277The author's return to Russia via Lithuania searching for The Gadfly in Moscow and St Petersburg.18 A CQR LIFE 302The life of Cambridge scientist, Sir Geoffrey Taylor, grandson of George Boole.19 THE HINTON GENUS 314Hinton descendants in Mexico in the mining industry and their plant collecting. The life of Howard Everest Hinton, entomologist.20 UNCOLLAPSIBLE HINTONS 337The Byrdcliffe arts community, Woodstock USA. Carmelita Hinton and the setting up of Putney School, Vermont USA.21 THE GADGET 356The Manhattan Project and Joan Hinton. Hiroshima.22 PEKING JOAN 374Joan Hinton and Sid Engst in Mongolia, Si'an and Beijing during the Cultural Revolution.23 FANSHEN 398The life of Bill Hinton and the rural revolution in China. Author's visit to Long Bow village.24 THE VERMONTER 417The author returns to the USA to visit Putney School.Afterword 422Notes 427Illustration Credits 453Index 456

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Book SynopsisThis volume is a collection of articles on themes related to the book Laws of Form by George Spencer-Brown. Laws of Form was first published in 1969 and brings forth a new articulation of the foundations of thought. In Laws of Form we have a mathematical formalism based on one symbol and an approach to the question how the world would appear if a distinction could be drawn. Laws of Form does not answer the question how, given nothing as a beginning, a distinction can, indeed must, inevitably take place. This second question must, in its own structure, be left to each individual thinker. Nevertheless, Laws of Form, beautifully written and content free (form is emptiness, emptiness is form) is the most powerful mathematical text on the edge of nothing that has been produced since Euclid''s Elements. These papers are a tribute to Spencer-Brown and his singular achievement.

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Book SynopsisTrade Review"This is not your father’s – or grandfather’s – standard collection of conic sections."---Jim Stein, New Books Network"Undoubtedly [this book], written in the same entertaining unmistakable style of the author and containing a lot of information - mathematical, historical and general - will attract, as the previous ones, a large audience."---S. Cobzas, Studia Mathematica"What a beautiful book!"---Jonathan Shock, Mathemafrica.org"A wonderful addition to libraries where the mathematically curious find their reading." * Choice *"Havil’s narrative for each curve is a cornucopia of fun facts and rigorous explanation."---Andrew J. Simoson, Mathematical Intelligencer"Overall, the book was a delight to read. The writing is witty and entertaining, the history is at times peculiar and surprising, and the mathematics is rich and engaging. It would make a fine addition to a classroom bookcase or home coffee table, but while there are plenty of elegant diagrams and intriguing stories to give every curious reader the chance to glimpse mathematical beauty, only those with the ability to dig beneath the surface will understand just how much beauty this book has to offer."---Samuel Hewitt, Mathematical Gazette

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Book SynopsisTrade ReviewHis book adopts a self-consciously neo-Quinean methodology, and argues that the theory that is developed helps to motivate and clarify Quine's naturalistic metaphysical picture. * MathSciNet *Table of ContentsIntroduction Chapter I. The Quinean legacy Chapter II. Propositions Chapter III. Predicates and predication Chapter IV. First-order modal logic, and a first-order theory of propositions Chapter V. Properties and relations Chapter VI. Possible worlds and possible individuals References

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Book SynopsisThink of a number between one and ten.No, hang on, let's make this interesting. Between zero and infinity. Even if you stick to the whole numbers, there are a lot to choose from - an infinite number in fact. Throw in decimal fractions and infinity suddenly gets an awful lot bigger (is that even possible?) And then there are the negative numbers, the imaginary numbers, the irrational numbers like pi which never end. It literally never ends.The world of numbers is indeed strange and beautiful. Among its inhabitants are some really notable characters - pi, e, the "imaginary" number i and the famous golden ratio to name just a few. Prime numbers occupy a special status. Zero is very odd indeed: is it a number, or isn't it?How Numbers Work takes a tour of this mind-blowing but beautiful realm of numbers and the mathematical rules that connect them. Not only that, but take a crash course on the biggest unsolved problems that keep mathematicians up at night, find out about the strange and unexpected ways mathematics influences our everyday lives, and discover the incredible connection between numbers and reality itself.ABOUT THE SERIESNew Scientist Instant Expert books are definitive and accessible entry points to the most important subjects in science; subjects that challenge, attract debate, invite controversy and engage the most enquiring minds. Designed for curious readers who want to know how things work and why, the Instant Expert series explores the topics that really matter and their impact on individuals, society, and the planet, translating the scientific complexities around us into language that's open to everyone, and putting new ideas and discoveries into perspective and context.

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Book SynopsisBayesian ideas have recently been applied across such diverse fields as philosophy, statistics, economics, psychology, artificial intelligence, and legal theory. Fundamentals of Bayesian Epistemology examines epistemologists'' use of Bayesian probability mathematics to represent degrees of belief. Michael G. Titelbaum provides an accessible introduction to the key concepts and principles of the Bayesian formalism, enabling the reader both to follow epistemological debates and to see broader implicationsVolume 1 begins by motivating the use of degrees of belief in epistemology. It then introduces, explains, and applies the five core Bayesian normative rules: Kolmogorov''s three probability axioms, the Ratio Formula for conditional degrees of belief, and Conditionalization for updating attitudes over time. Finally, it discusses further normative rules (such as the Principal Principle, or indifference principles) that have been proposed to supplement or replace the core five.Volume 2 gives arguments for the five core rules introduced in Volume 1, then considers challenges to Bayesian epistemology. It begins by detailing Bayesianism''s successful applications to confirmation and decision theory. Then it describes three types of arguments for Bayesian rules, based on representation theorems, Dutch Books, and accuracy measures. Finally, it takes on objections to the Bayesian approach and alternative formalisms, including the statistical approaches of frequentism and likelihoodism.Trade ReviewMichael G. Titelbaum provides an accessible introduction to the key concepts and principles of the Bayesian formalism, enabling the reader both to follow epistemological debates and to see broader implications * MathSciNet *Table of ContentsIII Applications 6: Confirmation 7: Decision Theory IV Arguments for Bayesianism 8: Representation Theorems 9: Dutch Book Arguments 10: Accuracy Arguments Challenges and Objections 11: Memory Loss and Self-Location 12: Old Evidence, Logical Omniscience 13: Alternatives to Bayesianism 14: Comparisons, Ranges, Dempster-Shafer

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Book SynopsisBayesian ideas have recently been applied across such diverse fields as philosophy, statistics, economics, psychology, artificial intelligence, and legal theory. Fundamentals of Bayesian Epistemology examines epistemologists'' use of Bayesian probability mathematics to represent degrees of belief. Michael G. Titelbaum provides an accessible introduction to the key concepts and principles of the Bayesian formalism, enabling the reader both to follow epistemological debates and to see broader implicationsVolume 1 begins by motivating the use of degrees of belief in epistemology. It then introduces, explains, and applies the five core Bayesian normative rules: Kolmogorov''s three probability axioms, the Ratio Formula for conditional degrees of belief, and Conditionalization for updating attitudes over time. Finally, it discusses further normative rules (such as the Principal Principle, or indifference principles) that have been proposed to supplement or replace the core five.Volume 2 gives arguments for the five core rules introduced in Volume 1, then considers challenges to Bayesian epistemology. It begins by detailing Bayesianism''s successful applications to confirmation and decision theory. Then it describes three types of arguments for Bayesian rules, based on representation theorems, Dutch Books, and accuracy measures. Finally, it takes on objections to the Bayesian approach and alternative formalisms, including the statistical approaches of frequentism and likelihoodism.Table of ContentsQuick Reference Preface I Our Subject 1: Beliefs and Degrees of Belief II The Bayesian Formalism 2: Probability Distributions 3: Conditional Credences 4: Updating by Conditionalization 5: Further Rational Constraints

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Book SynopsisCategory theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda''s lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists!This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study.Trade ReviewThe book is well organised and very well written. The presentation of the material is from the concrete to the abstract, proofs are worked out in detail and the examples and the exercises spread throughout the text mark a pleasant rhythm for its reading. In all, Awodey's Category Theory is a very nice and recommendable introduction to the subject. * Pere Pascual, EMS Newsletter *Table of ContentsPreface ; 1. Categories ; 2. Abstract Structures ; 3. Duality ; 4. Groups and Categories ; 5. Limits and Colimits ; 6. Exponentials ; 7. Naturality ; 8. Categories of Diagrams ; 9. Adjoints ; 10. Monads and Algrebras ; References ; Solutions to Selected Exercises ; Index

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Book SynopsisSince 1973, Galois theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fifth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for todayâs algebra students.New to the Fifth Edition Reorganised and revised Chapters 7 and 13 New exercises and examples Expanded, updated references Further historical material on figures besides Galois: Omar Khayyam, Vandermonde, Ruffini, and Abel A new final chapter discussing other directions in which Galois theory has developed: the inverse Galois problem, differential Galois theory, and a (very) brief introduction to p-adic Galois representations This bestseller continues to deliver a rigorous, yet engaging, treatment of the subject while keeping pace with current educational requirements. More than 200 exercises and a wealth of historical notes augment the proofs, formulas, and theorems.Trade Review"In mathematics, the fundamental theorem of Galois theory connects field theory and group theory, enabling certain mathematical problems in field theory to be reduced to group theory, making the problems simpler and easier to understand. The fifth updated edition of the textbook Galois Theory is an invaluable teaching text and resource for instructors of undergraduate mathematics students. Featuring more than 200 exercises and historical notes to enhance understanding of the proofs, formulas, and theorems, the fifth edition of Galois Theory is a "must-have" for university library mathematics collections, and highly recommended for instructors or for self-study"- Midwest Books ReviewPraise for the Previous Editions"… this book remains a highly recommended introduction to Galois theory along the more classical lines. It contains many exercises and a wealth of examples, including a pretty application of finite fields to the game solitaire. … provides readers with insight and historical perspective; it is written for readers who would like to understand this central part of basic algebra rather than for those whose only aim is collecting credit points."—Zentralblatt MATH 1322"This edition preserves and even extends one of the most popular features of the original edition: the historical introduction and the story of the fatal duel of Evariste Galois. … These historical notes should be of interest to students as well as mathematicians in general. … after more than 30 years, Ian Stewart’s Galois Theory remains a valuable textbook for algebra undergraduate students."—Zentralblatt MATH, 1049"The penultimate chapter is about algebraically closed fields and the last chapter, on transcendental numbers, contains ‘what-every-mathematician-should-see-at-least-once,’ the proof of transcendence of pi. … The book is designed for second- and third-year undergraduate courses. I will certainly use it."—EMS NewsletterTable of Contents1. Classical Algebra. 1.1. Complex Numbers. 1.2. Subfields and Subrings of the Complex Numbers. 1.3. Solving Equations. 1.4. Solution by Radicals. 2. The Fundamental Theorem of Algebra. 2.1. Polynomials. 2.2. Fundamental Theorem of Algebra. 2.3. Implications 3. Factorisation of Polynomials. 3.1. The Euclidean Algorithm. 3.2 Irreducibility. 3.3. Gauss’s Lemma. 3.4. Eisenstein’s Criterion. 3.5. Reduction Modulo p. 3.6. Zeros of Polynomials. 4. Field Extensions. 4.1. Field Extensions. 4.2. Rational Expressions. 4.3. Simple Extensions. 5. Simple Extensions. 5.1. Algebraic and Transcendental Extensions. 5.2. The Minimal Polynomial. 5.3. Simple Algebraic Extensions. 5.4. Classifying Simple Extensions. 6. The Degree of an Extension. 6.1. Definition of the Degree. 6.2. The Tower Law. 6.3. Primitive Element Theorem. 7. Ruler-and-Compass Constructions. 7.1. Approximate Constructions and More General Instruments. 7.2. Constructions in C. 7.3. Specific Constructions. 7.4. Impossibility Proofs. 7.5. Construction From a Given Set of Points. 8. The Idea Behind Galois Theory. 8.1. A First Look at Galois Theory. 8.2. Galois Groups According to Galois. 8.3. How to Use the Galois Group. 8.4. The Abstract Setting. 8.5. Polynomials and Extensions. 8.6. The Galois Correspondence. 8.7. Diet Galois. 8.8. Natural Irrationalities. 9. Normality and Separability. 9.1. Splitting Fields. 9.2. Normality. 9.3. Separability. 10. Counting Principles. 10.1. Linear Independence of Monomorphisms. 11. Field Automorphisms. 11.1. K-Monomorphisms. 11.2. Normal Closures. 12. The Galois Correspondence. 12.1. The Fundamental Theorem of Galois Theory. 13. Worked Examples. 13.1. Examples of Galois Groups. 13.2. Discussion. 14. Solubility and Simplicity. 14.1. Soluble Groups. 14.2. Simple Groups. 14.3. Cauchy’s Theorem. 15. Solution by Radicals. 15.1. Radical Extensions. 15.2. An Insoluble Quintic. 15.3. Other Methods. 16. Abstract Rings and Fields. 16.1. Rings and Fields. 16.2. General Properties of Rings and Fields. 16.3. Polynomials Over General Rings. 16.4. The Characteristic of a Field. 16.5. Integral Domains. 17. Abstract Field Extensions and Galois Groups. 17.1. Minimal Polynomials. 17.2. Simple Algebraic Extensions. 17.3. Splitting Fields. 17.4. Normality. 17.5. Separability. 17.6. Galois Theory for Abstract Fields. 17.7. Conjugates and Minimal Polynomials. 17.8. The Primitive Element Theorem. 17.9. Algebraic Closure of a Field. 18. The General Polynomial Equation. 18.1. Transcendence Degree. 18.2. Elementary Symmetric Polynomials. 18.3. The General Polynomial. 18.5. Solving Equations of Degree Four or Less. 18.6. Explicit Formulas. 19. Finite Fields. 19.1. Structure of Finite Fields. 19.2. The Multiplicative Group. 19.3. Counterexample to the Primitive Element Theorem. 19.4. Application to Solitaire. 20. Regular Polygons. 20.1. What Euclid Knew. 20.2. Which Constructions are Possible? 20.3. Regular Polygons. 20.4. Fermat Numbers. 20.5. How to Construct a Regular 17-gon. 21. Circle Division. 21.1. Genuine Radicals. 21.2. Fifth Roots Revisited. 21.3. Vandermonde Revisited. 21.4. The General Case. 21.5. Cyclotomic Polynomials. 21.6. Galois Group of Q(ζ)= Q. 21.7. Constructions Using a Trisector. 22. Calculating Galois Groups. 22.1. Transitive Subgroups. 22.2. Bare Hands on the Cubic. 22.3. The Discriminant. 22.4. General Algorithm for the Galois Group. 23. Algebraically Closed Fields. 23.1. Ordered Fields and Their Extensions. 23.2. Sylow’s Theorem. 23.3. The Algebraic Proof. 24. Transcendental Numbers. 24.1. Irrationality. 24.2. Transcendence of e. 24.3. Transcendence of π. 25. What Did Galois Do or Know? 25.1. List of the Relevant Material. 25.2. The First Memoir. 25.3. What Galois Proved. 25.4. What is Galois Up To? 25.5. Alternating Groups, Especially A5. 25.6. Simple Groups Known to Galois. 25.7. Speculations about Proofs. 25.8. A5 is Unique. 26. Further Directions. 26.1. Inverse Galois Problem. 26.2. Differential Galois Theory. 26.3. p-adic Numbers.

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Book SynopsisA Mathematical Tour introduces readers to a selection of mathematical topics chosen for their centrality, importance, historical significance, and intrinsic appeal and beauty.

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Book SynopsisAfter 30 years of research, the author of The History of Correlation organized his notes into a manuscript draft during the lockdown months of the COVID-19 pandemic. Getting it into shape for publication took another few years. It was a labor of love.Readers will enjoy learning in detail how correlation evolved from a completely non-mathematical concept to one today that is virtually always viewed mathematically. This book reports in detail on 19th- and 20th-century English-language publications; it discusses the good and bad of many dozens of 20th-century articles and statistics textbooks in regard to their presentation and explanation of correlation. The final chapter discusses 21st-century trends.Some topics included here have never been discussed in depth by any historian. For example: Was Francis Galton lying in the first sentence of his first paper about correlation? Why did he choose the word co-relation rather than correlation for his new coeff

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Book SynopsisImre Lakatos's influential and enduring work on the nature of mathematic discovery and development continues to be relevant to philosophers of mathematics. Including a specially commissioned preface written by Paolo Mancosu, and presented in a fresh twenty-first-century series livery, it is now available for a new generation of readers.Trade Review'For anyone interested in mathematics who has not encountered the work of the late Imre Lakatos before, this book is a treasure; and those who know well the famous dialogue, first published in 1963–4 in the British Journal for the Philosophy of Science, that forms the greater part of this book, will be eager to read the supplementary material … the book, as it stands, is rich and stimulating, and, unlike most writings on the philosophy of mathematics, succeeds in making excellent use of detailed observations about mathematics as it is actually practised.' Michael Dummett, Nature'The whole book, as well as being a delightful read, is of immense value to anyone concerned with mathematical education at any level.' C. W. Kilmister, The Times Higher Education Supplement'In this book the late Imre Lakatos explores 'the logic of discovery' and 'the logic of justification' as applied to mathematics … The arguments presented are deep … but the author's lucid literary style greatly facilitates their comprehension … The book is destined to become a classic. It should be read by all those who would understand more about the nature of mathematics, of how it is created and how it might best be taught.' Education'How is mathematics really done, and - once done - how should it be presented? Imre Lakatos had some very strong opinions about this. The current book, based on his PhD work under George Polya, is a classic book on the subject. It is often characterized as a work in the philosophy of mathematics, and it is that - and more. The argument, presented in several forms, is that mathematical philosophy should address the way that mathematics is done, not just the way it is often packaged for delivery.' William J. Satzer, MAA ReviewsTable of ContentsPreface to this edition Paolo Mancosu; Editors' preface; Acknowledgments; Author's introduction; Part I: 1. A problem and a conjecture; 2. A proof; 3. Criticism of the proof by counterexamples which are local but not global; 4. Criticism of the conjecture by global counterexamples; 5. Criticism of the proof-analysis by counterexamples which are global but not local. The problem of rigour; 6. Return to criticism of the proof by counterexamples which are local but not global. The problem of content; 7. The problem of content revisited; 8. Concept-formation; 9. How criticism may turn mathematical truth into logical truth; Part II: Editors' introduction; Appendix 1. Another case-study in the method of proofs and refutations; Appendix 2. The deductivist versus the heuristic approach; Bibliography; Index of names; Index of subjects.

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Book SynopsisThis truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.Trade Review'Hacking does not restrict himself to the foundations of mathematics, but dares to cover both the breadth and the depth of mathematical philosophy.' Literary Review of Canada'… readable, presented in easily digestible chunks, clearly explained, and just a lot of fun …' Danny Yee's Book Reviews'Show[s] non-specialists … the sort of distinctive contribution to science and maths that a brilliant, very well-informed, philosopher can bring … I thoroughly recommend this book.' Alan Weir, The Times Literary Supplement'Hacking has composed a great overview of our understanding of mathematics and of the historical turning points and philosophical basics.' Peeter Müürsepp, Mathematical ReviewsTable of ContentsForeword; 1. A Cartesian introduction; 2. What makes mathematics mathematics?; 3. Why is there philosophy of mathematics?; 4. Proofs; 5. Applications; 6. In Plato's name; 7. Counter-Platonisms; Disclosures.

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Book SynopsisFor Wittgenstein mathematics is a human activity characterizing ways of seeing conceptual possibilities and empirical situations, proof and logical methods central to its progress. Sentences exhibit differing ''aspects'', or dimensions of meaning, projecting mathematical ''realities''. Mathematics is an activity of constructing standpoints on equalities and differences of these. Wittgenstein''s Later Philosophy of Mathematics (19341951) grew from his Early (19121921) and Middle (192933) philosophies, a dialectical path reconstructed here partly as a response to the limitative results of Gödel and Turing.Trade Review'Brimming with ideas from, and analyses of, Wittgenstein's philosophy of mathematics, the book contains both material of interest for those new to the topic, as well as often subtle and novel interpretations aimed at experts … A rich and sophisticated take on an intricate topic, Floyd's book is to be warmly recommended to all those eager to deepen their understanding of Wittgenstein's philosophy of mathematics.' Sorin Bangu, Philosophia MathematicaTable of Contents1. Introduction; 2. Early Philosophy: Absolute Simplicity; 3. Middle Philosophy (1929–1933): Relative Simplicity; 4. Later Philosophy: Fluid Simplicity (1937–1951).

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Book SynopsisEven the most enthusiastic of maths students probably at one time wondered when exactly it would all prove useful in ‘real life’. Well, maths reaches so far and wide through our world that, love it or hate it, we’re all doing maths almost every minute of every day. David Darling and Agnijo Banerjee go in search of the perfect labyrinth, journey back to the second century in pursuit of ‘bubble maths’, reveal the weirdest mathematicians in history and transform the bewildering into the beautiful, delighting us once again. Trade Review‘The brilliant combination of an accomplished science writer and a young mathematical prodigy has resulted in page after page that oozes enthusiasm, clarity and intrigue… A weird but truly wonderful read.’ -- Bobby Seagull, author of The Life-Changing Magic of Numbers and co-presenter of Monkman & Seagull’s Genius Guide to Britain‘A grand tour of the most exotic locations in the mathematical cosmos. Weirder Maths is exhilarating and entertaining, and will leave you with a wide-eyed appreciation of the world of numbers.’ -- Michael Brooks, author of 13 Things That Don’t Make Sense and The Quantum Astrologer’s Handbook

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Book SynopsisMaths is everywhere, in everything. It’s in the finest margins of modern sport. It’s in the electrical pulses of our hearts and the flight of every bird. It is our key to secret messages, lost languages and perhaps even the shape of the universe of itself. David Darling and Agnijo Banerjee reveal the mathematics at the farthest reaches of our world – from its role in the plots of novels to how animals employ numerical skills to survive. Along the way they explore what makes a genius, why a seemingly simple problem can confound the best and brightest for decades, and what might be the great discovery of the twenty-first century. As Bertrand Russell once said, ‘mathematics, rightly viewed, possesses not only truth, but supreme beauty’. Banerjee and Darling make sure we see it right again.Trade Review‘The brilliant combination of an accomplished science writer and a young mathematical prodigy.’ -- Bobby Seagull, author of The Life-Changing Magic of Numbers and co-presenter of Monkman & Seagull’s Genius Guide to Britain

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Book SynopsisOur finances, politics, media, opportunities, information, shopping and knowledge production are mediated through algorithms and their statistical approaches to knowledge; increasingly, these methods form the organizational backbone of contemporary capitalism. Revolutionary Mathematics traces the revolution in statistics and probability that has quietly underwritten the explosion of machine learning, big data and predictive algorithms that now decide many aspects of our lives. Exploring shifts in the philosophical understanding of probability in the late twentieth century, Joque shows how this was not merely a technical change but a wholesale philosophical transformation in the production of knowledge and the extraction of value. This book provides a new and unique perspective on the dangers of allowing artificial intelligence and big data to manage society. It is essential reading for those who want to understand the underlying ideological and philosophical changes that have fueled the rise of algorithms and convinced so many to blindly trust their outputs, reshaping our current political and economic situation.Trade ReviewPraise for Deconstruction Machines: Writing in the Age of Cyberwar:Deconstruction machines provides a powerful insight into how cyberwar serves to militarize writing, threatens civic infrastructure and thereby brings war into the code and software that governs our everyday lives. * International Affairs *Praise for Deconstruction Machines: Writing in the Age of Cyberwar:This book makes an important contribution to the future of philosophy under the threat posed by cyberwar and calls us to reconsider and continue working on the philosophical and deconstructive notion of spatiality. -- Catherine MalabouRevolutionary Mathematics critically explores the mathematical systems behind the machinery of abstraction that drives capitalism today - in particular, the machine learning algorithms at the heart of the global platform economy. In linking Marxist theories of real abstraction to mathematical paradigms, such as Bayesian statistics, this challenging work equips critical thought with the tools to contest the hidden codes that run our world. -- Alex Williams, University of East AngliaI have never before read a book on statistics at one gulp, but Justin Joque's investigation into the mathematical premises of algorithmic capital is a tour de force of radical forensics as compelling as any noir thriller. -- Nick Dyer-Witheford, University of Western OntarioWhile almost everyone today is a critic of the capricious algorithms that saturate our lives, Joque takes us into the metaphysical underpinnings of these systems: the statistical methods that provide a phantom objectivity to machine learning, financial derivatives, and even scientific research writ large. Both intensely focused and rich with philosophical asides, Revolutionary Mathematics is an original and vital contribution towards understanding -- and overcoming -- digitized capitalism and its manifold oppressions. -- Gavin Mueller, author of Breaking Things at Work

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Book SynopsisI first had a quick look, then I started reading it. I couldn't stop. -Gerard 't Hooft (Nobel Prize, in Physics 1999)This is a book about the mathematical nature of our Universe. Armed with no more than basic high school mathematics, Dr. Joel L. Schiff takes you on a foray through some of the most intriguing aspects of the world around us. Along the way, you will visit the bizarre world of subatomic particles, honey bees and ants, galaxies, black holes, infinity, and more. Included are such goodies as measuring the speed of light with your microwave oven, determining the size of the Earth with a stick in the ground and the age of the Solar System from meteorites, understanding how the Theory of Relativity makes your everyday GPS system possible, and so much more. These topics are easily accessible to anyone who has ever brushed up against the Pythagorean Theorem and the symbol π, with the lightest dusting of algebra. Through this book, science-curious readers will come to appreciate the patterns, seeming contradictions, and extraordinary mathematical beauty of our Universe. Table of ContentsDedication Acknowledgements About the Author Prologue Preface 1. The Mystery of Mathematics Let us be reasonable All set Where is Mathematics? Fine tuning A blast from the past: Euclid’s geometry Taking the Fifth further Pi in the sky Off to Monte Carlo Smashed pi The divine Isoperimetric Inequality 2. From Here to Infinity Zeno’s Paradox Summing Up In what Universe is this true? The power of e Fast money What is normal? Multiplying ad infinitum 3. Imaginary Worlds The Strange Case of The ‘ ’s have it The God-like Euler identity Even more imaginaries – quaternions But wait, there is more – octonians The world’s hardest problem – the Riemann Hypothesis 4. Random Universe Going steady Brownian Motion Life is a gamble The dating game The world of entropy – order to chaos Information entropy 5. Order from Chaos Cellular Automata Life as a game Infectious disease model – SIR Mimicking Darwin One-dimensional CA The whole is greater than the sum of its parts Bees and termites … And ants Bacteria count A hive of Mathematics: Fibonacci Dynamical systems Messrs. Fatou, Julia and Mandelbrot The fractal Universe 6. Mathematics in Space Faster than a speeding bullet Down to Earth Heavens above Light-years The great recession The Universe is flat Measuring the invisible: Black holes A galaxy far, far away 7. The Unreality of Reality Miniature Universe Quantum world Infinite space Qubits It is all relative, Albert That equation What time is it anyway? Matters of gravity Time in motion Radiation Symmetry and groups 8. The Unknowable Universe Gödel incompleteness Halting problem EMX Where is it, Dr. Heisenberg? Summing up Appendix I: Being Reasonable Appendix II: Hyperbolic Geometry and Minkowski Spacetime Appendix III: The Uncountable Real Numbers Appendix IV: c2 = c: Square and Line have Same Cardinality Appendix V: Geometric Series Appendix VI: Cesàro Sums Appendix VII: Rotating a Vector via a Quaternion Appendix VIII: Quaternions q2 = −1 Appendix IX: Riemann Zeta Function Appendix X: Random Walk Code Appendix XI: Age of the Solar System Appendix XII: Chelyabinsk Meteoroid Appendix XIII: Logic Gates Appendix XIV: Galaxy Distance via Cepheids Appendix XV: Time Dilation Appendix XVI: Expansion of the Universe Bibliography Index

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Book SynopsisPresents excerpts from the travel diary of a Japanese mathematician, Yamaguchi Kanzan, who journeyed on foot throughout Japan to collect temple geometry problems. This book explains the sacred and devotional aspects of sangaku, and reveals how Japanese folk mathematicians discovered many theorems independently of mathematicians in the West.Trade ReviewWinner of the 2008 PROSE Award in Mathematics, Association of American Publishers "Now Fukagawa Hidetoshi, a mathematics teacher, and writer Tony Rothman present a collection of Sangaku problems in their book, Sacred Mathematics. The puzzles range from simple algebra within the grasp of any intermediate-school student, to challenging problems that require graduate-school mathematics to solve. Copious illustrations and many detailed solutions show the scope, complexity, and beauty of what was tackled in Japan during the Tokugawa shogunate."--Peter J. Lu, Nature "Fascinating and beautiful book."--Physics World "This book is the most thorough (and beautiful) account of Japanese temple geometry (sangaku) available."--Paul J. Campbell, Mathematics Magazine "The difficult problems with complete solutions and rich commentary that comprise the heart of this book will interest every mathematics student."--Choice "This is a marvelous book. Good books are not just written or compiled, they are crafted. Sacred Mathematics is a well crafted work that combines mathematics, history and cultural considerations into an intriguing narrative... The writing style is appealing and the organization of material excellent. Princeton University Press must be congratulated on producing this quality publication and offering it at an agreeable price. This book is highly recommended for personal reading and library acquisition. It should be especially appealing to problem solvers."--Frank J. Swetz, Convergence "A unique book in every respect. Sacred Mathematics demonstrates how mathematical thinking can vary by culture yet transcend cultural and geographic boundaries."--International Institute for Asian Studies NewsletterTable of ContentsForeword by Freeman Dyson ix Preface by Fukagawa Hidetoshi xiii Preface by Tony Rothman xv Acknowledg ments xix What Do I Need to Know to Read This Book? xxi Notation xxv Chapter 1: Japan and Temple Geometry 1 Chapter 2: The Chinese Foundation of Japanese Mathematics 27 Chapter 3: Japa nese Mathematics and Mathematicians of the Edo Period 59 Chapter 4: Easier Temple Geometry Problems 89 Chapter 5: Harder Temple Geometry Problems 145 Chapter 6: Still Harder Temple Geometry Problems 191 Chapter 7: The Travel Diary of Mathematician Yamaguchi Kanzan 243 Chapter 8: East and West 283 Chapter 9: The Mysterious Enri 301 Chapter 10: Introduction to Inversion 313 For Further Reading 337 Index 341

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Book SynopsisTrade Review"The field has been due for a general treatment accessible to undergraduates and to mathematicians in other areas. . . . With Reverse Mathematics, John Stillwell provides exactly that kind of introduction."—Carl Mummert, Notices of the American Mathematical Society"Stillwell carefully situates the field in the broader context of the history of mathematics and its foundations, and does a fine job of making the whole endeavor accessible to a general mathematical audience."—Jeremy Avigad, Carnegie Mellon University"Filling an important niche, this book gives readers a good picture of the basics of reverse mathematics while suggesting several directions for further reading and study."—Denis Hirschfeldt, University of Chicago"Stillwell's book is self-contained and includes much background material in analysis, mathematical logic, combinatorics, and computability. I heartily commend this very readable and accessible book."—Stephen Simpson, Vanderbilt University

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Book SynopsisTrade Review"Nahin’s style is entertaining, directly addressing his readers. . . . Highly recommended."---Adhemar Bultheel, MAA Reviews"This book will be both enjoyable and a rich source of useful as well as intriguing information to a wide range of readers."---Michael Th. Rassias, zbMATH Open"I thoroughly enjoyed this book!"---Jonathan Shock, Mathemafrica.org"N/A"---Andrew Simoson, The Mathematical Intelligencer

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