Bestselling author and astrophysicist Mario Livio examines the lives and theories of history’s greatest mathematicians to ask how—if mathematics is an abstract construction of the human mind—it can so perfectly explain the physical world.
Nobel Laureate Eugene Wigner once wondered about “the unreasonable effectiveness of mathematics” in the formulation of the laws of nature. Is God a Mathematician? investigates why mathematics is as powerful as it is. From ancient times to the present, scientists and philosophers have marveled at how such a seemingly abstract discipline could so perfectly explain the natural world. More than that—mathematics has often made predictions, for example, about subatomic particles or cosmic phenomena that were unknown at the time, but later were proven to be true. Is mathematics ultimately invented or discovered? If, as Einstein insisted, mathematics is “a product of human thought that is independent of experience,” how can it so accurately describe and even predict the world around us?
Physicist and author Mario Livio brilliantly explores mathematical ideas from Pythagoras to the present day as he shows us how intriguing questions and ingenious answers have led to ever deeper insights into our world. This fascinating book will interest anyone curious about the human mind, the scientific world, and the relationship between them.
Unknown
"Is God a mathematician? In his new book Mario Livio delves into this question, putting it into a scientific, historical and philosophical context. He steers skillfully through deep and tricky waters, but writes with clarity and ease...Read the book and decide for yourself what the answer is." -- Sir Michael Atiyah, recipient of the Fields Medal, 1966, and the Abel Prize, 2004
"This highly readable book explores one of the most fascinating questions that lies at the heart of fundamental physics -- why is mathematics so effective in describing nature and is mathematics an invention of the human mind or part of the fabric of physical reality? Livio provides a wonderful review of the various issues, presents a wide variety of opinions, and in addition some fascinating insights of his own. I strongly recommend this volume to anyone interested in these questions." -- David Gross, 2004 Nobel Prize Winner in Physics, Frederick W. Gluck Professor of Theoretical Physics and Director, Kavli Institute For Theoretical Physics, University of California, Santa Barbara
"All science proceeds from the assumption that the cosmos is ordered in an intelligible way. Beneath the bewildering richness of natural phenomena there lies an elegant mathematical unity. How astonishing that the human mind is attuned to this hidden subtext of nature! With elegance and clarity, Mario Livio charts how, through science and mathematics, we have come to glimpse the fundamental rules on which the universe runs." -- Paul Davies, author of The Goldilocks Enigma and Director of the Beyond Center for Fundamental Concepts in Science, Arizona State University
"An exhilarating foray into the founding premises of mathematical science." -- Booklist
"Theologians have God, philosophers existence, and scientists mathematics. Mario Livio makes the case for how these three ideas might be related...Livio's rich history gives the discussions human force and verve." -- Sam Kean, New Scientist
About the Author
Mario Livio is an internationally known astrophysicist, a bestselling author, and a popular speaker who has appeared on The Daily Show , 60 Minutes , and NOVA. He is the author of the bestsellers The Golden Ratio , Brilliant Blunders , and Galileo. He lives in Baltimore, Maryland.
Review
"Is God a mathematician? In his new book Mario Livio delves into this question, putting it into a scientific, historical and philosophical context. He steers skillfully through deep and tricky waters, but writes with clarity and ease...Read the book and decide for yourself what the answer is." -- Sir Michael Atiyah, recipient of the Fields Medal, 1966, and the Abel Prize, 2004
"This highly readable book explores one of the most fascinating questions that lies at the heart of fundamental physics -- why is mathematics so effective in describing nature and is mathematics an invention of the human mind or part of the fabric of physical reality? Livio provides a wonderful review of the various issues, presents a wide variety of opinions, and in addition some fascinating insights of his own. I strongly recommend this volume to anyone interested in these questions." -- David Gross, 2004 Nobel Prize Winner in Physics, Frederick W. Gluck Professor of Theoretical Physics and Director, Kavli Institute For Theoretical Physics, University of California, Santa Barbara
"All science proceeds from the assumption that the cosmos is ordered in an intelligible way. Beneath the bewildering richness of natural phenomena there lies an elegant mathematical unity. How astonishing that the human mind is attuned to this hidden subtext of nature! With elegance and clarity, Mario Livio charts how, through science and mathematics, we have come to glimpse the fundamental rules on which the universe runs." -- Paul Davies, author of The Goldilocks Enigma and Director of the Beyond Center for Fundamental Concepts in Science, Arizona State University
"An exhilarating foray into the founding premises of mathematical science." -- Booklist
"Theologians have God, philosophers existence, and scientists mathematics. Mario Livio makes the case for how these three ideas might be related...Livio's rich history gives the discussions human force and verve." -- Sam Kean, New Scientist --This text refers to an alternate kindle_edition edition.
Starred Review Four centuries after church inquisitors accused Galileo of dangerous skepticism, a modern astrophysicist hails the Italian scientist as the embodiment of bold faith: namely, faith that God himself inscribed the heavens in mathematics. Because mathematics now empowers research communities investigating everything from deep-space pulsars to genetic proteins, a secularized version of Galileo’s credo now defines the new orthodoxy of science. But Livio recognizes a profound mystery inherent in the formulas his colleagues employ so sedulously: Why does the universe harmonize so well with numbers accessible to human minds? Probing this mystery, Livio traces the evolution of mathematical reasoning from the ritual symbolism of the ancient Pythagoreans to the multilayered analyses of twenty-first-century string theorists. In the impressive parade of intellectual explorers, we encounter Archimedes pondering geometrical figures at the very moment of his death, Descartes overthrowing all of medieval philosophy with one audacious thought, and Gödel quashing the ambitions of system-building logicians. This wide-ranging inquiry, however, ultimately highlights far more than personalities. A sharp conflict emerges between platonically minded philosophers who view mathematical breakthroughs as transcendent discoveries and psychologically inclined thinkers who interpret these breakthroughs as merely human inventions. Testing the tensions between these views, Livio delivers an exhilarating foray into the founding premises of mathematical science. --Bryce Christensen --This text refers to an alternate kindle_edition edition.
Review
"Is God a mathematician? In his new book Mario Livio delves into this question, putting it into a scientific, historical and philosophical context. He steers skillfully through deep and tricky waters, but writes with clarity and ease...Read the book and decide for yourself what the answer is."-- Sir Michael Atiyah, recipient of the Fields Medal, 1966, and the Abel Prize, 2004
"This highly readable book explores one of the most fascinating questions that lies at the heart of fundamental physics -- why is mathematics so effective in describing nature and is mathematics an invention of the human mind or part of the fabric of physical reality? Livio provides a wonderful review of the various issues, presents a wide variety of opinions, and in addition some fascinating insights of his own. I strongly recommend this volume to anyone interested in these questions."-- David Gross, 2004 Nobel Prize Winner in Physics, Frederick W. Gluck Professor of Theoretical Physics and Director, Kavli Institute For Theoretical Physics, University of California, Santa Barbara
"All science proceeds from the assumption that the cosmos is ordered in an intelligible way. Beneath the bewildering richness of natural phenomena there lies an elegant mathematical unity. How astonishing that the human mind is attuned to this hidden subtext of nature! With elegance and clarity, Mario Livio charts how, through science and mathematics, we have come to glimpse the fundamental rules on which the universe runs."-- Paul Davies, author of The Goldilocks Enigma and Director of the Beyond Center for Fundamental Concepts in Science, Arizona State University
"An exhilarating foray into the founding premises of mathematical science."-- Booklist
"Theologians have God, philosophers existence, and scientists mathematics. Mario Livio makes the case for how these three ideas might be related...Livio's rich history gives the discussions human force and verve."-- Sam Kean, New Scientist --This text refers to an alternate kindle_edition edition.
A few years ago, I was giving a talk at Cornell University. One of my PowerPoint slides read: "Is God a mathematician?" As soon as that slide appeared, I heard a student in the front row gasp: "Oh God, I hope not!"
My rhetorical question was neither a philosophical attempt to define God for my audience nor a shrewd scheme to intimidate the math phobics. Rather, I was simply presenting a mystery with which some of the most original minds have struggled for centuries -- the apparent omnipresence and omnipotent powers of mathematics. These are the type of characteristics one normally associates only with a deity. As the British physicist James Jeans (1877-1946) once put it: "The universe appears to have been designed by a pure mathematician." Mathematics appears to be almost too effective in describing and explaining not only the cosmos at large, but even some of the most chaotic of human enterprises.
Whether physicists are attempting to formulate theories of the universe, stock market analysts are scratching their heads to predict the next market crash, neurobiologists are constructing models of brain function, or military intelligence statisticians are trying to optimize resource allocation, they are all using mathematics. Furthermore, even though they may be applying formalisms developed in different branches of mathematics, they are still referring to the same global, coherent mathematics. What is it that gives mathematics such incredible powers? Or, as Einstein once wondered: "How is it possible that mathematics, a product of human thought that is independent of experience [the emphasis is mine], fits so excellently the objects of physical reality?"
This sense of utter bewilderment is not new. Some of the philosophers in ancient Greece, Pythagoras and Plato in particular, were already in awe of the apparent ability of mathematics to shape and guide the universe, while existing, as it seemed, above the powers of humans to alter, direct, or influence it. The English political philosopher Thomas Hobbes (1588-1679) could not hide his admiration either. In Leviathan, Hobbes's impressive exposition of what he regarded as the foundation of society and government, he singled out geometry as the paradigm of rational argument:
Seeing then that truth consisteth in the right ordering of names in our affirmations, a man that seeketh precise truth had need to remember what every name he uses stands for, and to place it accordingly; or else he will find himself entangled in words, as a bird in lime twigs; the more he struggles, the more belimed. And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind), men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning.
Millennia of impressive mathematical research and erudite philosophical speculation have done relatively little to shed light on the enigma of the power of mathematics. If anything, the mystery has in some sense even deepened. Renowned Oxford mathematical physicist Roger Penrose, for instance, now perceives not just a single, but a triple mystery. Penrose identifies three different "worlds": the world of our conscious perceptions, the physical world, and the Platonic world of mathematical forms. The first world is the home of all of our mental images -- how we perceive the faces of our children, how we enjoy a breathtaking sunset, or how we react to the horrifying images of war. This is also the world that contains love, jealousy, and prejudices, as well as our perception of music, of the smells of food, and of fear. The second world is the one we normally refer to as physical reality. Real flowers, aspirin tablets, white clouds, and jet airplanes reside in this world, as do galaxies, planets, atoms, baboon hearts, and human brains. The Platonic world of mathematical forms, which to Penrose has an actual reality comparable to that of the physical and the mental worlds, is the motherland of mathematics. This is where you will find the natural numbers 1, 2, 3, 4,..., all the shapes and theorems of Euclidean geometry, Newton's laws of motion, string theory, catastrophe theory, and mathematical models of stock market behavior. And now, Penrose observes, come the three mysteries. First, the world of physical reality seems to obey laws that actually reside in the world of mathematical forms. This was the puzzle that left Einstein perplexed. Physics Nobel laureate Eugene Wigner (1902-95) was equally dumbfounded:
The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
Second, the perceiving minds themselves -- the dwelling of our conscious perceptions -- somehow managed to emerge from the physical world. How was mind literally born out of matter? Would we ever be able to formulate a theory of the workings of consciousness that would be as coherent and as convincing as, say, our current theory of electromagnetism? Finally, the circle is mysteriously closed. Those perceiving minds were miraculously able to gain access to the mathematical world by discovering or creating and articulating a treasury of abstract mathematical forms and concepts.
Penrose does not offer an explanation for any of the three mysteries. Rather, he laconically concludes: "No doubt there are not really three worlds but one, the true nature of which we do not even glimpse at present." This is a much more humble admission than the response of the schoolmaster in the play Forty Years On (written by the English author Alan Bennett) to a somewhat similar question:
Foster: I'm still a bit hazy about the Trinity, sir.
Schoolmaster: Three in one, one in three, perfectly straightforward. Any doubts about that see your maths master.
The puzzle is even more entangled than I have just indicated. There are actually two sides to the success of mathematics in explaining the world around us (a success that Wigner dubbed "the unreasonable effectiveness of mathematics"), one more astonishing than the other. First, there is an aspect one might call "active." When physicists wander through nature's labyrinth, they light their way by mathematics -- the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. Newton observed a falling apple, the Moon, and tides on the beaches (I'm not even sure if he ever saw those!), not mathematical equations. Yet he was somehow able to extract from all of these natural phenomena, clear, concise, and unbelievably accurate mathematical laws of nature. Similarly, when the Scottish physicist James Clerk Maxwell (1831-79) extended the framework of classical physics to include all the electric and magnetic phenomena that were known in the 1860s, he did so by means of just four mathematical equations. Think about this for a moment. The explanation of a collection of experimental results in electromagnetism and light, which had previously taken volumes to describe, was reduced to four succinct equations. Einstein's general relativity is even more astounding -- it is a perfect example of an extraordinarily precise, self-consistent mathematical theory of something as fundamental as the structure of space and time.
But there is also a "passive" side to the mysterious effectiveness of mathematics, and it is so surprising that the "active" aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons -- with absolutely no application in mind -- turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality! How is that possible? Take for instance the somewhat amusing case of the eccentric British mathematician Godfrey Harold Hardy (1877-1947). Hardy was so proud of the fact that his work consisted of nothing but pure mathematics that he emphatically declared: "No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." Guess what -- he was wrong. One of his works was reincarnated as the Hardy-Weinberg law (named after Hardy and the German physician Wilhelm Weinberg [1862-1937]), a fundamental principle used by geneticists to study the evolution of populations. Put simply, the Hardy-Weinberg law states that if a large population is mating totally at random (and migration, mutation, and selection do not occur), then the genetic constitution remains constant from one generation to the next. Even Hardy's seemingly abstract work on number theory -- the study of the properties of the natural numbers -- found unexpected applications. In 1973, the British mathematician Clifford Cocks used the theory of numbers to create a breakthrough in cryptography -- the development of codes. Cocks's discovery made another statement by Hardy obsolete. In his famous book A Mathematician's Apology, published in 1940, Hardy pronounced: "No one has yet discovered any war-like purpose to be served by the theory of numbers." Clearly, Hardy was yet again in error. Codes have been absolutely essential for military communications. So even Hardy, one of the most vocal critics of applied mathematics, was "dragged" (probably kicking and screaming, if he had been alive) into producing useful mathematical theories.
But this is only the tip of the iceberg. Kepler and Newton discovered that the planets in our solar system follow orbits in the shape of ellipses -- the very curves studied by the Greek mathematician Menaechmus (fl. ca. 350 BC) two millennia earlier. The...
--This text refers to an alternate kindle_edition edition.
From The Washington Post
From The Washington Post's Book World/washingtonpost.com Reviewed by Marc Kaufman Did you know that 365 -- the number of days in a year -- is equal to 10 times 10, plus 11 times 11, plus 12 times 12? Or that the sum of any successive odd numbers always equals a square number -- as in 1 + 3 = 4 (2 squared), while 1 + 3 + 5 = 9 (3 squared), and 1 + 3 + 5 + 7 = 16 (4 squared)? Those are just the start of a remarkable number of magical patterns, coincidences and constants in mathematics. No wonder philosophers and mathematicians have been arguing for centuries over whether math is a system that humans invented or a cosmic -- possibly divine -- order that we simply discovered. That's the fundamental question Mario Livio probes in his engrossing book Is God a Mathematician? Livio, an astrophysicist at the Hubble Space Telescope Science Institute in Baltimore, explains the invention-vs.-discovery debate largely through the work and personalities of great figures in math history, from Pythagoras and Plato to Isaac Newton, Bertrand Russell and Albert Einstein. At times, Livio's theorems, proofs and conundrums may be challenging for readers who struggled through algebra, but he makes most of this material not only comprehensible but downright intriguing. Often, he gives a relatively complex explanation of a mathematical problem or insight, then follows it with a "simply put" distillation. An extended section on knot theory is, well, pretty knotty. But it ultimately sheds light on the workings of the DNA double helix, and Livio illustrates the theory with a concrete example: Two teams taking different approaches to the notoriously difficult problem of how many knots could be formed with a specific number of crossings -- in this case, 16 or fewer -- came up with the same answer: 1,701,936. The author's enthusiasm is infectious. But it also leads him to refer again and again to his subjects as "famous" and "great" and to their work as "monumental" and "miraculous." He has a weakness as well for extended quotes from these men (and they are all men) that slow the narrative without adding much. There are exceptions, including the tale of how Albert Einstein and mathematician Oskar Morgenstern tried to guide Kurt Gödel, a fellow mathematician and exile from Nazi Germany, through the U.S. immigration process. A deep-thinking and intense man, Gödel threw himself into preparing for his citizenship test, including an extremely close reading of the U.S. Constitution. In his rigorously logical analysis, he found constitutional weaknesses that he thought could allow for the rise of a fascist dictatorship in America. His colleagues told him to keep that reading to himself, but he blurted it out during his naturalization exam. He was allowed to stay anyway. The interplay of mini-biography and the march of mathematical knowledge serves the author well. It does not, however, ultimately help him to answer the big question, Is God a mathematician? On one side of the debate are all those remarkable constants that crop up, the makings of the ideal yet hidden world posited by Plato. In addition, there's what the physicist Eugene Wigner, in a seminal 1960 essay, called the "unreasonable effectiveness" of mathematical theorems: the astounding ability of math to predict unimagined results. Wigner was picking up on ideas explored earlier by Einstein, and Einstein's general theory of relativity remains one of the best examples: His predictions about how gravity can cause ripples in space-time was recently corroborated by measuring radio waves from a distant set of compact, high-energy stars called double pulsars, using technology unknown in Einstein's day. Doesn't all this indicate that the mathematical structure of the world is out there waiting to be discovered? On the other hand, math cannot explain many situations, and chaos theory suggests that it may never be possible to predict the weather or the stock market with accuracy. Recent research has pointed to basic mathematical constructs in the human brain, suggesting that we impose numbers and forms on the world, not vice versa. In addition, mathematics is less stable than it appears to us in grade school. At the higher reaches of the field, there is constant ferment and debate. If the "truths" discovered through mathematics are always changing, doesn't that indicate they are a product of human study and manipulation, rather than something fixed and eternal? As explained by Livio, the history of mathematics is partly a struggle between these points of view: that math is how God (or nature) organizes the world, or it is simply a human tool to understand that world. Livio comes down in the middle, contending that math may well be both invented and discovered. He points, for instance, to the eternal truth contained in the geometry formulated by Euclid 2,400 years ago. By the 19th century, however, iconoclasts had posited and established a whole new world of non-Euclidian geometry. Livio writes about the symmetries of the universe: the immutable, if incompletely understood, laws of math and physics that make a hydrogen atom, for instance, behave in the same way on Earth as it acts 10 billion light years away. Another sign of universal structure, as teased apart with the help of math? No, Livio writes, it is more likely a sign that "to some extent, scientists have selected what problems to work on based on those problems being amenable to a mathematical treatment." The author acknowledges that some readers will find his inconclusive conclusion to be unsatisfying. I didn't. Sometimes the adventure, the intellectual ride, is more important than the final destination. Copyright 2009, The Washington Post. All Rights Reserved. --This text refers to an alternate kindle_edition edition.
Description:
Bestselling author and astrophysicist Mario Livio examines the lives and theories of history’s greatest mathematicians to ask how—if mathematics is an abstract construction of the human mind—it can so perfectly explain the physical world.
Nobel Laureate Eugene Wigner once wondered about “the unreasonable effectiveness of mathematics” in the formulation of the laws of nature. Is God a Mathematician? investigates why mathematics is as powerful as it is. From ancient times to the present, scientists and philosophers have marveled at how such a seemingly abstract discipline could so perfectly explain the natural world. More than that—mathematics has often made predictions, for example, about subatomic particles or cosmic phenomena that were unknown at the time, but later were proven to be true. Is mathematics ultimately invented or discovered? If, as Einstein insisted, mathematics is “a product of human thought that is independent of experience,” how can it so accurately describe and even predict the world around us?
Physicist and author Mario Livio brilliantly explores mathematical ideas from Pythagoras to the present day as he shows us how intriguing questions and ingenious answers have led to ever deeper insights into our world. This fascinating book will interest anyone curious about the human mind, the scientific world, and the relationship between them.
Unknown
"Is God a mathematician? In his new book Mario Livio delves into this question, putting it into a scientific, historical and philosophical context. He steers skillfully through deep and tricky waters, but writes with clarity and ease...Read the book and decide for yourself what the answer is." -- Sir Michael Atiyah, recipient of the Fields Medal, 1966, and the Abel Prize, 2004
"This highly readable book explores one of the most fascinating questions that lies at the heart of fundamental physics -- why is mathematics so effective in describing nature and is mathematics an invention of the human mind or part of the fabric of physical reality? Livio provides a wonderful review of the various issues, presents a wide variety of opinions, and in addition some fascinating insights of his own. I strongly recommend this volume to anyone interested in these questions." -- David Gross, 2004 Nobel Prize Winner in Physics, Frederick W. Gluck Professor of Theoretical Physics and Director, Kavli Institute For Theoretical Physics, University of California, Santa Barbara
"All science proceeds from the assumption that the cosmos is ordered in an intelligible way. Beneath the bewildering richness of natural phenomena there lies an elegant mathematical unity. How astonishing that the human mind is attuned to this hidden subtext of nature! With elegance and clarity, Mario Livio charts how, through science and mathematics, we have come to glimpse the fundamental rules on which the universe runs." -- Paul Davies, author of The Goldilocks Enigma and Director of the Beyond Center for Fundamental Concepts in Science, Arizona State University
"An exhilarating foray into the founding premises of mathematical science." -- Booklist
"Theologians have God, philosophers existence, and scientists mathematics. Mario Livio makes the case for how these three ideas might be related...Livio's rich history gives the discussions human force and verve." -- Sam Kean, New Scientist
About the Author
Mario Livio is an internationally known astrophysicist, a bestselling author, and a popular speaker who has appeared on The Daily Show , 60 Minutes , and NOVA. He is the author of the bestsellers The Golden Ratio , Brilliant Blunders , and Galileo. He lives in Baltimore, Maryland.
Review
"Is God a mathematician? In his new book Mario Livio delves into this question, putting it into a scientific, historical and philosophical context. He steers skillfully through deep and tricky waters, but writes with clarity and ease...Read the book and decide for yourself what the answer is." -- Sir Michael Atiyah, recipient of the Fields Medal, 1966, and the Abel Prize, 2004
"This highly readable book explores one of the most fascinating questions that lies at the heart of fundamental physics -- why is mathematics so effective in describing nature and is mathematics an invention of the human mind or part of the fabric of physical reality? Livio provides a wonderful review of the various issues, presents a wide variety of opinions, and in addition some fascinating insights of his own. I strongly recommend this volume to anyone interested in these questions." -- David Gross, 2004 Nobel Prize Winner in Physics, Frederick W. Gluck Professor of Theoretical Physics and Director, Kavli Institute For Theoretical Physics, University of California, Santa Barbara
"All science proceeds from the assumption that the cosmos is ordered in an intelligible way. Beneath the bewildering richness of natural phenomena there lies an elegant mathematical unity. How astonishing that the human mind is attuned to this hidden subtext of nature! With elegance and clarity, Mario Livio charts how, through science and mathematics, we have come to glimpse the fundamental rules on which the universe runs." -- Paul Davies, author of The Goldilocks Enigma and Director of the Beyond Center for Fundamental Concepts in Science, Arizona State University
"An exhilarating foray into the founding premises of mathematical science." -- Booklist
"Theologians have God, philosophers existence, and scientists mathematics. Mario Livio makes the case for how these three ideas might be related...Livio's rich history gives the discussions human force and verve." -- Sam Kean, New Scientist --This text refers to an alternate kindle_edition edition.
From Publishers Weekly
The title of astrophysicist Livio's latest wide-ranging science survey is a teaser since God rarely makes an appearance; along with the French astronomer Laplace, Livio has no need of that hypothesis. Rather, Livio ( The Golden Ratio ) is concerned with the contentious question: is mathematics a human invention? Or is it the intricate design of the universe that we are slowly discovering? Scientists in past centuries have argued for the latter, Platonist position. In the last 50 years, however, many scientists, calling into question the whole idea of scientific discovery, maintain that we have invented mathematics. Livio gives as one example the famous golden ratio, which has fascinated Western mathematicians for millennia and was originally emphasized for its mystical symbolism. But Chinese mathematicians, not sharing that outlook, didn't discover it—or maybe they just didn't need to invent it. Livio hedges his bets, unsatisfyingly arguing that mathematics is partly discovered and partly invented. But Livio is a smooth writer. His fans will enjoy this book, and new ones may discover him. B&w illus. (Jan. 6)
Copyright © Reed Business Information, a division of Reed Elsevier Inc. All rights reserved. --This text refers to an alternate kindle_edition edition.
From Booklist
Starred Review Four centuries after church inquisitors accused Galileo of dangerous skepticism, a modern astrophysicist hails the Italian scientist as the embodiment of bold faith: namely, faith that God himself inscribed the heavens in mathematics. Because mathematics now empowers research communities investigating everything from deep-space pulsars to genetic proteins, a secularized version of Galileo’s credo now defines the new orthodoxy of science. But Livio recognizes a profound mystery inherent in the formulas his colleagues employ so sedulously: Why does the universe harmonize so well with numbers accessible to human minds? Probing this mystery, Livio traces the evolution of mathematical reasoning from the ritual symbolism of the ancient Pythagoreans to the multilayered analyses of twenty-first-century string theorists. In the impressive parade of intellectual explorers, we encounter Archimedes pondering geometrical figures at the very moment of his death, Descartes overthrowing all of medieval philosophy with one audacious thought, and Gödel quashing the ambitions of system-building logicians. This wide-ranging inquiry, however, ultimately highlights far more than personalities. A sharp conflict emerges between platonically minded philosophers who view mathematical breakthroughs as transcendent discoveries and psychologically inclined thinkers who interpret these breakthroughs as merely human inventions. Testing the tensions between these views, Livio delivers an exhilarating foray into the founding premises of mathematical science. --Bryce Christensen --This text refers to an alternate kindle_edition edition.
Review
"Is God a mathematician? In his new book Mario Livio delves into this question, putting it into a scientific, historical and philosophical context. He steers skillfully through deep and tricky waters, but writes with clarity and ease...Read the book and decide for yourself what the answer is."-- Sir Michael Atiyah, recipient of the Fields Medal, 1966, and the Abel Prize, 2004
"This highly readable book explores one of the most fascinating questions that lies at the heart of fundamental physics -- why is mathematics so effective in describing nature and is mathematics an invention of the human mind or part of the fabric of physical reality? Livio provides a wonderful review of the various issues, presents a wide variety of opinions, and in addition some fascinating insights of his own. I strongly recommend this volume to anyone interested in these questions."-- David Gross, 2004 Nobel Prize Winner in Physics, Frederick W. Gluck Professor of Theoretical Physics and Director, Kavli Institute For Theoretical Physics, University of California, Santa Barbara
"All science proceeds from the assumption that the cosmos is ordered in an intelligible way. Beneath the bewildering richness of natural phenomena there lies an elegant mathematical unity. How astonishing that the human mind is attuned to this hidden subtext of nature! With elegance and clarity, Mario Livio charts how, through science and mathematics, we have come to glimpse the fundamental rules on which the universe runs."-- Paul Davies, author of The Goldilocks Enigma and Director of the Beyond Center for Fundamental Concepts in Science, Arizona State University
"An exhilarating foray into the founding premises of mathematical science."-- Booklist
"Theologians have God, philosophers existence, and scientists mathematics. Mario Livio makes the case for how these three ideas might be related...Livio's rich history gives the discussions human force and verve."-- Sam Kean, New Scientist --This text refers to an alternate kindle_edition edition.
Excerpt. © Reprinted by permission. All rights reserved.
CHAPTER 1
A MYSTERY
A few years ago, I was giving a talk at Cornell University. One of my PowerPoint slides read: "Is God a mathematician?" As soon as that slide appeared, I heard a student in the front row gasp: "Oh God, I hope not!"
My rhetorical question was neither a philosophical attempt to define God for my audience nor a shrewd scheme to intimidate the math phobics. Rather, I was simply presenting a mystery with which some of the most original minds have struggled for centuries -- the apparent omnipresence and omnipotent powers of mathematics. These are the type of characteristics one normally associates only with a deity. As the British physicist James Jeans (1877-1946) once put it: "The universe appears to have been designed by a pure mathematician." Mathematics appears to be almost too effective in describing and explaining not only the cosmos at large, but even some of the most chaotic of human enterprises.
Whether physicists are attempting to formulate theories of the universe, stock market analysts are scratching their heads to predict the next market crash, neurobiologists are constructing models of brain function, or military intelligence statisticians are trying to optimize resource allocation, they are all using mathematics. Furthermore, even though they may be applying formalisms developed in different branches of mathematics, they are still referring to the same global, coherent mathematics. What is it that gives mathematics such incredible powers? Or, as Einstein once wondered: "How is it possible that mathematics, a product of human thought that is independent of experience [the emphasis is mine], fits so excellently the objects of physical reality?"
This sense of utter bewilderment is not new. Some of the philosophers in ancient Greece, Pythagoras and Plato in particular, were already in awe of the apparent ability of mathematics to shape and guide the universe, while existing, as it seemed, above the powers of humans to alter, direct, or influence it. The English political philosopher Thomas Hobbes (1588-1679) could not hide his admiration either. In Leviathan, Hobbes's impressive exposition of what he regarded as the foundation of society and government, he singled out geometry as the paradigm of rational argument:
Seeing then that truth consisteth in the right ordering of names in our affirmations, a man that seeketh precise truth had need to remember what every name he uses stands for, and to place it accordingly; or else he will find himself entangled in words, as a bird in lime twigs; the more he struggles, the more belimed. And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind), men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning.
Millennia of impressive mathematical research and erudite philosophical speculation have done relatively little to shed light on the enigma of the power of mathematics. If anything, the mystery has in some sense even deepened. Renowned Oxford mathematical physicist Roger Penrose, for instance, now perceives not just a single, but a triple mystery. Penrose identifies three different "worlds": the world of our conscious perceptions, the physical world, and the Platonic world of mathematical forms. The first world is the home of all of our mental images -- how we perceive the faces of our children, how we enjoy a breathtaking sunset, or how we react to the horrifying images of war. This is also the world that contains love, jealousy, and prejudices, as well as our perception of music, of the smells of food, and of fear. The second world is the one we normally refer to as physical reality. Real flowers, aspirin tablets, white clouds, and jet airplanes reside in this world, as do galaxies, planets, atoms, baboon hearts, and human brains. The Platonic world of mathematical forms, which to Penrose has an actual reality comparable to that of the physical and the mental worlds, is the motherland of mathematics. This is where you will find the natural numbers 1, 2, 3, 4,..., all the shapes and theorems of Euclidean geometry, Newton's laws of motion, string theory, catastrophe theory, and mathematical models of stock market behavior. And now, Penrose observes, come the three mysteries. First, the world of physical reality seems to obey laws that actually reside in the world of mathematical forms. This was the puzzle that left Einstein perplexed. Physics Nobel laureate Eugene Wigner (1902-95) was equally dumbfounded:
The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
Second, the perceiving minds themselves -- the dwelling of our conscious perceptions -- somehow managed to emerge from the physical world. How was mind literally born out of matter? Would we ever be able to formulate a theory of the workings of consciousness that would be as coherent and as convincing as, say, our current theory of electromagnetism? Finally, the circle is mysteriously closed. Those perceiving minds were miraculously able to gain access to the mathematical world by discovering or creating and articulating a treasury of abstract mathematical forms and concepts.
Penrose does not offer an explanation for any of the three mysteries. Rather, he laconically concludes: "No doubt there are not really three worlds but one, the true nature of which we do not even glimpse at present." This is a much more humble admission than the response of the schoolmaster in the play Forty Years On (written by the English author Alan Bennett) to a somewhat similar question:
Foster: I'm still a bit hazy about the Trinity, sir.
Schoolmaster: Three in one, one in three, perfectly straightforward. Any doubts about that see your maths master.
The puzzle is even more entangled than I have just indicated. There are actually two sides to the success of mathematics in explaining the world around us (a success that Wigner dubbed "the unreasonable effectiveness of mathematics"), one more astonishing than the other. First, there is an aspect one might call "active." When physicists wander through nature's labyrinth, they light their way by mathematics -- the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. Newton observed a falling apple, the Moon, and tides on the beaches (I'm not even sure if he ever saw those!), not mathematical equations. Yet he was somehow able to extract from all of these natural phenomena, clear, concise, and unbelievably accurate mathematical laws of nature. Similarly, when the Scottish physicist James Clerk Maxwell (1831-79) extended the framework of classical physics to include all the electric and magnetic phenomena that were known in the 1860s, he did so by means of just four mathematical equations. Think about this for a moment. The explanation of a collection of experimental results in electromagnetism and light, which had previously taken volumes to describe, was reduced to four succinct equations. Einstein's general relativity is even more astounding -- it is a perfect example of an extraordinarily precise, self-consistent mathematical theory of something as fundamental as the structure of space and time.
But there is also a "passive" side to the mysterious effectiveness of mathematics, and it is so surprising that the "active" aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons -- with absolutely no application in mind -- turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality! How is that possible? Take for instance the somewhat amusing case of the eccentric British mathematician Godfrey Harold Hardy (1877-1947). Hardy was so proud of the fact that his work consisted of nothing but pure mathematics that he emphatically declared: "No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." Guess what -- he was wrong. One of his works was reincarnated as the Hardy-Weinberg law (named after Hardy and the German physician Wilhelm Weinberg [1862-1937]), a fundamental principle used by geneticists to study the evolution of populations. Put simply, the Hardy-Weinberg law states that if a large population is mating totally at random (and migration, mutation, and selection do not occur), then the genetic constitution remains constant from one generation to the next. Even Hardy's seemingly abstract work on number theory -- the study of the properties of the natural numbers -- found unexpected applications. In 1973, the British mathematician Clifford Cocks used the theory of numbers to create a breakthrough in cryptography -- the development of codes. Cocks's discovery made another statement by Hardy obsolete. In his famous book A Mathematician's Apology, published in 1940, Hardy pronounced: "No one has yet discovered any war-like purpose to be served by the theory of numbers." Clearly, Hardy was yet again in error. Codes have been absolutely essential for military communications. So even Hardy, one of the most vocal critics of applied mathematics, was "dragged" (probably kicking and screaming, if he had been alive) into producing useful mathematical theories.
But this is only the tip of the iceberg. Kepler and Newton discovered that the planets in our solar system follow orbits in the shape of ellipses -- the very curves studied by the Greek mathematician Menaechmus (fl. ca. 350 BC) two millennia earlier. The...
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From The Washington Post
From The Washington Post's Book World/washingtonpost.com Reviewed by Marc Kaufman Did you know that 365 -- the number of days in a year -- is equal to 10 times 10, plus 11 times 11, plus 12 times 12? Or that the sum of any successive odd numbers always equals a square number -- as in 1 + 3 = 4 (2 squared), while 1 + 3 + 5 = 9 (3 squared), and 1 + 3 + 5 + 7 = 16 (4 squared)? Those are just the start of a remarkable number of magical patterns, coincidences and constants in mathematics. No wonder philosophers and mathematicians have been arguing for centuries over whether math is a system that humans invented or a cosmic -- possibly divine -- order that we simply discovered. That's the fundamental question Mario Livio probes in his engrossing book Is God a Mathematician? Livio, an astrophysicist at the Hubble Space Telescope Science Institute in Baltimore, explains the invention-vs.-discovery debate largely through the work and personalities of great figures in math history, from Pythagoras and Plato to Isaac Newton, Bertrand Russell and Albert Einstein. At times, Livio's theorems, proofs and conundrums may be challenging for readers who struggled through algebra, but he makes most of this material not only comprehensible but downright intriguing. Often, he gives a relatively complex explanation of a mathematical problem or insight, then follows it with a "simply put" distillation. An extended section on knot theory is, well, pretty knotty. But it ultimately sheds light on the workings of the DNA double helix, and Livio illustrates the theory with a concrete example: Two teams taking different approaches to the notoriously difficult problem of how many knots could be formed with a specific number of crossings -- in this case, 16 or fewer -- came up with the same answer: 1,701,936. The author's enthusiasm is infectious. But it also leads him to refer again and again to his subjects as "famous" and "great" and to their work as "monumental" and "miraculous." He has a weakness as well for extended quotes from these men (and they are all men) that slow the narrative without adding much. There are exceptions, including the tale of how Albert Einstein and mathematician Oskar Morgenstern tried to guide Kurt Gödel, a fellow mathematician and exile from Nazi Germany, through the U.S. immigration process. A deep-thinking and intense man, Gödel threw himself into preparing for his citizenship test, including an extremely close reading of the U.S. Constitution. In his rigorously logical analysis, he found constitutional weaknesses that he thought could allow for the rise of a fascist dictatorship in America. His colleagues told him to keep that reading to himself, but he blurted it out during his naturalization exam. He was allowed to stay anyway. The interplay of mini-biography and the march of mathematical knowledge serves the author well. It does not, however, ultimately help him to answer the big question, Is God a mathematician? On one side of the debate are all those remarkable constants that crop up, the makings of the ideal yet hidden world posited by Plato. In addition, there's what the physicist Eugene Wigner, in a seminal 1960 essay, called the "unreasonable effectiveness" of mathematical theorems: the astounding ability of math to predict unimagined results. Wigner was picking up on ideas explored earlier by Einstein, and Einstein's general theory of relativity remains one of the best examples: His predictions about how gravity can cause ripples in space-time was recently corroborated by measuring radio waves from a distant set of compact, high-energy stars called double pulsars, using technology unknown in Einstein's day. Doesn't all this indicate that the mathematical structure of the world is out there waiting to be discovered? On the other hand, math cannot explain many situations, and chaos theory suggests that it may never be possible to predict the weather or the stock market with accuracy. Recent research has pointed to basic mathematical constructs in the human brain, suggesting that we impose numbers and forms on the world, not vice versa. In addition, mathematics is less stable than it appears to us in grade school. At the higher reaches of the field, there is constant ferment and debate. If the "truths" discovered through mathematics are always changing, doesn't that indicate they are a product of human study and manipulation, rather than something fixed and eternal? As explained by Livio, the history of mathematics is partly a struggle between these points of view: that math is how God (or nature) organizes the world, or it is simply a human tool to understand that world. Livio comes down in the middle, contending that math may well be both invented and discovered. He points, for instance, to the eternal truth contained in the geometry formulated by Euclid 2,400 years ago. By the 19th century, however, iconoclasts had posited and established a whole new world of non-Euclidian geometry. Livio writes about the symmetries of the universe: the immutable, if incompletely understood, laws of math and physics that make a hydrogen atom, for instance, behave in the same way on Earth as it acts 10 billion light years away. Another sign of universal structure, as teased apart with the help of math? No, Livio writes, it is more likely a sign that "to some extent, scientists have selected what problems to work on based on those problems being amenable to a mathematical treatment." The author acknowledges that some readers will find his inconclusive conclusion to be unsatisfying. I didn't. Sometimes the adventure, the intellectual ride, is more important than the final destination.
Copyright 2009, The Washington Post. All Rights Reserved. --This text refers to an alternate kindle_edition edition.