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.

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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 national bestseller *Brilliant Blunders* and other books. He lives in Baltimore, Maryland.

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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**Book Description **SIMON SCHUSTER, United States, 2010. Paperback. Book Condition: New. ed. Language: English . Brand New Book. 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. Bookseller Inventory # AA89780743294065

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**Book Description **SIMON SCHUSTER, United States, 2010. Paperback. Book Condition: New. ed. Language: English . Brand New Book. 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. Bookseller Inventory # AA89780743294065

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