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Infinite Powers: How Calculus Reveals the Secrets of the Universe Hardcover – April 2, 2019
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“Marvelous . . . an array of witty and astonishing stories . . . to illuminate how calculus has helped bring into being our contemporary world.”—The Washington Post
From preeminent math personality and author of The Joy of x, a brilliant and endlessly appealing explanation of calculus – how it works and why it makes our lives immeasurably better.
Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket.
Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down-to-earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real-world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous.
Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greece and bringing us right up to the discovery of gravitational waves (a phenomenon predicted by calculus). Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes “backwards” sometimes; how to make electricity with magnets; how to ensure your rocket doesn’t miss the moon; how to turn the tide in the fight against AIDS.
As Strogatz proves, calculus is truly the language of the universe. By unveiling the principles of that language, Infinite Powers makes us marvel at the world anew.
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A New York Times Bestseller Featured on NPR's Science Friday Shortlisted for the 2019 Royal Society Science Book Prize Named one of 10 Best Books to Read this Spring by Amazon's Chris Schluep “Marvelous . . . an array of witty and astonishing stories . . . to illuminate how calculus has helped bring into being our contemporary world and so many of the instruments whose role we now blithely assume.” —The Washington Post “Fortunately, we live in an era when a top mathematician can write a book about calculus that is accessible to the mathematically challenged...it is the historical detail in the book that not only allows me to follow the math by taking me through how it was discovered, but also sticks in my mind. [Strogatz] makes me want to get out a textbook and start studying calculus.” —Five Books “Wonderful . . . bringing the insights of calculus—among the most important of all developments in the history of mathematics—to everyone.” —Brian Greene, on Twitter “I've never read a clearer explanation of calculus or the significant powers we gained by harnessing infinity. Michael and I highly recommend it!” —Vsauce, on Twitter “Fascinating anecdotes abound in Infinite Powers . . . Strogatz uses the right amount of technical detail to convey complex concepts with clarity . . . evocatively conveys how calculus illuminates the patterns of the Universe, large and small.” —Nature “A brilliant, appealing explanation of how calculus works and why it makes our lives so much better.” —Amazon's Chris Schluep, for the Saturday Evening Post “Strogatz does a great job of explaining a difficult subject . . . he lays out the case that calculus is fundamental to the way we live today . . . a solid choice for readers who want to know what calculus is all about, and for teachers who wish to improve their presentation.” —Library Journal “An energetic effort that successfully communicates the author’s love of mathematics.” —Kirkus Reviews “Far-ranging survey . . . clear and accessible . . . Strogatz successfully illuminates a notoriously complex topic and this work should enhance appreciation for the history behind its innovations." —Publishers Weekly “A tale of how mathematics has changed all of our lives… Strogatz is a terrific storyteller and patient teacher…. By bringing infinity down to earth… and coupling those stories with some periodic excursions back out to the stars, Infinite Powers does a marvelous job of bringing calculus to life.” —Dan Rockmore, LitHub “Are you one of those people who always said you’d someday learn calculus? Well, someday is here, thanks to Steven Strogatz’s wide-ranging, humane, thoroughly readable take on one of the greatest ideas our species has ever produced.” —Jordan Ellenberg, author of How Not to Be Wrong “This is a glorious book. Steven Strogatz manages to unmask the true hidden wonder and delightful simplicity of calculus. Infinite Powers is a master class in accessible math writing and a perfect read for anyone who feels like they never quite understood what all the fuss was about. It had me leaping for joy.”  —
About the Author
- Publisher : Mariner Books; Later prt. edition (April 2, 2019)
- Language : English
- Hardcover : 384 pages
- ISBN-10 : 1328879984
- ISBN-13 : 978-1328879981
- Item Weight : 1.28 pounds
- Dimensions : 9 x 6 x 9 inches
- Best Sellers Rank: #393,711 in Books (See Top 100 in Books)
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But as the author notes in the beginning, “For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that way. Or maybe it’s the only way a universe with us in it could be, because nonmathematical universes can’t harbor life intelligent enough to ask the question.” How often do you hear a Professor of Applied Mathematics, at an Ivy League school no less, say something even remotely so self-reflective?
Steven Strogatz is a great communicator who is both a great mathematician and who, it is easy to tell, gets goose bumps every time he thinks about the wonders of calculus. I am not a professional mathematician but have always found mathematics to be both fascinating and, well, not easy, but very relatable. It’s predictable, and that’s comforting once you can see the pattern.
If you don’t feel quite that way you may – spoiler alert – find this book to be a bit more like a textbook than advertised. There are plenty of equations and symbols and the like. That is, after all, the alphabet of calculus. But here’s the thing. Unless you are also a math professor, you can ignore all of that. Just go with the prose. It tells the same story, but in a far more relatable form to the average lover of the written word. Just ignore the symbols. If you do you will miss nothing and you will find the professor’s enthusiasm to be quite contagious.
The beauty of the book is that it is written from a perspective of humility. Both in terms of the enormity of calculus (Most people will relate to the subject matter simply as science.), and in terms of how far we have yet to go in terms of truly understanding the universe and the reality that defines it. We’ve only explained the tip of the iceberg.
Math is a human convention. It’s not hydrogen or oxygen. It’s not even dark matter, which we “know” makes up most of the universe but which no one has ever isolated, although the Chinese are close. It is very accurate at deciphering reality if getting close to the “real” explanation is close enough. But close is only close. It isn’t reality itself. Reality is, after all, by definition, real.
That is, ultimately, the problem with the promise of AI. Because AI is ultimately dependent on calculus and other disciplines of mathematics, it will get very smart, but it will never be human. What it will do, however, if we let it, is dumb down what it means to be smart to a standard perfectly suited to its abilities but ignorant of its shortcomings.
That’s why, despite the promises of the silicon gods, we are very unlikely to see fully autonomous vehicles for decades to come. The only way that could happen is if we take all human drivers off the road overnight (The AI isn’t the problem; it’s us. We are unpredictable.), switch every vehicle to an autonomous vehicle all at the same time, and rebuild our infrastructure to accommodate the vulnerabilities of the various disciplines of mathematics on which the technology is based. And that’s obviously not going to happen. Nor do we want it to.
Pi, as but one example, despite what you were taught in school, is not a number; it’s a range. It’s a small range, to be sure, but it’s a range nonetheless. In other words, it is precise enough for most things, but it is NOT the fabric of the universe.
Science is a methodology for understanding reality; it is not, in the most literal sense, reality itself. Reality is not “waiting” to be discovered. It is. And just as an artist can draw a landscape, science can draw reality. Neither, however, IS reality.
The history of calculus is truly fascinating. And that, to me, as a reader, makes it entertaining. Newton and Galileo and all the rest were truly amazing people. It boggles the mind to think of what they concluded when they did.
Perhaps the book’s greatest contribution, however, is that it will put Silicon Valley in perspective. You may think your smart phone has changed your life in ways that nothing else possibly could. You’re wrong. I am a great admirer of Steve Jobs but James Clerk Maxwell (a Scot in the 1860s) changed your life in ways that Steve did not come close to.
And that is why this book is so timely. Calculus is changing our world, and not entirely in good ways. If ever we needed perspective we need it now. Math is elegant. It was designed that way. (Remember that it is not of the universe, like rain or sunshine.) And it does have an uncanny reliability that helps us to understand the world around us.
Take GPS. We all use it. We all rely on it. But did you know that GPS is all about time, not navigation. Those GPS satellites don’t “see” you; they time you. It only works because scientists came to understand the mathematics of what we call time at a very precise level. That’s not reality, of course, because time is a concept (time, even as we understand it, varies with altitude), but it is close enough to give us GPS. And isn’t that an amazing thing.
I think so. And that’s why I found this to be such an enjoyable book, beyond the fact that I am simply stimulated by really enthusiastic people and Professor Strogatz is one of the most enthusiastic people I have had the privilege to read in a really long time.
If, on the other hand, you prefer a good murder mystery, or something with a little romance, at least, you won’t find it in this book. But that’s just my opinion. A little like pi, if you will. Pretty accurate, but not reality itself.
Decide for yourself. You won’t be wasting your time.
Western authors have a penchant for writing sloppy history. The history of math in most western books jumps from Euclid, Archimedes, Pythagoras (500-200 BCE) straight to Descartes, Euler, Newton and Leibniz (1500-1900 CE) as if nobody else was doing math on the planet between that period or before that period. I think they simply have a blind spot (or purposeful ignorance) for anything outside Greece (and by extension Europe). For example, even going by the current dating for the Vedas given by western historians to between c. 1500 BCE and c. 800 BCE, which I think may be incorrect, those recitations have infinite progressions in them as well. Pythagoras' theorem was known to many civilizations around the world yet it is credited to Pythagoras. A lot of "historians" love to stick to the biblical timeline and retrofit chronology for other civilizations to that one conveniently ignoring anything that refutes the narrative. Western history of science is baloney. If you want the real history of mathematics, ask people from other civilizations.
We start with the work of Archimedes from about the 3rd century BCE. We see here the beginnings of integral calculus, where triangles and parabolic regions are apparently and mysteriously equivalent. Eighteen hundred years passed until a new Archimedes appeared, whom we know as Galileo Galilei. It was interesting to learn about the law of odd numbers rule, which led Galileo to conclude that the total distance fallen is proportional to the square of the time elapsed. What Galileo did for the motion of objects, Johannes Kepler did for the motion of the planets. Both channeled the spirit of Archimedes, “carving solid objects in their minds into imaginary thin wafers, like so many slices of salami.”
We see the arrival of algebra in Europe around 1200 from Asia and the Middle East. Hindu mathematicians invented the concept of zero and the decimal place system. Algebraic techniques for solving equations came from Egypt, Iraq, Persia, and China. But the study of algebra as a symbolic system began to flourish in Renaissance Europe around the 1500s. Analytic geometry makes its appearance with Pierre de Fermat, and Rene Descartes. Fermat actually invented the ideas that led to the concept of derivatives.
From here we delve into functions – power and exponential, for example. There are some interesting basics of the relationship of logarithms to exponents. And then there is the natural logarithm, which grows as a rate precisely equal to the function itself. The author notes that “exponential functions expressed in base e are always the cleanest, most elegant, and most beautiful.” This leads into a more detailed discussion of the derivative. By the time we get to Newton, we see the concept of the fundamental theorem. Newton’s brainstorm was to invite time and motion into the picture and let the area flow. And now we are into integral calculus. The author notes that the reason integration is so much harder than differentiation relates to the distinction between local and global, which he clearly demonstrates in the book. I think the author has done a great job of showing us just how these concepts arose and how to make sense of them. You won’t get this is in your typical calculus book.
After this, we delve into differential equations – ordinary and partial. The author gives a clear explanation of what these beasts are and some real-world examples to help us understand. In talking about the future of calculus, the author discusses some applications, such as nonlinearity (biology, sociology) and chaos, where you have an inherent sensitivity to initial conditions. He concludes by taking us to the “Twilight Zone” for three examples of the eerie effectiveness of calculus.
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"To shed light on any continuous shape, object, motion, process, or phenomenon - no matter how wild and complicated it may appear - reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole."
Due to my consistent successes in learning varied topics, I never made much sense to me that math should be looked at as this special field that requires unique skills to understand and be proficient in. Like any other field, it is honed through lots of practice, practical applications, and attention. Steven's book really inspired in me that that hunch was more than just a hunch, and a year later I have gone well beyond my stunted grade school math education and can see myself learning math for the pure love of it for the rest of my life. Saying nothing of the fact that I know it will improve my hard skills across every domain I work in today and into the future.
Steven really has a gift not only for doing math itself, but for expressing exactly how and why learning math is, like any skill or topic worth devoting effort to, one of the deepest and most beautiful struggles. And, regardless of our learning stage or status, that struggle and its rewards belongs to all of us equally.
Te deja claro su historia, de donde vienen los conceptos, su utilidad, todo sin usar muchas fórmulas.
Este libro no es para aprobar un examen sino para entender de verdad de que va ésta útil herramienta de las matemáticas.
Lo recomiendo para cualquiera que quiera comprender de verdad sus bases y no solo memorizar una serie de fórmulas y gráficas, como pasó en mi curso.