Infinite Powers: How Calculus Reveals the Secrets of the Universe

Prof. R. Feynman, after an interview with a novelist who was doing research for a book about WW. II, as they were departing, asked him if he knew calculus. No, was the reply, he didn't. ''You had better learn it'' said Feynman. ''It's the language God talks''.Mr. Strogatz starts the Introduction part of his wonderful book by telling this anecdote. He continues, ''For reasons no body understands, the universe is deeply mathematical. May be God made it that way. Or maybe it's the only way a universe with us in it could be, because non-mathematical universes can't harbor life intelligent enough to ask the question. In any case, it's a mysterious and marvelous fact that our universe obeys laws of nature that always turn out to be expressible in the language of calculus as sentences called differential equations.''''Such equations describe the difference between something right now and the same thing an instant later. The details differ depending on what part of nature we are talking about, but the structure of the laws is always the same... There seems to be something like a code to the universe, an operating system... Calculus taps into this order and express it.''''Isaac Newton was the first to glimpse this secret of the universe... If anything deserves to be called the secret of the universe, calculus is it.''Mr. Strogatz tells the story behind how humanity discovered this strange language, how they learned to speak it fluently and finally harnessing its forecasting powers, how they used it to remake the world.He has written this book 'in an attempt to make the greatest ideas and stories of calculus accessible to everyone'. I can say that he has greatly achieved this.First, he shows that calculus is one of humankind's most inspiring collective achievements, roots going back to Archimedes, even Zeno, to the concept of infinity. He tells the development of ideas in a comfortable, casual way, demanding only average mathematical knowledge. He gives examples of applications from our time, which is familiar to most of us. He has furnished his story with very informative drawings.What is very important is that, he tries to navigate the reader within the story of development of calculus in the historical, natural order of the development, which makes it much easier for the reader to grasp the ideas. He has added some wise, sense of humor here and there which makes the reading easy and fun. The rich bibliography at the end is very useful. I was able to meet another wonderful book 'The Calculus Gallery' of William Dunham from within that bibliography.I loved the book. No wonder it was a bestseller. I highly recommend this book to everyone who is or was scared of calculus and who wants to approach to understanding 'the language of God.'

In this book, the author intends to explain everything with the help of pictures, metaphors, and anecdotes. We also get exposed to some of the finest equations and proofs in human history. One of the first things introduced is the “infinity principle,” where things are broken down into infinite simpler parts, analyzed, and then added back together to produce the whole. Calculus can be thought of as a methodological theme consisting of a mystery of curves, the mystery of motion, and the mystery of change.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.