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Infinite Powers: The Story of Calculus - The Language of the Universe Paperback – February 6, 2020
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A magisterial history of calculus (and the people behind it) from one of the world's foremost mathematicians.
This is the captivating story of mathematics' greatest ever idea: calculus. Without it, there would be no computers, no microwave ovens, no GPS, and no space travel. But before it gave modern man almost infinite powers, calculus was behind centuries of controversy, competition, and even death.
Taking us on a thrilling journey through three millennia, professor Steven Strogatz charts the development of this seminal achievement from the days of Archimedes to today's breakthroughs in chaos theory and artificial intelligence. Filled with idiosyncratic characters from Pythagoras to Fourier, Infinite Powers is a compelling human drama that reveals the legacy of calculus on nearly every aspect of modern civilisation, including science, politics, medicine, philosophy, and much besides.
From the Publisher
- Publisher : Atlantic Books; Main edition (February 6, 2020)
- Language : English
- Paperback : 384 pages
- ISBN-10 : 1786492970
- ISBN-13 : 978-1786492975
- Item Weight : 11.6 ounces
- Dimensions : 5.08 x 1.06 x 7.8 inches
- Best Sellers Rank: #699,306 in Books (See Top 100 in Books)
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Examples few and limited to 6th grade level. Repeats notions but never builds, as chapters drag on.
Insufficient figures, and those offered neither titled nor numbered!
Intuitive address of "limits", but fails to translate into application.
Impression: little editing before publication.
"Made a good throw but never snared the critter's feet ... got away."
Most of us believe that Isaac Newton and Gottfried Leibniz invented Calculus. In this book, Dr. Strogatz traces the ideas of infinitesimal calculus from the days of Archimedes. We then meet Galileo, Kepler, Descartes and Fermat through to Newton and Leibniz. Along the way, we find the contributions of Arabic and Indian mathematicians. This book explains the beauty of calculus to any interested person, even if one is not well-versed in high-school mathematics. We follow the history of calculus. We find its key ideas with engrossing anecdotes of the scientists and mathematicians who developed the field.
What is calculus? Calculus is a branch of Mathematics which solves problems in three steps. It splits a complicated but continuous problem into an infinite number of simpler pieces. Then it solves each of the simple piece separately. Finally it puts them together as the solution. For instance, we can visualize motion at changing speed as composed of infinitely many, infinitesimally brief motions at constant speed. Similarly, one could pretend that curves comprise an infinite number of infinitesimally short straight lines. We call the splitting ‘differentiation’ and the putting together ‘integration’. Together, it becomes the Infinitesimal calculus. The author explains these ideas beautifully with many examples.
The first chapters in the book lay the foundation by showing us how ancient Greece approached the computation of Pi. Archimedes calculated the perimeter of a many-sided polygon inscribed in a circle. The polygon approached the circle as the number of sides approached infinity and the length of those sides approached zero. He also used the same approach in representing a smooth segment of a parabola. He did it through a mosaic of an infinite number of triangular shards. From here, we move on to see how this principle plays a big role in our life experience. Modern-day animators at DreamWorks used tens of thousands of polygons in creating Shrek’s round belly and trumpet-like ears. In the film ‘Avatar’, animators used millions of polygons to render each plant in the imaginary world of Pandora. Scientists have used the same principle in doing facial surgery for patients with misaligned jaws and other congenital malformations by creating Virtual surgery simulators. In modern-day CT Scanners, x-rays pass through tissue, bone and organs producing different levels of absorption. To compute the reduction of intensity in this process, the CT software calculates step by infinitesimal step as the x-rays travel through the tissue, and then combines all the results as the integral.
Calculus provides the language for describing waves and the tools for analyzing them. The vibration of strings produces music. It led to the discovery of the wave equation. James Clerk Maxwell used it to predict the existence of electromagnetic waves. It is calculus which enabled the invention of vacuum tubes, transistors, computers, radar, microwave ovens, drugs for HIV and the design of Boeing 787 wings.
The book contains many philosophical musings of the author which are important to understand the illusions which calculus creates to resolve reality. Calculus models Reality through an infinite number of divisions. If we look at any phenomenon in excruciatingly fine detail in time or space, we will see the breakdown of smoothness on which calculus depends on. The book illustrates this in explaining Usain Bolt’s record-breaking 100-meter dash in the 2009 World Championships in Berlin. Conversely calculus also takes its creative license by treating discrete objects as if they are continuous. The modeling is approximate but useful. For example, the DNA is a discrete collection of atoms and not continuous. Molecular Genetics treats it as a continuous curve, like a perfect elastic band. This enables Genetics to apply two spin offs of calculus - elasticity theory and differential geometry - to calculate how DNA deforms when subjected to the forces from proteins, from the environment and from interactions within itself.
The author is at his best in the final two chapters, namely ‘The Future of Calculus and Conclusion’. He begins with a defence of what he believes the essence of calculus is. The 19th century saw infinity, and the infinitesimal expunged from calculus. Scientists clarified what limits, derivatives, integrals and real numbers meant. They believed that they arrested the rampage of infinitesimal calculus. Dr. Strogatz differs. He is firm in his belief that the credo of calculus is the Infinity Principle. It is this principle that allows us to study any pattern, curve, motion, natural process, system or phenomenon that changes smoothly and continuously. He explains the role of calculus in linear, non-linear and chaotic systems using everyday experiences with amazing simplicity. Biology and sociology are nonlinear. When a system is nonlinear, its behavior may be impossible to forecast with formulas, even though that behavior is deterministic. Determinism does not imply predictability.
In this chapter, he further discusses Complex systems and higher Dimensions. He says that natural selection has tuned our nervous systems to perceive directions as up, down, front, back, left and right of ordinary space. We cannot visualize the fourth dimension, try as we might. We can only comprehend it through mathematical abstractions. This is a huge hurdle in Complex systems where we meet higher dimensions. Hence, the author says that it will be hard, if not impossible, to make progress on the most difficult problems of our time. They are in the behavior of economies, societies, cells, workings of the immune system, genes, brains and consciousness. Reading this, I felt that the Earth’s climate is a highly complicated, nonlinear system. Hence, humility and caution is necessary in making predictions for a hundred years.
Dr. Strogatz concludes the book by contemplating on what he calls the greatest mystery of all. Why is the Universe comprehensible and why is calculus in sync with it? He provides three eerie examples of the effectiveness of calculus. The differential equations and integrals of Quantum Electrodynamics (QED) is one. They help predict the properties of electrons and other particles to a precision of eight decimal places. Such precision is comparable to planning to snap one’s finger 3.17 years from now and getting it right to the nearest second without a clock or an alarm. The second example takes us back to 1928. Then, Paul Dirac wrote a differential equation for the electron which implied something new, true and beautiful about nature. Its logic and beauty demanded the existence of a new particle, the Positron. In 1932, the experimental physicist, Carl Anderson, found it, albeit unaware of Dirac’s prediction. The last example takes us even further back to 1915, to the differential equations of General Relativity. These equations predicted the expansion of the Universe, existence of black holes, gravitational waves, gravity’s effect on time, how matter curves space-time and how the curvature tells matter to move. It is a celestial dance of calculus and nature, in the author’s words.
This marvellous book leaves us in no doubt that Calculus is the language that God speaks. Everyone interested in the beauty of Mathematics, and Nature must read this book.
Luckily it wasn’t a termite, so the book survived fine!
I hadn’t realized that calculus had such a long history
all the way back to Archimedes, wow.
Steven writes well in an interesting style.
Not too much math, so it’s suitable for most people.
Top reviews from other countries
Archimides proof of the area of the circle is the highlight.
Just after the Newton - Liebnitz schism. The book changes form, abandons diagrams and equations and turns into a general history of maths albeit an interesting one.
So for people who want to go a bit further than BBC documentary level maths, I would recommend Ian Stewarts very excellent 17 Equations that Changed the World.
A word of warning for Kindle users - be careful which font you chose, otherwise some of the mathematical symbols can disappear! I found empty space where 1/3 and pi (the symbol) should have been, until I experimented with the fonts, and up they popped.
However, I couldn’t resist the Kindle edition when it was advertised briefly in June at a special offer price. I’m very glad I bought it!
The book is a celebration of the power of calculus and its centrality to so much of the modern world. Strogatz has a lively and engaging writing style, and deploys some nice turns of phrase, such as ‘calculus was the Cambrian explosion for mathematics’. He covers a remarkably wide range of topics.
‘Infinite Powers’ contains lots of fascinating insights into the development of calculus (e.g. Archimedes’ fundamental contributions to its pre-history) that I’d never come across before. There are stories – new to me -- about such masters as Galileo, Kepler, Fermat, Newton (whom we can now ‘watch at play’ thanks to his college notebooks being on the Web) and Leibniz.
The end notes contain many further gems, with references that I enjoyed following up. For example I learnt for the first time about Kovalevskaya’s curious integrable top, and encountered some new (to me) ideas concerning John Couch Adams and Britain’s non-discovery of Neptune.
I wish this book had been available when I first met calculus some 57 years ago!
Reading Infinite Powers is my small attempt at taking an interest in something I dismissed in my youth. It wasn't easy to get through, but I enjoyed it far more than I thought I would. Now the only downside is that Amazon keeps recommending books about maths and physics to me... one was enough, thank you.
One can go through life without understanding eg. the periodic table, basic biology, chemistry, physics, astronomy etc but nothing beats a fundamental toolbox to really appreciate and understand how things work. This book adds to the puzzle, for sure, math is amazing, precise, delightful - and spans across all these science fields. Exciting.
All is described in a rather accessible way, for most people. It takes a bit of stamina to pull through it, but hang in there, it is worth it.
You will emerge wiser and look at things differently.
I think I will go back at some time and re-read part of it again.
The author is a great story teller and explains complicated matters in a accessible way.
I will read more from the same author.