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Steven Strogatz

Infinite Powers

The Story of Calculus
The Language of the Universe

Book review by Anthony Campbell. The review is licensed under a Creative Commons Licence.
Strogatz's enthusiasm for calculus knows virtually no bounds. "My goal in this book has been to show calculus as a whole, to give a feeling for its beauty, unity and grandeur." He sees it as a major part of mathematics, with a long history. In this he is unusual. The conventional view is that calculus burst on the scene in the seventeenth century with the work of Leibniz and Newton, entered a golden period of wild expansion in the 1700s, and consolidated its achievements in the 1800s; by 1900 the story was practically finished.

Strogatz, in contrast, sets it in a much wider historical perspective. He doesn't want to say that it was invented in the late seventeenth century. Rather, he sees its roots extending back to the Greek thinkers, especially Archimedes, and he compares the seventeenth-century development by Newton and Leibniz to the dramatic biological evolutionary event known as the Cambrian explosion that occurred half a billion years ago, gave rise to new types of animals, and shaped the subsequent course of life on earth, Biological evolution and the evolution of calculus are still continuing.

From this perspective much of higher mathematics becomes a development of calculus. As he acknowledges, holding this view places him in a minority, and a small one at that.

Actually, a minority of one. None of my colleagues in the math department would agree that all of this is calculus, and for good reasons: it would be absurd. Half the courses in the curriculum would have to be renamed. … So instead we give different names to each offshoot of calculus and obscure the continuity among them.
This is therefore a historical and at times even a philosophical treatment of its subject. For example, an important concept in calculus is that of infinitesimals. An infinitesimal "is supposed to be the tiniest number you can possibly imagine that isn't actually zero". Moreover, you can have an infinitesimal part of an infinitesimal, which "is incomparably smaller still". But do infinitesimals really exist?
It depends on what you mean by really. Physicists tell us infinitesimals don't exist in the real world… Within the ideal world of mathematics, infinitesimals don't exist in the real world of number systems, but they do exist in certain nonstandard number systems that generalize the real numbers.
Calculus has the reputation of being difficult but Strogatz writes in an accessible style and provides plenty of examples and analogies to help comprehension. However, his aim is to teach us about calculus rather than to make us confront calculus itself. This isn't Calculus for Dummies or Teach Yourself Calculus.
It isn't necessary to know how to do calculus to appreciate it, just as it isn't necessary to learn how to prepare fine cuisine to enjoy eating it. I'm going to try to explain everything we need with the help of pictures, metaphors, and anecdotes. I'll also walk us through some of the finest equations and proofs ever created, because how could we visit a gallery without seeing its masterpieces?
In fact, this description is a little misleading. It's correct for the first half of the book, but thereafter the diagrams and equations cease and we get a straightforward history of calculus with its ramifications and practical applications, right down to the present day. This is certainly interesting but possibly a little anticlimactic.

I should say the first half of the book is demanding but rewarding. I felt I was really acquiring new knowledge. This part is accessible to anyone with no formal training in calculus who remembers a certain amount of basic geometry and algebra from school or is prepared to refresh their memory of these things if it has become hazy.

The reader isn't expected to perform any actual calculus. This is a relief, no doubt, but paradoxically I found myself rather regretting it. I was left feeling curious and wanting more—not to start on a full-scale calculus course, obviously, but perhaps the chance to sample the subject a lttle at first hand. To use Strogatz's own analogy, I felt no ambition to become a Cordon Bleu chef but I thought I might at least try my hand at an omelette. Perhaps this was the effect Strogatz secretly intended. If so, for me he succeeded.


%T Infinite Powers
%S The Story of Calculus: The Language of the Universe
%A Strogatz, Steven
%I Atlantic Books
%C London
%D 2019
%G e-book ISBN 978 1 78649 295 1
%P xxiii+359pp %K mathematics
%O bibliography, index
%O kindle version, downloaded from Amazon 2020

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