Infinite Powers: How Calculus Reveals the Secrets of the Universe

Description

MR3929748

This book is a love letter about calculus. Its author is a long-time mathematician. Its language is prose sprinkled spartanly with symbols and equations. It's for novices and experts. It's for people like the celebrated historian Herman Wouk, who after interviewing Richard Feynman about the Manhattan Project tried his advice about "learning the language of God''—namely, calculus—and failed because the learning curve via conventional, available means for those with math phobia was just too steep. It's for people like me who have taught calculus for over 40 years and scored AP calculus exams for ten years. Author Steven Strogatz claims his narrative is "accessible to everyone''. And indeed, its style is lively, full of very human discovery stories and vivid idea analogies. Strogatz allows the masters to speak, with foresight and hindsight and with brashness or humility, as if they were alive today, conversing over tea or greater spirits one on one with the reader.
Here are a few examples, along with some commentary.
"If anything deserves to be called the secret of the universe, calculus is it.''
"… the idea of using infinity to solve difficult geometry problems has to rank as one of the best ideas anyone ever had.'' At this juncture in the book, Strogatz is commenting on Archimedes' exact value for the quadrature of the parabola by slicing a parabolic cap into an infinite family of parallel line segments. He then lets Archimedes' lost words of challenge gleaned from an old palimpsest (a story told in detail in Reviel Netz & William Noel's The Archimedes Codex [De Capo Press, Philadelphia, PA, 2007]) express the hope that future mathematicians using his methods would "find other theorems which have not yet fallen to our share''.
Now jump ahead by nearly two thousand years to the time of Galileo and Johannes Kepler. "In science, being off by a hairsbreadth is acceptable. Being off by a ship's cable is not.'' Here Strogatz is paraphrasing Galileo popularizing the idea that falling terrestrial objects do so with constant acceleration rather than the Archimedean idea that heavier objects fall faster than lighter objects. "Where Galileo was rational, Kepler was mystical.'' In this vein about deriving rates of change and rates of rates of change and using Archimedes' method for computing areas, Strogatz contrasts Galileo and Kepler, saying that "Galileo was the intellectual descendant of Archimedes, entranced by mechanics'', while Kepler was "the heir to Pythagoras'', and prone to numerological arguments. Strogatz quotes Arthur Koestler, who concludes that Kepler's approach to discovering the planetary laws of motion was "an antidote to the pious belief that the Progress of Science is governed by logic.''
"Art'', said Picasso, "is a lie that makes us realize the truth.'' Strogatz describes himself as an applied mathematician, and he points out that every abstract model of a physical phenomenon is an approximation of reality, and as such borders on being a lie. And he marvels time after time in the spirit of Eugene Wigner's oft repeated phrase about "the unreasonable effectiveness of mathematics in the natural sciences''. In particular, Strogatz at this point highlights the world's fascination with Usain Bolt's Olympic running prowess in finishing the 100-meter dash well over a meter in front of the competition. A graph of Bolt's run plotting his distance along the track (at, say, each meter) versus his speed suggests that the relation is a continuous one. But if that run is plotted continuously, due to leaping and breaking during each stride, that plot now contains multitudinous jitters, belying any appearance of continuity. The applied mathematician then must be an artist, eschewing noise, so as to brush the signal. It is "close to wishful thinking and intellectual dishonesty''. Yet "scientists, like Galileo and Kepler, somehow manage to walk along that precipice''. Moreover, "there is simply too much insight to be gained from the continuum hypothesis not to use it''.
As for the two people credited with discovering what is commonly called calculus, Newton "rarely laughed''. At this turning point in his story, Strogatz contrasts Archimedes and Isaac Newton. Whereas Archimedes used series of numbers such as $\frac{4}{3}= 1+ \frac{1}{4} + \frac{1}{16}+\cdots$, Newton dynamically used the symbols $\frac{1}{1-x}=1+x+x^2+\cdots$ in brilliantly generalizing Archimedes' method at the age of 22. Leibniz said of himself, "I lack polished manners and thereby often spoil the first impression of my person.'' In somewhat the same way, he apologized for his use of infinitesimals in presenting the calculus, saying, "I consider both infinitesimals and infinituples as fictions of my mind for succinct ways of speaking.'' Yet Leibniz's more intuitive notation rather than Newton's more formal fluxion notation in representing derivatives and integrals is standard in today's texts.
Strogatz continues the story with differential equations, multivariable expressions, partial differential equations, Fourier series, and nonlinear equations—all without using symbols and equations. When introducing Joseph Fourier, Strogatz says that Fourier "often felt cold … and kept his room overheated and swathed himself in a heavy overcoat, even in the summer''. In a future edition of his book, perhaps Strogatz could give more of this story, for Fourier was part of the scientific crew who accompanied Napoleon's ill-fated Egyptian campaign of 1798 to map out a potential Suez canal, and who inadvertently uncovered the wonders of ancient Egypt. Fourier was in charge of assembling the final French report on these discoveries [see Description de l'Égypte, ou recueil des observations et des recherches qui ont été faites en Égypte pendant l'expédition de l'armée Française, second edition, Imprimerie de C. L. F. Panckoucke, Paris, 1821–1829]. But Egypt of course was hot. I myself spent a year teaching mathematics in Dar es Salaam at the equator, and upon my return to the States, it took my body three years to readjust and not feel cold all the time. Fourier must have had a more severe reaction. He never did warm up back in Paris. Nevertheless, it is remarkable that Fourier was able to channel a curse of ever being cold into the blessing of solving the PDE heat equation using the basic idea $\sin''x =-\sin x$. As Strogatz observes, for the heat and wave equation, "the calculus gets stripped out and replaced by multiplication [by $-1$]''.
Throughout Strogatz's narrative, he heaps up calculus victories. One of the more amazing stories is his account of the AIDS plague. For many years, the medical community often elected to delay administering medication during the seemingly mild second phase of the disease because the virus tended to mutate so quickly, and it would therefore be useless in the terminal third stage. But in 1995, Dr. David Ho and Alan Perelson via a simple differential equation showed that "a titanic struggle was taking place during [this second] phase in the patient's body''. So they proposed a three-drug cocktail treatment, a triple whammy, so as to outfox the virus; it simply could not mutate that fast against three different agents. Other victories include the FBI's use of wavelets to encode fingerprint whorls tractably, Paul Dirac's calculus solution predicting the existence of antimatter, which in turn has applications in positron emission tomography using x-rays to see inside soft tissue in the body such as the brain, cell phones and GPS, and the 2015 detection of gravity waves.
The accolades for calculus go on and on. Read the book. You'll remember afresh why calculus is so cool. Reviewed by Andrew James Simoson