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Math as a Language in Its Own Right

Alfred Goldhaber

Einstein's Heroes: Imagining the World Through the Language of Mathematics. Robyn Arianrhod. xii + 323 pp. Oxford University Press, 2005. $28.

Human language is a wonderful mixture of nature and nurture. Our capacity for learning language, so far eclipsing that of other creatures, is a clear indication of the inherited language hardware we have. That hardware, though, is both more vulnerable and more elastic than computer hardware. Rare cases where children have been isolated from other humans suggest that after a certain age the capacity for language may wither away. On the other hand, with stimulation, language and other mental capacities actually can grow. Imagine a computer, which upon reaching the end of its three-year extended warranty, begins to enlarge its internal networks—that's what our brains can do. For most of us, our language or languages give the framework for our thinking, so that learning language is at least a large part of learning how to think.

Albert Einstein may have been an exception. By his own account he began to speak late, definitely after the age of three. He said his thinking was largely in images, and in tactile or muscular sensations, not in verbal form. Nevertheless, as an expositor of his scientific ideas for the general public he has yet to be surpassed, and he was an eloquent advocate on all other matters dear to him. It's not that his language was deficient, just overshadowed by some of his other, outsized talents. Thus for Einstein as much as for the rest of us, the notion of a human being without language would seem a contradiction in terms.

Robyn Arianrhod's theme in Einstein's Heroes: Imagining the World Through the Language of Mathematics is that mathematics is a language, with its own grammar and (implicitly) a number of dialects. Her view implies that mathematics, like more familiar languages, is something characteristically human, an idea appealing to anyone fond of math. The notion of mathematics as a language is not new, but what distinguishes her take on it is that she focuses on a particular, critical event in the use of mathematics, where we can see mathematical language growing in front of our eyes until it reveals a brand-new piece of physics.

She starts her account with a riff on Remembering Babylon, David Malouf's novel in which a young English boy has been marooned in an aboriginal community in Australia and suffused with its language and culture. On rejoining British society he feels strange—and seems strange to those around him—having been virtually transformed into an aboriginal thinker by being steeped in that language. With this prelude Arianrhod makes a point of the power of language, which she proceeds to bring home with her mathematical exploration.

Who are the heroes of the title? The first is Isaac Newton, who created the earliest grand vista of mathematically encapsulated physics through his universal theory of gravitation. Then comes Michael Faraday, who replaced Newton's notion of forces acting instantly between separated objects with a new concept, a field generated by an object in one place, flowing from there throughout space to influence the motion of anything that encounters it. Finally, James Clerk Maxwell reformulated the field concept, which Faraday had conceded was not properly mathematical, by using a new language—(differential) vector calculus. This led to a spectacular deduction, the existence of electromagnetic waves traveling at the speed of light. Maxwell's reformulation invited scientists to identify light as an electromagnetic wave and also to try generating in the laboratory new waves of much lower frequency. Heinrich Hertz later achieved this feat, and today these waves are that commonplace of daily life, radio.

If the Maxwell moment is the peak of Arianrhod's  narrative, it is just the highest point in a fully articulated terrain of physics and mathematics, going back to the Greeks and coming forward through Chinese and Indian and Arab mathematicians, then on to Copernicus, Galileo and Newton, and even past Maxwell to 20th-century physics. Arianrhod tells many tales that should be comfortably familiar to people already acquainted with modern physics, and these, along with a number of surprises, make the book attractive for that kind of audience.

For me, it was most enlightening to learn more about Maxwell, whose life was not easy but whose optimism, kindness and humility made him a happy man despite a string of difficulties that might have crushed a weaker character. For those who are interested in scientific subjects but have no experience with math or physics, the author takes particular care to include simple descriptions and drawings to illustrate the ideas. Thus this really is a book for all who would like to know the essentials of a key part of modern science.

Turning the "language lens" on Einstein himself, we may ask, Whence came the language of his thought, neither verbal nor conventionally mathematical? Surely there must have been an inborn capacity, but how was it educated? Could we discern elements of this process to help nurture new creative minds today? That Arianrhod's perspective can provoke such questions illustrates its value.

In my opinion, this book is so good that there should in due course be a second edition. If so, here are three things worth adding to reshape and strengthen the "summit" of the story. First, despite Faraday's diffidence about his contribution, in modern thinking it is just as mathematical as Maxwell's, only Faraday's version was integral calculus. Arianrhod mentions that a manuscript discovered recently shows that Archimedes understood the integral form of calculus, a subject whose development usually is credited to Newton and Leibniz nearly two millennia later. In fact, the current view of theoretical physicists, and perhaps also mathematicians, is that integral calculus, although mathematically equivalent to differential calculus, is more intuitively accessible. That's why Faraday could express his concepts in words and pictures rather than more abstruse symbols.

Second, Maxwell's mathematical contribution that completed the equations of electromagnetism was the inclusion of the electric field in the time-dependent version of Ampère's law (which gets only one paragraph in the current edition). Maxwell showed that this was necessary for the mathematical consistency of the theory, an excellent example of Arianrhod's point that the grammar of mathematics has powerful implications. In her next edition, the author could work through this example using the integral form of Maxwell's equations so that nonmathematicians could easily understand what's going on.

Finally, even though the integral form of Maxwell's equations is equivalent to the differential form, it is the latter (but only with Maxwell's extra piece!) that makes electromagnetic waves "pop out" of his equations, yielding the crucial prediction of radio waves. Arianrhod could cap the story by explaining that it was by judiciously switching to (and developing further) a new mathematical dialect that Maxwell reached the summit. These additions would put extra sheen on a magnificent saga, already well worth reading in the book's current edition.

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