BOOK REVIEW
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.