The Universe in Zero Words: The Story of Mathematics as Told Through Equations
“. . . a very handsome book . . . will sit with quiet elegance on your coffee table for guests to peruse if ever the conversation should lag.”
Printed on white, heavy, almost glowing card stock, this book is lushly illustrated and though slightly smaller than a typical coffee table book has the appearance of one.
The title is a misnomer as well: Inside there are plenty of words, and a reader might infer that Zero Words means that the book contains mathematical equations, when in fact it provides essays on 24 historically important equations organized into four periods: Equations of antiquity, Equations in the Age of Exploration (the not-so-ancient), Equations in a Promethean Age (the 1800s), and Equations in Our Own Time.
The essays are light and anecdotal as what would be expected of a coffee table book, though anyone not already having some mathematical background might find portions of the text oblique. The intended audience will most likely not care—Ah, what gorgeous typography and pictures! The host must be highly intelligent to provide the coffee table with so beautiful a book!
The very first equation is 1 + 1 = 2, and the accompanying essay provides the reader a history of ancient mathematics.
The second equation is not an equation but the number zero. The essay addresses zero’s importance in our modern numbering system, second only in importance to numbers themselves.
The third is the Pythagorean theorem, and the corresponding essay covers mathematics in Pythagoras’ time, which was geometric not algebraic. It also touches on the discovery of prime numbers, the concept of proof, and rational and irrational numbers; the author, Dana Mackenzie, points out to us that Pythagoras most likely did not discover (nor prove) irrational numbers.
The next equation is the number pi, which may be the most famous constant in mathematics. Pi is both a constant and irrational. Pi can be defined to ever more precision by the summing of terms of an equation that never ends. The calculation of pi combines geometry, arithmetic and analysis of the infinite, which leads the author to address the ancient concept of infinity, starting with Zeno’s paradox and ending with Archimedes two centuries later. It is up to the mathematician Georg Cantor to give us modern concepts and equations for handling infinity, addressed in the later section Equations in our own time.
The next equation covers the laws of levers with the aphorism of Archimedes, “Give me a place to stand, and I will move the Earth.” Levers are followed by the “not-so-ancient” period that includes Cardano’s formulae for quadratics, cubics, and quartics, and whose solutions point to the need for imaginary numbers.
Next are Kepler’s laws of planetary motion—One: planets orbit the sun in ellipses not circles. Two: planets speed up when closer to the Sun in a precisely quantifiable way, and three, the length of a planet’s year is proportional to the square of its orbital period and the cube of its mean distance to the Sun.
Then we are treated to Fermat’s last theorem, technically not a theorem but a conjecture, and something that Fermat scribbled in the margin of a mathematics textbook (“I have discovered a truly marvelous proof of this, which this margin is too narrow to contain“), leaving others to puzzle out the details.
The author notes that “Fermat seems to have been torn between the desire for recognition and a nearly pathological fear of revealing too many of his secrets.” The quest for a proof opens up a new area of mathematics, including algebraic number theory.
The following pair of equations belong to Newton: his laws of motion and calculus.
The very last equation(s) in The Age of Exploration are Euler’s theorems. Euler was an author of more than 800 articles, 50 books and memoirs, and was so prolific that the Academy of Sciences in St. Petersburg continued publishing his articles for a half-century after his death. Euler provides us with what the author calls “the most beautiful equation in history”, which this reviewer cannot show because Microsoft Word™ does not provide the ability to produce mathematical symbols apart from what is present on the keypad – something that is very wrong on many levels.
The third section, Equations in the Promethean age, is so titled because the 1800’s were a time of questioning Science with a capital S, a concern over the danger of scientists playing God. The title was taken from Mary Shelley’s Frankenstein: Or, The Modern Prometheus. For those of you who are truly fascinated by mad scientists, please read Margaret Atwood’s contribution to Seeing Further: The Story of Science, Discovery and the Genius of the Royal Society, Edited by Bill Bryson, previously reviewed here.
And the first equation of the Promethean age is Hamilton’s quaternions. Quaternions may be used to multiply number triplets, which can be used to algebraically manipulate three dimensions. The author provides perhaps the best easy to understand explanation of quaternions and hyper-complex numbers this reviewer has ever read. Quaternions were notoriously difficult to use and Hamilton was ahead of his time—physics wasn’t ready for dimensional manipulation until Einstein. Quaternions’ more successful, easier to use alternative is vector analysis.
The next equation is for group theory, a branch of abstract algebra whose inventors died tragically young. Niels Able died at the age of 26 of tuberculosis, and Evariste Galois died at 20, shot under unclear circumstances.
The subsequent equation is for non-Euclidean geometry, geometry that may have different forms. Hyperbolic geometry is the geometry of positive curvature. There is also negative curvature, and the geometry of the sphere. These all have a constant curvature. There can be geometries of varying curvature as well. The first mathematicians to understand the geometry of varying curvature in 2-dimensional space were Gauss and Riemann, who also were also ahead of their time, anticipating the 4D space-time of Einstein.
Next up is Gauss’ prime number theorem. Dana Mackenzie repeatedly describes his amazement at Gauss’ theorem’s successful predictions. However, The Universe in Zero Words has no equations of probability and statistics, a study of which would put a damper on the author’s amazement. It’s only math, after all.
We get the Fourier series, and what appears to be a not insignificant number of feuds among mathematicians over attribution for naming rights for their equations: Cardano versus Tartaglia, Newton versus Leibniz, and here Fourier versus LaGrange.
Maxwell’s equations unify magnetism, electricity, and the nature of light and have been voted as the greatest equations ever. And again due to limitations in Word™ this equation sadly cannot be provided in this review. But here’s a fun fact as consolation: Since 1983 the meter has been defined as the distance light travels in 1/299792485 of a second; meaning that the speed of light is now a definition and no longer an experimental constant.
For our modern era, the most famous equation can only belong to one scientist, Einstein with his E=mc^2, the equation showing the equivalence of matter and energy. Einstein has another equation, shared with Planck and just as important, E=hv, which shows that light is both a wave and a particle.
Dirac’s wave equation demonstrates that for very small objects the observer affects the observation, that a photon will exhibit either particle or wave behavior depending on the experiment the observer chooses to perform. Dirac’s equation makes use of Hamilton’s quaternions as 4 x 4 matrices, and led to the prediction of the existence of anti-matter, which changed the rules for theoretical physics. A theoretical physicist no longer has to wait for an experiment to provide answers to new questions such as, why is there more matter than anti-matter, or why isn’t the universe empty? A theoretical physicist could make predictions based on mathematical symmetry.
The Chern-Gauss-Bonnet equation permits extrapolation of global shape from a local curve or patch. The implication of Chern-Gauss is astounding: From a small measurement of local space one can determine the global shape of the Universe. Chern-Gauss shows that you cannot do geometry without physics just as Einstein and Dirac shows you cannot do physics without geometry.
And that brings us to the penultimate equation. The Lorenz equation shows us that the impact of small nonlinear calculations over time may grow to great significance. Mathematical chaos has been called the “butterfly effect,” the possibility of a hurricane evolving from something as insignificant as a butterfly flapping its wings.
Nonlinearity doesn’t guarantee mathematical chaos but opens up the possibility but when the significance of nonlinearity was first raised by Poincaré in 1893, the idea generated incredulity. This was from preconceived notions due to mathematics education of the time. Math was taught with simplifying assumptions, which either skipped or simplified nonlinear equations to make them teachable. Mathematical analysis of complexity only became tractable after the advent of computers.
The last equation in The Universe in Zero Words is the Black-Scholes. The Black-Scholes equation provides an appearance of controlled risk in accordance with the laws of physics. Black-Scholes permits a stock trader to make a profit (or limit loss) whether a stock goes up or down through use of computer trading, derivatives, and hedging. Black-Scholes can only work in a volatile market—if a share price doesn’t move then the option (to buy or sell a derivative) can’t be exercised and the buyer is out the cost of the option.
The equation turns out to be very brittle—it works exceedingly well until it doesn’t, as experienced in the recent U.S financial meltdown. Programmed trading can also be gamed, as the spring 2012 debacle of J.P. Morgan illustrates.
The author provides some interesting points in the concluding chapter. One, for example, is on the difference between how a computer and how a human plays chess. Depending on the position in an end game, a computer could actually search and find a 223-move checkmate, where a human would simply give up and draw.
Another is the failure of modeling in certain domains: Simplifying assumptions works for physics but doesn’t necessarily work for biology, whether modeling proteins or consciousness.
The Universe in Zero Words covers the same ground as In Pursuit of the Unknown: 17 Equations that Changed the World by Ian Stewart and previously reviewed here. A reader may wonder about what goes on behind the presence of this bounteous multiplicity of science books covering nearly identical material. Of course there will always be many perspectives from which to convey the wonder of the world, and there will always be new discoveries in math (though both conclude with Black-Sholes).
Perhaps this expression is a right of passage for author Dana Mackenzie? The difficulty for a reviewer is that once a writer reaches the level of quality that makes a popular science text good—and both Dana Mackenzie and Ian Stewart are very, very good, it then becomes very difficult to compare the significant differences of X by author Smith against Y by author Jones.
That being said, The Universe in Zero Words is a very handsome book and will sit with quiet elegance on your coffee table for guests to peruse if ever the conversation should lag.