Romancing the Equations

The symbols flanking an equal sign can be profound, utilitarian, beautiful--or even funny.

The history of discovery in the physical sciences forms a continuous braid woven of triumphs in theory and experiment. Occasionally, a scientist is talented at both, but more likely, one's expertise is either as a theorist or an experimentalist. In the astrophysical sciences, where laboratory tests of cosmic phenomena are few, experimentalists are more accurately described as "observers."

A fundamental difference between theorists and observers is that if an observer's data have a history of being flawed (through inferior methodology or because nobody can reproduce the observations), future data published by that person are regarded as suspect--especially if the results hint at a brand-new phenomenon or overthrow well-tested ideas. Conversely, when armed with pencil and paper and some equations, the theorist can be wrong many times, as long as an interesting path is taken. Interesting paths often contain keys to further discovery.

In pure math, an equation simply needs to have its left side equal its right side, and the equation need not apply to the real world. In the physical universe, however, equations connect measured quantities, such as temperature, energy, velocity, and force. Someone locked away in a closet-can therefore derive all kinds of mathematical theorems (if so inclined) but would be unlikely to walk out as a leading theoretical physicist. Why? Nature has unlimited power of veto on the ideas of the physicist, while mathematics is accountable only to its own, self-contained logic. Behold the primary reason why child prodigies exist among mathematicians but not among physicists.

The mathematics of cosmic discovery is expressed in a language of complicated-looking equations. But these equations are nothing more than the mathematical representations of ideas. What distinguishes theories rooted in equations from theories rooted in armchair speculation is that the mathematical image of your ideas forces those ideas--and the deductions drawn from them--to be logically constructed. The most amazing thing about mathematics, which is a pure construct of the human mind, is that it actually works as a tool to help us decode the universe. There was no tablet in the sky that declared the universe to be mathematically describable. We just figured out that it could be. Without math, science would not exist as we know it today. Note that abstract logic cannot possibly be natural to the human mind. If it were, then mathematics would be everybody's easiest course in school and our species would not have taken several millennia to figure out the scientific method.

If you fear equations, you are not alone. In the preface of his book A Brief History of Time, Cambridge physicist Stephen Hawking reflects on a comment from a publisher that for every equation included in the book, the list of people who would buy it would get cut in half. If the publisher's advice were taken literally, the inclusion of just ten equations would reduce the readership by a factor of one-half raised to the tenth power, leaving just 1/1024 of the potential readers. A Brief History of Time was not published equationless--it contains, for instance, E = [mc.sup.2], but it might have included many more. As we all know, Hawking wrote what came to be one of the best-selling science books of all time.

If the mere sight of equations upsets you, consider that they are generally no more complicated than anything else you might not understand on first sight. For example, the following equation--known as a Maxwellian distribution of speeds and named for the famous Scottish physicist James Clerk Maxwell (1831-79)--contains a healthy assortment of symbols:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Like many key equations that describe the universe, it is a distribution function, which is a slightly more sophisticated version of the colorful bar charts that are common in USA Today. Such equations are invaluable tools that organize information about the universe. The Maxwellian distribution of velocities tells us the fraction of all gas molecules that happen to be moving within a designated range of speed. When applied to the molecular activity in Earth's lower atmosphere, the equation can be used to calculate the speed of the largest number of air molecules. It's about 1,400 feet per second. Quantities connected to this speed (through other formulas, of course) include air pressure, atmospheric viscosity, and the speed of sound.

To appreciate the full depth and soul of the equation requires that you study some basic calculus and physics. But I submit to you that this equation is no more cryptic than terminology found in other disciplines. Take Pachycephalosaurus. Any eight-year-old child knows it is a dinosaur with a dome-shaped head. But understanding its designation as a genus requires some training far beyond the simple memorization of its name. An equally cryptic term is oxymetazoline hydrochloride, which happens to be the active ingredient in my twelve-hour nasal spray. It clears my stuffy nose. But 111 need to take a course in pharmacology to understand how and why it works. And the following four lines from Chaucer's Canterbury Tales, penned in Middle English, require no small amount of homework to decode and understand: