I’m pleased to introduce a new semi-monthly column by writer, physicist, and Princeton University lecturer Tony Rothman.  His most recent book, with Fukagawa Hidetoshi, is called SACRED MATHEMATICS: Japanese Temple Geometry.  Please enjoy his inaugural post!

“Do The Math”

Tony Rothman

The word “metaphysics” derives from the Greek meta ta physika. It was originally used by Aristotle’s Hellenistic editors merely to refer to his books that came after the books on physika—the things of nature. Thus “metaphysics”—after the things of nature. In this series I do not intend primarily to discuss the things of nature, the latest and most dazzling scientific discoveries, trends and fashions. I would like instead to explore how our world looks through the eyes of a professional physicist, one trained in mathematics and steeped in analytical habits. My particular area of expertise is cosmology, the study of the early universe, but like any physical scientist I value facts and data over opinion, pay close attention to the logic of an argument and show an appreciation for a carefully designed experiment or an elegant mathematical demonstration. To those of us raised in the scientific community such an outlook seems reasonable. When we listen to the news, we learn we do not think much like journalists, talk show hosts or politicians. Sometimes we wonder whether we are space aliens.

Hearing “Do the math” does frequently make me ask what planet I inhabit. Over the past few years, “Do the math” has become an American catch phrase. As far as I can tell, it usually refers to counting: “Do the Democrats have enough votes to pass this bill in Congress? You do the math.” It is a sad commentary on twenty-first century America that an activity human beings are supposed to have mastered five or six thousand years ago is considered higher mathematics. Rarely do I hear “Do the math” applied to something as advanced as multiplication; division is out of the question.

I speak seriously. For the past several years I have taught introductory physics at Princeton University. Two years ago we gave our usual final exam at the end of the second semester, which is devoted to electricity and magnetism. Princeton freshmen are easily the best undergraduates I have taught in twenty-five years of teaching and they are far better trained than I was at their age. By the end of the course we have covered some sophisticated material, including an introduction to Maxwell’s equations and even a nontrivial topic in calculus known as surface integrals. On the final exam we decided to give them a break with an easy problem. From some basic quantities they needed to arrive at an equation that amounted to A = B²C. Almost everyone got that far. We next asked if B were lowered by a factor of one thousand, how much would C have to change to keep A the same.

Of the two hundred or so students who took the exam, approximately twenty used their brains. If B goes down a thousand times, B² goes down a million times; therefore to keep A constant C must increase by a factor of one million. That is all that was required. The vast majority of students went back to their calculators, numerically recomputed B and C from the information provided and found a new value for A. With all the arithmetic mistakes that could and did occur by doing the problem on a calculator, about 60% of the students arrived at the wrong answer. Those who got it right often said that C had to increase by a factor of 999,999.9998 and rounded their answer off to one million. This is a little like the engineer who said that two plus two equals four to within a tolerance of .0002, and got the job.

A formal way of stating the problem is to say that since A is remaining constant, the ratio A/A = 1. Therefore, the ratio of the old value of B²C to the new value of B²C must be also be 1 and consequently the old value of B²C must equal the new value of B²C. Reasoning by proportions, or ratios, was known to the ancient Chinese and is something I was taught in sixth or seventh grade. It is one of the basic tools in the arsenal of any scientist. Not only has the tool been lost to the calculator generation but so has the concept of an exact answer.

The act of dividing one quantity A by another quantity B to get a ratio that compares the sizes of A and B may be the single most important act a numerate person can perform. Yet the ability to make that comparison is vanishing before our eyes. My favorite recent example is the Cingular/AT&T ad “Fewest dropped calls.” The ad is great because it is meaningless. Clearly the company with zero customers will have the fewest dropped calls. It goes without saying that the entire advertising industry is based on ignoring standards of comparison, in other words, by omitting the denominator of a fraction. “Doctors recommend…” How many doctors? What percentage?

Less amusing examples of denominator omission now affect us all. Before me is an article on plug-in hybrid cars that claims 100 mpg for the prototype. It is true, by the odometer one can drive 1,000 miles on 10 gallons of gasoline: 100 mpg. What the claim ignores is the energy needed to charge the battery and the energy lost in transmission from the power plant, both of which are significant. Without entering the debate on the merits of plug-in hybrids, what is clearly called for is a proper ratio to measure vehicle efficiency: the number of miles driven per total amount of energy required, perhaps with an adjustment for emissions.

More seriously, about a month ago the New York Times reported that the total money worldwide tied up in derivatives is roughly 550 trillion dollars. Whether this number is accurate, I don’t know, but I do know that it is about ten times the world’s gross product. One does not have to be a scientist to realize that there is simply not enough money in the world to cover such bets and that the system must collapse.

The moral of these anecdotes is that one number in isolation means little. Only when it is compared with a standard does it give us knowledge. Unfortunately, this elementary truth is ignored on a daily basis, not only by Princeton students, proponents of future cars and financial wizards, but by the news media as a whole. NPR, the BBC and the NY Times routinely presents figures without comparison. “The number of unemployed Americans this year has increased by two million.” “Six thousand million tons of carbon were released into the atmosphere in 2006.” Are these large numbers or are they small numbers? How does one know?

Only when we learn what fraction two million people is of the total workforce does the fact acquire meaning. The simple act of quoting numbers as percentages rather than absolute figures, a procedure known to Chinese peasants three thousand years ago, would instill a great deal of numerical hygiene into public discourse.