No one can rival a beginning teacher for focusing on details at the expense of the big picture, much to the frequent chagrin of their students. One of my office mates in graduate school used to take points off of his students’ assignments if they wrote things like “2 π = 6.28”, since (as he said) that meant they were saying that π = 3.14 exactly. He wanted them to use the symbol for approximate equality, which is two wiggly lines: ≈.
Of course, as several of us pointed out to him, the students didn’t really mean that π = 3.14 to all levels of precision. They were using the equals sign in a common sense, as a way to assign a number to a symbol. The truth is that π is very close to 3.14 in value; if you leave out all those extra digits, your numerical result will still be correct to much greater than 1% accuracy. My office mate was holding them to a level of mathematical rigor that wasn’t really necessary for the environment in which they used it. To that standard, we can never write “π = ” and finish the expression, since the number for which the symbol π stands is irrational, requiring an infinite string of digits.
(By the way, I was guilty of similar focusing on details, though a different set, so I’m not saying I was better than he was. Graduate students in physics often earn their keep by teaching introductory physics classes, but they’re frequently given little or no training on good practices, and little oversight to make sure they’re doing their jobs properly.)
The truth is that almost nothing is equal in the mathematical sense, once real numbers, real physical systems, and real life comes into play. Even the number π, ubiquitous and important as it is, stands as a kind of mathematical extension of realistic things: no matter how perfect your circle or sphere is, it will always have little imperfections such that the ratio of circumference to diameter isn’t always going to be π. It will be close enough for many things—the shapes of stars, the orbits of planets, interactions in quantum electrodynamics—all involve π, and their deviations from perfection aren’t strong enough that we stop using the equals sign.
When we speak of equality in society, we use the realistic rather than the formal mathematical version of =. The problem is that many opponents of (for example) same-sex marriage seem to think that the formal mathematical version is what applies. Two men or two women can’t be equal to a man and a woman in marriage, in this way of thinking, because the “numbers” aren’t the same to arbitrary precision. However, no marriages—no relationships—are ever equivalent in that sense.
Among my family, friends, and acquaintances, there are a variety of relationships, whether recognized by law or not. In a certain sense, though, these relationships have a symmetry. No two of them are alike to all decimal places, but who would really expect them to be? From the point of view of the law—and, you know, human decency—these relationships should be considered equal. The people within those relationships should also, as individuals, be considered equal.
The mathematical symbol “=” evokes two parallel lines, which in Euclidean geometry will never meet, no matter how far you extend them. The Greek letter π as I drew it above is a equals sign turned on its side and connected with a curve. That’s a good symbol of equality for me, in a real-world sense.
One response to “Equality, mathematical and practical”
There are tons of limits, including infinite series, all writable with finitely many symbols, which are exactly equal to \pi . Much else could be said. I think scientists lose some really top-notch students to mathematics sometimes, and unfortunately often, when they sound like they don’t know what they are talking about. These toung students may be accused of philosophical naivety, but they know that the equality sign means ‘are the same thing’.
“the students didn’t really mean that π = 3.14 to all levels of precision.” You think you know this in every case, or even the majority of cases? Taking marks off is probably a mistake, but asking for precision and pointing out errors is not.