Usage Deltas

Terms from mathematics have a habit of working their way into common usage, especially among business types. Mike Pope, a technical writer and editor at Microsoft, takes a look at what happens when math terms with precise meanings turn fuzzy in extended use. The deltas (changes) can be significant.

A colleague who's got a math background recently noticed that email missives from our upper management often include the term inflection point. (For example, one of the emails discussed "driving an inflection point" toward something.) This isn't new in business-speak — in 2005, Robert Cringley noted that Intel's Andy Grove had popularized the term, and it appears in the American Heritage Dictionary as "a moment of dramatic change, especially in the development of a company, industry, or market."

But my colleague just noticed the term, and to his math-trained sensibilities, the authors were using it wrong. As he carefully explained to us, in calculus, an inflection point is where the second derivative of a function equals zero, which is to say, the point at which a line changes direction. A line bending upward (we hope) seems like an irresistible image, though, so it doesn't surprise me that inflection point is now biztalk for any change, never mind the actual math.

While my colleague pondered the ostensible lexical abuse of his beloved calculus, I wondered how many other terms are used differently in mathematics and in common usage. Not long ago, Nancy Friedman did a similar exercise with terms from science, with an emphasis on how the terms are misused, and she included a few math terms, like exponential. In the list we came up with, I wouldn't say that popular usage is a misuse of mathematical terms, but instead that mathematics (usually) has very precise definitions whereas popular usage is more metaphorical. See what you think.

Calculus. In math, calculus deals with measuring change. In the wilds of English, you find people talking about their "personal calculus," which seems to mean something like "assessment"  or "choice"— for example, about student loans, residence, and running for president. The philosopher discussing a "hedonistic calculus" might be thinking of quantification. A poet who talks about "a calculus of the emotions" might simply be invoking complexity, or perhaps (stretching the mathematical connection) change within limits.

Formula. Math people understand a formula to be a mathematical statement, an expression "used to calculate a desired result." An equation is a formula. The rest of us use formula more loosely to mean a set of instructions, a methodology, a recipe – "a formula for success." In this generic sense, a formula can involve words, numbers, or ideas.  

Orthogonal. In math, orthogonal refers to a number of things, among them right angles (perpendicularity) and functions that are linearly independent. In general usage, orthogonal largely just preserves the idea of things that are unrelated. Certainly among computer people, it would not be surprising to hear someone say (as cited in the Jargon File) "This may be orthogonal to the discussion, but...".

Tangent. One definition in geometry of a tangent is a line that touches a curve but doesn't intersect it. (Picture a chopstick balanced on a basketball.) In popular usage, "on a tangent" means to go off in some different direction, at least conversationally. The metaphoric relationship is reasonably clear, although it's possible that a mathematician might have been happier to describe the tangential discussion topic as simply being an angle – "a figure formed by two lines diverging from a common point."

Permutation. In general usage, permutation refers to change or transformation, to variation ("Civil disobedience has many permutations"), or to reordering ("Lineup permutations continue," about a baseball team). Mathematically, the last of these is the closest: arranging elements in ways where order matters, which is interesting for calculating things like the number of ways you can seat people at a dinner party. Or for that matter, the ways in which you can arrange batting order.

Congruent. Triangles are congruent only if they match when superimposed, meaning that they have the same shape and size. Outside geometry, however, your behavior can be congruent with your principles, which just means that they agree.

Random sample. A guy on the sidewalk asking likely-looking voters to take a political survey might look to you and me like he's performing a random sampling – he is, after all, grabbing strangers as they walk by. A mathematician might disagree, however, because in statistics, random sampling must draw from a population in a way that every individual has an equal chance of being chosen.

Diametric, diametrically. Here we might say that the term has a more specific meaning in general usage than it does in math. In math, diametric is simply of or about a diameter. In popular usage, though, diametric and diametrically refer to extreme opposites. That is, popular usage focuses on just those points where the diameter touches the opposite sides of the circle.

We came up with more terms as well, like corollary, function of, and regression, where common usage and mathematical definitions diverge to greater or lesser degrees. While I take my colleague's point that our ordinary usage of these terms is not mathematically rigorous, I wouldn't agree with him that the terms are therefore used incorrectly. And I can certainly assure him that we will continue to get emails sprinkled with terms that will raise his mathematical hackles.