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.


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Mike Pope has been a technical writer and editor for nearly 30 years. He has worked at Microsoft and Amazon, and currently works at Tableau Software. You can read more at Mike's Web Log and Evolving English II. Click here to read more articles by Mike Pope.

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Comments from our users:

Wednesday May 18th 2011, 1:11 AM
Comment by: Jonathan B.
How could you possibly leave out "point differential" as used by basketball announcers when the phrase "point difference" would suffice? They use this phase ss if we need the language of differential equations to understand a basketball score.
Wednesday May 18th 2011, 2:44 AM
Comment by: Mike P. (Seattle, WA)Visual Thesaurus Contributor
Jonathan -- heh, excellent example. I guess that those of us who were kicking around terms at work don't watch enough basketball to know this phrase. I bet it wouldn't be too hard to come up with list of vocabulary that sports announcers seem to have either appropriated or invented to sound more ... what? Professional? Precise? Same as biztalk, I suppose.
Wednesday May 18th 2011, 7:38 AM
Comment by: Orin Hargraves (CO)Visual Thesaurus Contributor
The great thing about borrowing a term from math for extended use (as opposed to, say, farming or sewing) is that it makes you sound smarter! So I think these kinds of borrowings will always be fashionable. Nice article.
Wednesday May 18th 2011, 8:24 AM
Comment by: Roger Dee (Haslett, MI)Top 10 Commenter
A quagmire of obtuse language serves the point of obfuscating an issue and exponentiating the orator's status. We do it all the time.
Wednesday May 18th 2011, 9:47 AM
Comment by: Gordon W. (Jonesboro, GA)
Is it possible that at least some of the terms that mathematicans hold to be limited to precise use were actually in general use first and that it is they who "stole" the term and now want to limit their use in other contexts?
Wednesday May 18th 2011, 9:47 AM
Comment by: David D.
Also, there was recently a certain secretary of war or something, who spoke of "the metrics" of the military campaign in Afghanistan, and who, as Orin Hargraves noted above, badly needed to sound smarter.
Wednesday May 18th 2011, 1:15 PM
Comment by: Mike P. (Seattle, WA)Visual Thesaurus Contributor
@Gordon -- that's a good question that we might be able to get Ben to comment on. Most of the terms we were playing with do seem to have roots in classical languages, which might sort of perhaps suggest that they were coined for specific technical purposes. But that's not to say even then that they were coined specifically by mathematicians. And even if they were borrowed into math from other fields or from common usage, I suppose that wouldn't stop at least a few mathematicians (perhaps my colleague among them) from bemoaning that the common usage was not rigorous. :-)
Wednesday May 18th 2011, 5:16 PM
Comment by: Licia C. (Milano Italy)
This fascinating process, where “a lexical item that was once confined to a fixed meaning within a specialized domain is taken up in general language”, was called de-terminologization by Ingrid Meyer and Kristen Mackintosh in a very interesting article, “When terms move into our everyday lives: An overview of de-terminologization” (2000).
Thursday May 19th 2011, 10:42 AM
Comment by: Russell M. B. (Toronto Canada)
"Parameters" was much overused and very often misused for a while. I'm not hearing it so much any more.
Thursday May 19th 2011, 4:12 PM
Comment by: Mike P. (Seattle, WA)Visual Thesaurus Contributor
@Russell -- "parameters" is one of the terms that Nancy covered in her article on misused terms from science. I'm not so sure that it's any less popular than it was before, but that's just based on subjective impression. I do, as should be clear from this, work in an environment where people love their business-speak.
Thursday May 19th 2011, 4:29 PM
Comment by: Gary S. (Golden, CO)
While I may be going off on a tangent by entering an orthogonal topic into this discussion, is anyone else disturbed by the general lack of people using adverbs in our language? To wit "But my colleague just noticed the term, and to his math-trained sensibilities, the authors were using it WRONG. Wrong Mr. Pope, it should be WRONGLY.Never-the-less, I enjoyed your article immense-ly!

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