Here’s a trend I find worrisome enough to consider petitioning the Word Police. The word “exponential” has been growing in media usage — pretty much, well, exponentially.
Do a keyword search in The New York Times’s archives for the various forms of the word and you’ll see a rough doubling each decade: 105 hits for the 1970s, 279 for the 1980s, 604 for the 1990s and 1,375 for the 2000s (the rise continues with 1,753 hits so far in the 2010s, though it might not quite double this decade). Crude and unscientific evidence, but the hits for “linear” (for example) don’t reveal comparable growth.
You would think such popularization of a technical term would warm the cockles of my mathematician heart. How wonderful if the surging acceleration of our times (in digital capacity, epidemics, chaos in general) is being described mathematically!
Alas, some recent hits reveal that the word is often used just to mean “lots.” Tasks are “exponentially more difficult” if one is blind, space war “exponentially more dangerous” than that on earth. The Washington Post reports on “exponentially richer private-sector jobs” and even an e-sports industry that in 2015 was “exponentially more nascent than today.” Hyperbole, more than acceleration, seems to define our times.
[Never be uninteresting. Read the most thought-provoking, funny, delightful and raw stories from The New York Times Opinion section.]
Examples like the above violate a basic criterion: “Exponential” is applicable to a trend, not a single comparison. Specifically, one needs a quantity that grows by the same factor in successive time intervals (something that doubles each decade, or triples each hour). The misuse arises from a freewheeling interpretation that seems to have hijacked the word.
The remarkable effects of exponential growth were recognized at least as far back as 1256 when the Arab scholar Ibn Khallikan recorded the classical tale of an Indian king hoodwinked into granting a doubling progression of grains for each square on a chessboard — resulting in a gift of more wheat than grown in the entire world. And yet, Webster’s 1828 dictionary struggled with the definition, resorting to gobbledygook that had exponential curves “partake both of the nature of algebraic and transcendental ones.”
It was only near the end of that century, when the algebra had been better codified and the language (and lexicographers!) had caught up, that the first edition of the Oxford English Dictionary got it right. “Involving the unknown quantity or variable as an exponent” refers to formulas such as “2 to the power x” (2ˣ) for doubling or “3 to the power x” (3ˣ) for tripling.
The current online version of the Oxford Living Dictionaries relegates this definition to second place and lists “becoming more and more rapid” as its leading entry. Since dictionaries don’t invent new meanings, this is really a reflection of usage. The problem is that this new criterion, while easy to understand, muddles the exponential with a host of lesser types of growth.
Take the humble quadratic (typified by the formula “x to the power 2,” or x²) which, for instance, tracks how the distance traveled by an object in free-fall increases with x, the time. Quadratic increases also “become more and more rapid,” but the exponential wins out — strikingly so — once enough time has elapsed. In fact, an exponential increase will eventually rocket past any fixed power of x.
These distinctions aren’t just academic subtleties — they have important real-world ramifications. For instance, at the start of the AIDS epidemic, it was found that the total number of cases in the United States was growing only cubically (that is, like x³), not exponentially as might be expected for a viral illness, a finding crucial for planning.
As another example, the practical running time of computer algorithms that slave away unseen for us (minimizing travel time, distance, cost, etc.) needs to grow far less than exponentially even if they are working on problems with an exponentially increasing number of possible solutions (for example, ever-growing possible combinations of routes and fares). An exponential rise in the time needed to calculate solutions would make programs running these algorithms so slow as to be essentially useless. You can earn million from the Clay Mathematical Institute if you resolve the “P versus NP problem,” a key issue in which is whether a certain class of problems can always be solved in reasonable, non-exponential time.
There’s something more fundamental at stake here beyond such million-dollar distinctions. It’s true that English has a long history of borrowing specialized words for other purposes — for example, “catalyst” from chemistry being applied to people. But an essential characteristic of mathematics, one it arguably lives and dies by, is precision.
To be true to the subject, to aspire to the goal of living in a quantitatively literate society, the language we use to transmit mathematical ideas needs to preserve this precision. This isn’t just a matter of philosophy or aesthetics; lack of linguistic precision is what leads to commonly encountered phrases like “five times bigger,” which rarely means “six times as big” as it should, or “five times smaller,” which means — I’m honestly not sure what.
Everyday language, as early lexicographers most likely found, is an imperfect medium to transfer this precision. Learning the specialized vocabulary of math is one of the big challenges for schoolchildren. As those of us on the front lines will tell you, a word like “percent” can trip up even college students. Imprecise usage of math terms (such as “200 percent certain,” “divide by half,” “infinite” instead of “many,” “order of magnitude” more often than not) just adds to the confusion.
Math is one of the few institutions we have left free of doublespeak or embellishment or biased opinion. Its words are supposed to mean exactly what they say. Let’s keep them that way.
Manil Suri (@manilsuri), the author of the novel “The City of Devi,” is a mathematics professor at the University of Maryland, Baltimore County.
201781期开马结果【这】【应】【该】【是】【兵】【勇】【们】【最】【后】【的】【意】【力】【箭】【了】，【普】【通】【箭】【支】【对】【高】【峰】【这】【等】【身】【手】【根】【本】【没】【用】。【等】【他】【们】【射】【完】【这】【一】【拨】，【高】【峰】【就】【可】【以】【从】【容】【地】【离】【开】。 【高】【峰】【转】【过】【身】【都】【不】【用】【出】【手】，【直】【接】【施】【展】【意】【力】【控】【制】【意】【力】【箭】，【却】【不】【料】【其】【中】【有】【两】【只】【来】【势】【迅】【猛】，【竟】【让】【他】【这】【个】【意】【教】【气】【生】【凝】【滞】。 【高】【峰】【暗】【叫】【苦】【矣】，【朝】【廷】【高】【手】【终】【于】【赶】【到】。 【一】【般】【的】【意】【力】【箭】，【威】【力】【都】【在】【意】【生】【一】【二】【段】
【洛】【熙】【的】【心】【情】【有】【些】【沉】【重】。 【没】【有】【什】【么】【岁】【月】【静】【好】，【只】【是】【有】【人】【为】【你】【负】【重】【前】【行】，【这】【句】【话】【说】【的】【真】【的】【一】【点】【都】【没】【错】。 【原】【来】【她】【这】【些】【年】【的】【平】【静】【安】【和】【都】【是】【自】【己】【的】【亲】【弟】【弟】【洛】【冰】【带】【给】【自】【己】【的】。 【再】【看】【看】【自】【己】【的】【弟】【弟】，【明】【明】【年】【纪】【应】【该】【比】【自】【己】【小】，【却】【早】【早】【的】【成】【为】【了】【一】【个】【顶】【天】【立】【地】【的】【男】【人】，【为】【了】【救】【自】【己】【救】【大】【家】【变】【成】【了】【现】【在】【这】【幅】【模】【样】。 【洛】【熙】【心】【中】201781期开马结果“【这】…”**【惊】【道】：“【既】【然】【大】【兄】【有】【危】【险】，【还】【不】【尽】【起】【大】【军】【前】【去】【营】【救】，【管】【什】【么】【袁】【无】【殇】？” “【胡】【闹】！”【贾】【仁】【怒】【道】：“【听】【命】【行】【事】！【若】【坏】【了】【主】【公】【的】【大】【事】，【就】【算】【砍】【了】【你】【那】【颗】【头】【都】【没】【用】！” “【哦】！”**【讪】【讪】【而】【退】，【他】【也】【是】【一】【时】【情】【急】。【其】【实】【以】【袁】【无】【殇】【的】【军】【势】，【就】【算】【要】【救】【援】【北】【城】，【也】【不】【能】【直】【接】【率】【兵】【去】【闯】。 “【苏】【启】，【你】【率】【兵】【与】【我】
【欧】【老】【先】【生】【又】【睡】【了】【一】【个】【舒】【舒】【服】【服】【的】【觉】，【醒】【来】【的】【时】【候】，【陈】【旭】【不】【在】【了】，【云】【溪】，【也】【不】【在】【了】，【他】【有】【些】【失】【落】，【出】【门】【的】【时】【候】，【看】【到】【自】【己】【最】【爱】【的】【荷】【花】【池】【真】【的】【被】【填】【平】【了】，【他】【怒】【吼】【一】【声】：“【我】【的】【荷】【花】【和】【鲤】【鱼】【呢】？” “【老】【爷】【子】，【不】【是】【您】【吩】【咐】【把】【水】【塘】【填】【了】【吗】？”【老】【管】【家】【见】【老】【爷】【子】【暴】【跳】【如】【雷】，【吓】【得】【差】【点】【不】【敢】【站】【出】【来】【解】【释】。 “【谁】【让】【你】【们】【填】【了】【我】【的】
【等】【到】【那】【一】【抹】【蓝】【色】【已】【经】【完】【全】【消】【失】【的】【时】【候】，【莫】【锦】【逸】【忍】【不】【住】【心】【中】【的】【气】【恼】，【用】【力】【地】【锤】【了】【一】【下】【禁】【锢】【着】【自】【己】【的】【屏】【障】。 【随】【着】【他】【这】【一】【拳】，【本】【来】【无】【形】【的】【屏】【障】【浮】【现】【出】【了】【如】【同】【水】【波】【纹】【一】【样】【的】【波】【纹】，【随】【后】【又】【平】【静】【了】【下】【去】，【再】【次】【消】【失】【在】【了】【无】【形】【之】【中】。 【顾】【子】【衿】【立】【刻】【提】【高】【了】【警】【惕】。 【然】【而】【对】【方】【在】【发】【出】【那】【一】【阵】【声】【音】【之】【后】【很】【快】【就】【安】【静】【了】【下】【去】，【顾】【子】【衿】