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 `pageok` [Eugene Volokh, January 5, 2008 at 2:00pm] Trackbacks Careful With Those Scientific Allusions: I just noticed this item from Andrew Sullivan in The Atlantic: Consider this hypothetical. It's November 2008. A young Pakistani Muslim is watching television and sees that this man — Barack Hussein Obama — is the new face of America. In one simple image, America's soft power has been ratcheted up not a notch, but a logarithm. Now I've been trying hard to stifle my natural temptation towards mathematico-linguistic pedantry. Really, I have. "A number of" to mean "many" still annoys me — zero is a number; so is one — as does "to the nth degree" (depends on the n, no?). But I have to acknowledge that these are established English idioms, governed by the rules of English idiom, not of mathematics. I don't like 'em, but that's my problem, not the speaker's. Still, if you're going to try to come up with new figurative usages, it seems to me that the figure of speech should fit rhetorically. "A number of" at least sounds large, but "a logarithm" doesn't. Logarithms, I think, generally seem small. In all the commonly used bases, they are smaller than the original number, often much smaller. A million is a big number; comparatively, the base-ten logarithm of a million (six) is much smaller. A logarithmic scale does have the property that small steps can correspond to large increases, which is what I take it Sullivan is referring to. But "ratcheted up ... a logarithm" doesn't quite capture that, I think. "Exponential increases" does communicate "large increases," in a way I have to grudgingly accept (down, math pedant self, down!). But logarithm is the opposite of exponential, not a synonym. And when new terms are coined, the correspondence to the original referent does matter, especially given that most people who even know what a logarithmic scale is will likely think of the original referent. My sense is that scientific allusions, like classical allusions, tempt people into error — they sound cool, and people use them because of that rather than because they're apt. So think twice before you ratchet things up a logarithm. UPDATE: The winner is commenter Elmer: "To summarize, ratchet and soft and logarithm just don't go together well, though Soft Ratchet Logarithm would, of course, be an OK name for a band." (link) John (mail): Andrew is not a uniter. He is a divisor. 1.5.2008 2:07pm (link) MarkField (mail): After all your posts criticising grammarians and the "that's not a word" crowd -- with which I agreed -- I'm surprised to find this bothers you. I guess we all have little things that just bug us (is that a scientific allusion?). 1.5.2008 2:08pm (link) Richard Riley (mail): I think Andrew was trying to say something like "not a notch, but an order of magnitude," and order-of-magnitude suggested "exponent" and hence "logarithm" in his mind. Not saying he's right, obviously - I'm just guessing that was how he got to "logarithm," which I agree is an odd locution. 1.5.2008 2:13pm (link) sbron: The phrase "exponentially increasing in" a quantity, e.g. a function f(x) exponentially increasing in x is generally accepted in math/science. But it makes no sense to speak of an exponential increase in reputation without stating what reputation is a function of. But the statement is mathematically and factually idiotic. If the Pakistani interprets the middle name Hussein as implying Muslim at birth, then Senator Obama is an apostate, since he is now a practicing Christian. Being an apostate is viewed as being far worse than an infidel among many Muslims. 1.5.2008 2:14pm (link) Eugene Volokh (www): MarkField: As I mentioned, it's hard to make anything but esthetic objections to well-settled usages, even if they don't correspond to the logic of the discipline from which they're drawn. But when it comes to new figures of speech, the likely effect on listeners will be much more closely tied to the phrase's literal meaning. That, I think, is what's likely to happen here; if readers see "logarithm," their first thought is likely to be small, not big, even if they aren't closet math pedants. And, of course, it does bug me. 1.5.2008 2:18pm (link) MarkField (mail): And, of course, it does bug me. I was having exactly that conversation with my daughters last night. We agreed that it's usually rude to jump in and correct someone, except in cases where the speaker uses the wrong word and it's unclear what s/he means. But we all admitted that some things just bug. 1.5.2008 2:25pm (link) AK (mail): How about this one: "The probability that Andrew Sullivan will next week write an article identifying Obama as the 'conservative' choice for President varies directly with Obama's margin of victory in the NH primary" ? 1.5.2008 2:27pm (link) Christopher M (mail): "if readers see 'logarithm,' their first thought is likely to be small, not big" I disagree. Most readers won't know what a logarithm is, and the context makes clear that he's talking about a big increase. Also, it's not hard to construe what he said so that it makes even literal sense. If x = a certain (positive) quantity, and then the logarithm of x increases, then it follows that x itself has increased exponentially. Start with 100 (where the base-10 logarithm is 2); then it increases "by a logarithm" so the logarithm is now 3; well then now the original number has increased to 1000. 1.5.2008 2:28pm (link) notalawyer: My favorite example of this kind of rhetorical misstep comes from literature, not mathematics. In the novel The Ugly American the culturally sensitive hero is an American who is physically ugly. But by now "ugly American" has passed into common use denoting the opposite of what it meant at first. 1.5.2008 2:37pm (link) Steve2: My guess is that it's from knowing enough about logarithmic scales to know that an increase of one on a logarithmic scale is greater than an increase of one on a linear scale but not knowing enough about them to know how they work and thinking that they must have units called logarithms. 1.5.2008 2:42pm (link) dearieme: That's a small step for man, a quantum leap for EV-kind. Relatively speaking. 1.5.2008 2:59pm (link) Marc : My favorite is the use of the word fraction in phrases like "Now for a mere fraction of the price!"* _____ *5/4ths 1.5.2008 3:05pm (link) Marc : My favorite is the use of the word fraction in phrases like "Now for a mere fraction of the price!"* _____ *5/4ths 1.5.2008 3:05pm (link) alias: I think he meant to say something like "order of magnitude," as that term is used in figurative discourse 1.5.2008 3:08pm (link) Kevin Murphy: Consider the common phrase "a quantum change." To a physicist that's a really really really tiny change. The smallest possible change. But it seems to mean something like "an utter and complete change" to the layman. 1.5.2008 3:10pm (link) Cornellian (mail): I'm not sure you're using the term "allusion" correctly either. It's more like a reference than an allusion. 1.5.2008 3:10pm (link) AnonCoward: I think Steve2 is on the right track. Perhaps order of magnitude is closer to what was intended. I'm not sure a number of can be considered an idiom. If you look up number in the dictionary, one of its meanings is many. 1.5.2008 3:11pm (link) John Jenkins (mail): @Christopher M: The construction still doesn't make sense. It might make sense in the way you describe, had he originally just said that it increased "logarithmically" rather than by "a logarithm." You are trying to say that the "logarithm of x increased by 1" = "x increased by a logarithm." Those are not equivalent expressions. I couldn't even begin to evaluate the second one because it is nonsensical. There is no unit of measure called a "logarithm." I suspect that you have never seen logarithm used that way, but would like to see where you have, if you can provide some source for it (other than Sullivan). 1.5.2008 3:12pm (link) Sean O'Hara (mail) (www): I have bigger problems with that passage than the badly-chosen rhetorical expression -- electing a guy who converted away from Islam is going to anger as many Muslims as it mollifies. 1.5.2008 3:18pm (link) MatthewM (mail): This is off-topic, but the gross ignorance of this Sullivan quote is a perfect example of why I shun his articles like the plague. Electing Obama will do nothing to stem jihadist violence; young Arab men are just as racist against blacks as any other subset of the world population; and the cultural and political reason for the jihadist war will not go away because of a pretty face. The quote is emblematic of the delusions that govern a huge percentage of our populace; it bodes very ill for the future. 1.5.2008 3:27pm (link) CDU (mail) (www): The real problem with isn't that it bugs Eugene, it's that it doesn't clearly communicate his intent to the reader. Writing that someone finds aesthetically displeasing may be bad, but it isn't wrong. Writing that fails to accomplish its purpose, on the other hand, is wrong. 1.5.2008 3:50pm (link) Hoosier: THIS is why I love VC. notalawyer—Your peeve is my peeve, my brother. When people use the phrase "ugly American" now, they really mean someone like Graham Greene's "Quiet American." Greene's Alden Pyle has some similarities with the (stereotypical) neocon and Wilsonian: A guy with big ideas from books and no real-world experience tries to apply them to a foreign culture he knows nothing about, and causes only suffering. Both novels have ironic titles, which is something that gets missed by people who cite them but don't read them. (Another classic example: "Silence" by Shusako Endo. Actually, this might make for an interesting thread, Professor . . .). Adding insult to injury, both novelsare well-known to be based on the life and legend of Edward Lansdale. Which is unfortunate, since they are not. (Credit where due: Thanks to Jonathan Nashel's "Edward Landsdale's Cold War" for setting me straight on this. Lederer and Burdek cleary is not. But I had thought that Pyle and Lansdale had some connection. As it turns out, that's not possible.) I agree with some of the comments here re: "descriptivist" use of vocabulary. So if Porf. V is being inconsistent, I am too. Because "quantum leap" is just terrible as currently used. I mean, can't a descriptivist at least ask that metaphors not mean *precisely the opposite* of what they factually imply? Orwell asked this in the late '40s: If you speak of two conflicting forces being "like hammer and anvil," then try to keep in mind that it's the *hammer* that is eventually worn down by the clash. In the long run, the anvil will be just fine. The name of the Volkswagen Quantum thus always indicated to me (6' 1" and 200+ lbs) that I probably couldn't get inside. (Just the OPPOSITE of what the Mazda LaPuta suggests. --Rimshot--) Christopher M—"Most readers won't know what a logarithm is, and the context makes clear that he's talking about a big increase. " I think that's exactly right. And those of us who are subjected at work to constant references to logarithms tend to think of them as vague predictions about future numbers (In my case, numbers of minority applicants five years from now; average feshman gpa's next fall; number of female undergrads who will drop Engineering before the end of sophomore year . . . ). I don't equate the word with any specific quantity or degree, large or small. But the context did make clear what Sullivan meant for the reader to understand. Re: Objection to "Exponential"—This may be a cavil. But, again, I'm with Orwell: I don't want rules for English usage to be unchanging. I don't like it, however, when changes reduce either clarity of expression, or the very ability to expresss differences. "Exponential growth" is really, really, REALLY big growth. Really. Ask a cosmologist. (Or "cosmetologist." I don't want to get picky here.) Now, the growth of yearly manufacturing output in the Rhine-Ruhr region since 1370 can be failry described as "exponential." But since 1970? Nope. Despite what a recently-published comparative government textbook claims. My point? If I want to express the 1370 to 2008-dimension of change effectively and succinctly, what adjective can I now use? The hassles of working in a living language, I suppose. Which is why my Church, two generations after Vat II, insists that the normative statement of any Church teaching is the Latin version: 200 years hence, it's not likely that the adjectives will have taken on a different sense. 1.5.2008 4:31pm (link) Alan Gunn (mail): The real problem with isn't that it bugs Eugene, it's that it doesn't clearly communicate his intent to the reader. Writing that someone finds aesthetically displeasing may be bad, but it isn't wrong. Writing that fails to accomplish its purpose, on the other hand, is wrong. I doubt that anyone didn't know what Sullivan intended, but that's not really relevant. He could have communicated his intent by saying something like "a whole lot." What he was trying to do was communicate that in a forceful, interesting way, and he failed. His "error" (if that's the right word for a poor choice of wording) was similar in principle to that made by those who use "transpire" to mean "take place." They could simply say "take place," but they strive for a sort of elegance that doesn't work for readers who know what "transpire" originally meant. Sullivan's mistake may have been even worse, as there are probably a lot of people who, like me, have forgotten most of what they once knew of logarithms and so responded mostly by being temporarily puzzled. As for bugging Eugene, I'm delighted to see that there are usages that can do that. As someone with inclinations toward pedantry (though not the mathematical sort) I welcome him to the club. 1.5.2008 4:40pm (link) Laura S.: Eugene, you're wrong here. "A logarithm" is reasonable short-hand for "an order of magnitude" 1.5.2008 4:44pm (link) CLS (mail) (www): Andrew has a view that things are what he wants them to be. He's rather full of himself on such matters so no doubt words can mean what he wants them to mean. He defined Christianity according to his personal whims. He praises Catholicism but only on the basis of how it corresponds with his views. He is the center of his own little universe and everything circles around that. 1.5.2008 4:50pm (link) Curt Fischer: A logarithm" is reasonable short-hand for "an order of magnitude As an engineering Ph.D. student, I'm inclinded to agree. Logarithm seems a reasonable abbreviation of "order of magnitude". Of course, both phrases still annoy me because the base of the logarithm is unspecified! In base 2 an order of magnitude is only a factor of 2. In base 10 it is 10. I prefer "power(s) of ten". 1.5.2008 5:05pm (link) r78: He is the center of his own little universe and everything circles around that. Why not? Believing that the theory of evolution is a scientific fact would seem to disqualify one from becoming the (Republican) nominee for the presidency - if the Iowa results are any guide. So perhaps it is time to return to pre-Coperinican ideas. . . 1.5.2008 5:07pm (link) Elmer: Obama and Huckabee are good public speakers, and that is a rare skill. Mr. Sullivan should try to emulate his hero by improving his writing. He wanted to say that America's soft power would increase a whole bunch when people saw Obama's picture. He used the ratchet analogy. Ratchets are quite common and understood by nearly everyone. Click, click, click; forward, not backwards, one step at a time, with firm resistance to attempted reversal. There's a problem already. Soft power is supposed to be soft, and ratchet is hard, both in sound and function. I have done some unusual things, but I have never put a jack under my pillow. Now Mr. Sullivan explicitly rejects his own analogy: "...not a notch, but a logarithm." Depending on the reader, that is understood as either "...not the inapt analogy I just made, but I don't understand math," or perhaps "... not the tool something, but you never understood logarithms in high school, and that Pakistani guy probably didn't either." To summarize, ratchet and soft and logarithm just don't go together well, though Soft Ratchet Logarithm would, of course, be an OK name for a band. Mr. Obama doesn't make such rhetorical mistakes, and if his acolytes study his speeches until they understand how to use both the sound and sense of words, our nation will set aside partisan differences in celebration. 1.5.2008 5:18pm (link) Elmer: Sorry for the repition. The other posts making the same points arrived while I was typing. 1.5.2008 5:27pm (link) Elmer: Consider the common phrase "a quantum change." To a physicist that's a really really really tiny change. The smallest possible change. But it seems to mean something like "an utter and complete change" to the layman. It depends on the context. In the blackbody radiation context, it's pretty small, but in the Volkswagen context, it's one car. 1.5.2008 5:38pm (link) Thoughtful (mail): EV: So think twice before you ratchet things up a logarithm. More, I will think about it a number of times... 1.5.2008 5:49pm (link) hw: While we're hammering out the correct usage, I'll add that exp(x) is not the 'opposite' of log(x), but rather the inverse. That is, composition of exp and log yields the identity (i.e. 'return-the-input-unaltered') operation: log(exp(x)) = x for all x, and exp(log(x)) = x for all x>0 (restricting ourselves to the real numbers, of course). It doesn't make much sense to talk about opposites, as it's an ill-defined notion. For instance, what's the opposite of a cat, or the opposite of a sweater? 1.5.2008 6:40pm (link) wb (mail): Sullivan's usage is way off based if he means a large increase or an order of magnitude increase. When one say that f(x) increases as log x (i.e., logarithmically) one is saying that f increases slowly as x increases. If f(x)~log log x then f increases extremely slowly as x increases. What Sullivan has done is to take a mathematical phrase that has an unequivocal meaning and transformed it into a metaphor with precisely the opposite sense of the mathematical meaning. But then that is vintage Sullivan viewing the world in a topsy-turvy way. 1.5.2008 6:52pm (link) Elmer: It doesn't make much sense to talk about opposites, as it's an ill-defined notion. For instance, what's the opposite of a cat, or the opposite of a sweater? The square root of -1 doesn't make sense until you think of a multi-dimensional space. Researchers in sartorio-feline linguistic mathematics are attempting to answer those very questions. 1.5.2008 7:02pm (link) Important Person: If "a logarithm" is reasonable short-hand for "an order of magnitude," then "a square root" should be reasonable too. 1.5.2008 7:11pm (link) Bleepless (mail): Put these in numerical order: some, a few, several, a lot, lots, quite a few, many. Some fun, hey? 1.5.2008 7:21pm (link) JohnAnnArbor: My fave is when someone means that the situation had turned in the opposite direction and they say "it's turned 360 degrees." 1.5.2008 8:17pm (link) Brian G (mail) (www): Something tells me that Obama is smart enough not to say or even imply a message of "vote for me and the Muslim world will like us better." I dopn't give one whit what the world thinks. I would never vote for Obama because he is a one-world leftist liberal, just like Hillary and Edwards. 1.5.2008 8:18pm (link) Duffy Pratt (mail): A sweater IS the opposite of a cat. 1.5.2008 8:37pm (link) Hoosier: hw: The opposite of "cat" is "pet". The opposite of "sweater," oddly enough, is "slide rule." Which is why no one remembers the answer these days. And Hamlet seems to suggest that the opposite of "hawk" is "handsaw." But he was a bit off, so I'm not sure he can be fully relied upon in these matters. 1.5.2008 8:38pm (link) Tony Tutins (mail): I agree with EV that, although technically correct, using the word logarithm completely destroys the point Sullivan is trying to make, showing he has zero feel for what he's trying to say. (Although a logarithm is an exponent, by speaking in terms of logarithms the apparent magnitude of the change is minimized (6 vs. one million), as if he was looking the wrong way through a telescope. But a quantum leap, although small, is nothing to sneeze at. The electron leaps over an infinity of disallowed states, and, after falling back down, can cause a photon to be emitted. Finally, my strong impression, starting with reading the description of making the hajj in the Autobiography of Malcolm X, to my experience with Muslims from South and East Asia, is that Muslims are not racist to their brother Muslims. 1.5.2008 9:39pm (link) Laura S.: If "a logarithm" is reasonable short-hand for "an order of magnitude," then "a square root" should be reasonable too. To borrow some phrasing: one notch on a (common) log scale is one order of magnitude (power of 10). Therefore "a logarithm" makes perfect sense and does not means "the log of". Notice in particular the difference between "a" and "the" "a square root" would be plausible but sadly that's not a common scale (axis) to use. Thus it would be much more idiosyncratic. Conversely, order-of-magnitude is a commonly used approach to comparing things (in approximation). So referencing it via "a logarithm" is more sensible. 1.5.2008 10:36pm (link) Thorley Winston (mail) (www): This is off-topic, but the gross ignorance of this Sullivan quote is a perfect example of why I shun his articles like the plague. I quit reading him because in general, he's not very bright, doesn't bring any specialized knowledge to his writing (unlike say a milblogger or a law blogger who know a lot about their field), and he's utterly lacking in wisdom and reason. There are a lot of good writers that are worth reading in the blogosphere -- Andrew Sullivan isn't one of them. 1.5.2008 10:49pm (link) Elmer: This is off-topic, According to the laws of quantum blogging, cats, Barack Obama, exponential functions, Graham Greene, Islam, grammar, sweaters, Hamlet, and your post are fundamentally interconnected. 1.5.2008 11:29pm (link) Elmer: This is one of my favorite threads, and quite competitive. Still, I would award the thread to John. 1.5.2008 11:33pm (link) Rich Rostrom (mail): Kevin Murphy: in physics, a "quantum change" is a change from one level to a different level with no intermediate state. It can be large or small, relative to the previous state. Tony Tutins: Islam, like Christianity, is non-racial; anyone can be a good Moslem. However, Arab racism is very real. Arabs have always been central to Islam; and the flood of oil wealth has led to the Arabization of other Moslem cultures. In any case, Pakistani Moslems are quite racist. If anything, I suspect that electing a hubshi President would make the U.S. more despised. 1.6.2008 12:11am (link) Marian Kechlibar: Rich, Tony, etc. I would like to direct you to this cartoon, that was published in Egyptian daily Al-Ahram. It was published with a caption saying "'Hillary' and 'Obama' -- A Woman and a Negro are Participating in the Campaign for the American Presidency" and the man in the foreground is saying: "This is another sign of the collapse of the Western civilization" The image is here Seems pretty racist to me :-) 1.6.2008 8:40am (link) Kevin P. (mail): Tony Tutins: Finally, my strong impression, starting with reading the description of making the hajj in the Autobiography of Malcolm X, to my experience with Muslims from South and East Asia, is that Muslims are not racist to their brother Muslims. Tony, I don't know if this is necessarily true. Black people are looked down upon by Arabs, and to a lesser extent in the Indian sub-continent. Witness the case of Darfur, where black Sudanese Muslims have been slaughtered, raped and enslaved by Arab Muslim raiders. There is probably some credit that an American black would receive for being a Muslim, but only some. At the end of the day, he's still be black and inferior. 1.6.2008 9:56am (link) seadrive: In my view, a complete misuse of 'logarithm'. If you are going to write clearly, and you want to use a metaphor with different possible interpretations, you have to be sure your readers are going to take the one you mean. And if you are going to read for comprehension, you have to try to pick the meaning intended by the author. Quibbling doesn't even count in horseshoes. E.g., the "nth degree" clearly suggests to a mathematician expressions of the general form "for i equal 1 to n", so the nth degree is the last or highest. To suggest that it might mean "any integer" is petty. Similarly (as noted above), the use of a "quantum change" outside of physics means a discontinuous jump and not a increment of continuous change. The size of the jump is irrelevant. 1.6.2008 10:05am (link) pst314 (mail): "'A logarithm' is reasonable short-hand for 'an order of magnitude'." I have never heard a scientist or engineer use 'logarithm' in that sense--not when I began studying physics in University 35 years ago, and not when conversing with the scientists and engineers I know today. We use "increase exponentially" to mean "very fast" and "increase logarithmically" to mean "very slow", but we never speak as Sullivan did, and I suspect that any student who did so would get a quick correction (along with some strange looks and some silent doubts about that student's ability.) But I believe Sullivan has no training in mathematics or the physical sciences, as his biography in Wikipedia only reports that he has degrees in history, public administration, and government. It seems likely that he has seen "logarithm" used but has guessed incorrectly as to its meaning and usage without bothering to actually check. 1.6.2008 10:31am (link) pst314 (mail): I am reminded of how Americans used "nimrod" to mean "fool" or "jerk", all because of a popular misunderstanding of what Bugs Bunny meant when he mocked the rabbit-hunting Elmer Fudd. "Nimrod" properly means "mighty hunter" and refers to a character in the Old Testament. When Bugs called Fudd a "Nimrod" he was being sarcastic. But most children had no idea what the word meant, and simply assumed it was a synonym for "fool". Poor Andrew had no idea either, but fortunately it is unlikely that he will be able to change the language. 1.6.2008 10:41am (link) pst314 (mail): "In the blackbody radiation context, it's pretty small, but in the Volkswagen context, it's one car." I'd like to have a car like that: The driver punches the destination into the navigation system and starts the engine. Nothing happens until the energy expended is equal the amount required to reach the destination, at which time the vehicle jumps from one location to the other without traversing any intervening roads. Of course, if anybody else was already there, the Pauli Exclusion Principle would dictate that the jump would not take place...unless we're all bosons on this bus. 1.6.2008 10:59am (link) Hoosier: pst--Good point. But don't take your cat. 1.6.2008 11:42am (link) scooby (mail): Sully is right. Proof: Given y = b ^ x Solve for x: (Assuming real numbers) ln y = x ln b (take natural log of both sides) ln y / ln b = x (divide by ln b) x = log b y (definition of log) ... which is just another way of saying x equals is log base b of y. So x really *is* a logarithm and it's reasonable to call it that, just as it's reasonable to call it an exponent. This is why my high school math teacher had us chanting "a logarithm *is* an exponent." You got it confused because you saw "logarithm" and *assumed* logarithmic curve or function or scale. But he never said any of that, just logarithm. 1.6.2008 1:23pm (link) Joshua: Re: "quantum leap" - I blame the TV series for propagating this misconception. Re: Obama's middle name "Hussein" implying he is an apostate - Traditional Islamic theology (as I understand it at least) holds that all human beings, including you, me, the good Professor, Andrew Sullivan, and everyone else on the planet, are Muslim by birth. If this is the case, then to someone who subscribes to the traditional view, there is really no distinction - not even a quantum one - between an infidel and an apostate. Not that that would improve Obama's standing much, or even at all, in their eyes. 1.6.2008 5:41pm (link) David Chesler (mail) (www): Obama and Huckabee are good public speakers What are you, some kind of racist? 1.6.2008 5:52pm (link) Nelson Lund (mail): Some very fine mathematicians--e.g., Euclid--have not considered either zero or one to be a number. So maybe "a number of" to mean "many" is not so much technically wrong as charmingly old-fashioned. And maybe it's worth defending for just that reason. 1.6.2008 6:03pm (link) Syd Henderson (mail): Joshua: Re: "quantum leap" - I blame the TV series for propagating this misconception. I think the misconception came first and the the TV took its title from it. 1.6.2008 8:09pm (link) neurodoc: "A number of" to mean "many"...If one means "many," then "a number of" is certainly not the way to express it. I understand "a number of" to signify simply more than none and fewer than all, with even less certainty than "few" on the low side, "most" when greater than the median, or "almost all" on the high side. Saying "a number of" may reflect an unpreparedness or inability to be very precise, or an attempt to bluff with the implication of "many" when not willing to commit to "many." But "many" is "many," "a number of" isn't "many," at least not when I hear/see it. I accept "exponentially" as signifying "much greater" or "growing very quickly." But since exponents need not be positive integers, why not the possibility of "exponentially" sometimes signifying "much smaller" or "shrinking rapidly," as would be the case with a quantity "raised" to the -1 or -2 power? (For example, how improbable is it that Huckabee's numbers will change "exponentially" in just a couple of days?) 1.7.2008 12:55am (link) Thales (mail) (www): Just because I can't resist: Careful with that [mathematico-linguistic] axe, Eugene. But point well taken--most Americans, thoughtful writers like Sullivan (and occasionally, myself) included, are innumerate. 1.7.2008 3:30pm `pageok` `pageok`