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."

Not saying he's right, obviously - I'm just guessing that was how he got to "logarithm," which I agree is an odd locution.

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.

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.

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.

"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" ?

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.

The Ugly Americanthe 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._____

*5/4ths

_____

*5/4ths

reallytiny change. The smallest possible change. But it seems to mean something like "an utter and complete change" to the layman.order of magnitudeis closer to what was intended.I'm not sure

a number ofcan be considered an idiom. If you look upnumberin the dictionary, one of its meanings is many.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).

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.

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.

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".

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. . .

It depends on the context. In the blackbody radiation context, it's pretty small, but in the Volkswagen context, it's one car.

More, I will think about it a number of times...

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?

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.

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.

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.

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.

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.

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.

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.

According to the laws of quantum blogging, cats, Barack Obama, exponential functions, Graham Greene, Islam, grammar, sweaters, Hamlet, and your post are fundamentally interconnected.

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

hubshiPresident would make the U.S. more despised.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 :-)

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.

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.

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.

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.

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.

Re: Obama's middle name "Hussein" implying he is an apostate - Traditional Islamic theology (as I understand it at least) holds that

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.allObama and Huckabee are good public speakersWhat are you, some kind of racist?

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.

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?)

But point well taken--most Americans, thoughtful writers like Sullivan (and occasionally, myself) included, are innumerate.