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"Times Less Than":

An e-mail from a reader reminded me again of this debate — some people argue that "A times less than B" is "mathematically incorrect," "simply wrong," and so on. The theory is that "times" refers to multiplication, so "5 times less than B" to mean "B/5" is mistaken, though "5 times more than" to mean "5xB" (or possibly "6xB") would be fine.

This prompted me to do some more searching, and discover not only a usage of this phrase by Jonathan Swift (via Merriam-Webster's Dictionary of English Usage), but also by Isaac Newton ("If the Diameters of the Circles ... be made three times less than before, the Mixture will be also three times less; if ten times less, the Mixture will be ten times less"), Sir William Herschel ("remember that the sun on Saturn appears to be a hundred time less than on the earth"), Erasmus Darwin, Robert Boyle, John Locke, and more. Nor is this some archaic usage; it remains routine today.

What's going on here? The correspondent whose message prompted me to repost about this suggested that "A times less than B" might be a calque — "a loan translation, esp. one resulting from bilingual interference in which the internal structure of a borrowed word or phrase is maintained but its morphemes are replaced by those of the native language, as German halbinsel for peninsula" — from my native Russian, where "X raz men'she [or men'eye] chem" is routine. But that hardly explains Newton and Herschel, I think.

Rather, I think what's going on in the critics' minds is itself a sort of calque, though a calque from mathematics to human language. It's true that if you view "times" as "x" and "less" as "-," then "A times less than B" is either literally meaningless, or corresponds to "B-AxB." But of course in English, including the English used by scientists of the highest caliber, "times" doesn't always mean "x" and "less" doesn't always mean "-." We see that from the very examples I just gave, as well as from observed common usage.

Nor can you somehow disprove my assertion by "logic" of the "but 'times' means multiplication!" sort. That is the logic of the calque, and while calques sometimes do create usage (in Russian, for instance, the word for "rhinoceros" is "nosorog," since "rhino-" translates as "nos" [nose] and "-ceros" translates as "rog" [horn]), sometimes they don't. If you want to know what is an acceptable form (though just one of several acceptable forms) in English, including scientific English, is, the actual usage of Newton and Herschel — and, I suspect, countless lesser lights of today — tells us more than the abstract logic of literal translation from mathematical symbols.

This having been said, it may well be that "A times less than B" is suboptimal usage, precisely because it annoys enough people. (I am skeptical that it genuinely confuses a considerable number of people.) But to say that the usage is "simply wrong" or "mathematically incorrect" is to misunderstand the connection between mathematics and English, including the English used by people who are masters of mathematics.

Finally, a request for people who want to argue the contrary: Please preface your comments with "Isaac Newton was wrong about how to talk in English about mathematics, and I am right, because ...."

UPDATE: A comment, which regrettably failed to follow the eminently reasonable request in the preceding paragraph:

It's disappointing to see Mr. Volokh make this argument. The formulation is confusing. It could reasonably be argued that while 3X10 equals 30 then 3Xless than 10 would be a minus 20. People who use math in their work would never use the subject formulation. I thought it was limited to journalists.

First, it could reasonably be argued that "three times less than ten would be a minus twenty" -- if one has no idea about how actual humans talk. Of course no-one would use "three times less than ten" to mean that. Perhaps it's distracting or annoying, but it would take a lot to persuade me that anyone would actually think "'"If the Diameters of the Circles ... be made three times less than before'; does that mean 1/3 of the original, or negative two times the original?." Maybe Data, but then again he seems to have had some troubles with contractions, too.

Second, "people who use math in their work would never use the subject formulation"? Really? Might there be some evidence against this assertion available, I don't know, somewhere? I'm not sure, but I could have sworn I saw some ....

Related Posts (on one page):

  1. "Times Less Than":
  2. Ten Times Lower:
FantasiaWHT:
If I tried to literally translate "A times less than B" into a mathetmatical formula, my first thought would be

A x < B

Or put another way, A x X, where X < B.

Which is not a very helpful formula.
9.25.2009 12:51pm
Fedya (www):
If "three times less" means "one-third as much", then if two things are of equal amount, isn't one of them "one time less" than the other?

And I reject your argument from authority. Just because presidents of both parties have said nucular doesn't make it a correct pronunciation. Or to mix metaphors, if Sir Isaac Newton told you to jump off a bridge, would you do it? ;-)
9.25.2009 12:59pm
Bruce F. Webster (mail) (www):
"A times more than B" means (A/1) * B
"A times less than B" means (1/A) * B

People who get upset by this are, I think, largely people who struggle with mathematics in the first place. :-) ..bruce..
9.25.2009 1:01pm
Bleh:

Finally, a request for people who want to argue the contrary: Please preface your comments with "Isaac Newton was wrong about how to talk in English about mathematics, and I am right, because ...."


Thread has already been won.
9.25.2009 1:04pm
Eugene Volokh (www):
Fedya: What I don't understand is your definition of "correct[ness]." Current common usage doesn't qualify. The usage of some of the greatest scientific luminaries doesn't qualify.

You seem to be saying that "correctness" is defined by mathematical logic. But we're not talking about the rules of mathematics, or of gravity plus medicine (the subject of the bridge jumping hypothetical), which may follow such logic. We're talking about the rules of English; where's your support for the proposition that English is defined by mathematical logic? Isaac Newton didn't adopt such a proposition; why should we.

As to "one time less," no-one would say "X is one time less than Y," just as no-one would say "X is one time more than Y." But that just reflects the fact that English idioms, being idioms, can't always be stretched to all possible applications -- again, English is not mathematics.
9.25.2009 1:05pm
Laura(southernxyl) (mail) (www):
Bruce has it. Division is simply multiplication by the reciprocal.
9.25.2009 1:10pm
Dan Weber (www):
Saying "we have 10 times fewer murders than we did last year" is perfectly clear, since everyone who doesn't have a stick up his klein jar will know you mean "we have one-tenth the murders we did last year."

If people understand, effective communication happens.

It would probably work with "we have one-tenth times fewer murders," although that feels a little clunky. People would understand what that means.

I wouldn't use this in a computer program on either input or output, but for talking with human beings it's just fine.
9.25.2009 1:17pm
John Armstrong (mail) (www):
Honestly, us mathematicians are alternately amused and bemused by what the rest of you people think is "mathematically correct".

English is shot through with imprecision, which is why we talk in mathematics in the first place. If you want to pick nits, pick 'em. Just don't drag a whole field you have only the haziest conception of into the matter.
9.25.2009 1:18pm
Alex Blackwell (mail):
Interestingly, the Columbia Journalism Review had a brief article several days ago on how journalists often have trouble writing about mathematical relations.
9.25.2009 1:21pm
pot meet kettle (mail):

Please preface your comments with "Isaac Newton was wrong about how to talk in English about mathematics, and I am right, because ...."


Genuinely curious whether Newton was considered a particularly gifted explainer of scientific ideas in English. Haven't read Principia, but was it highly rated on this count, as opposed to being comprehensive and pathbreaking?
9.25.2009 1:22pm
Eugene Volokh (www):
I mention Newton chiefly because anyone who says that Newton was "mathematically incorrect" has a hard task ahead of him; I'm not saying that he's particularly elegant or gifted as a writer in English, just that he's unlikely to be "wrong" in how he talks in English about mathematics. Likewise, I mentioned several other scientists precisely to show that this has been common English usage among noted scientists -- not necessarily the most elegant or even clear explanation (though I see nothing at all unclear about it), but common enough among great scientists that it's hard to see how it could be "wrong."
9.25.2009 1:29pm
Anonymouse Troll:

Isaac Newton was wrong about how to talk in English about mathematics, and I am right, because

real English is American English, his good math stuff was written in Latin, and his beta-version English has some weird letter "s" that looks like an "f", goofy pronouns that sound like bible talk and distinct meaning for the words will and shall.
9.25.2009 1:32pm
kietharch (mail):
It's disappointing to see Mr. Volokh make this argument. The formulation is confusing. It could reasonably be argued that while 3X10 equals 30 then 3Xless than 10 would be a minus 20. People who use math in their work would never use the subject formulation. I thought it was limited to journalists.
9.25.2009 1:37pm
pintler:

It would probably work with "we have one-tenth times fewer murders," although that feels a little clunky. People would understand what that means.


Doesn't 'one tenth times fewer' == 'ten times more' :-)

It sounds like a great way for the mayor to spin a massive
crime wave.
9.25.2009 1:42pm
Kevin Lynch (mail):

Haven't read Principia, but was it highly rated on this count, as opposed to being comprehensive and pathbreaking?


The Principia, or Philosophiæ Naturalis Principia Mathematica, was written/published in Latin. And I don't remember it being highlighted as a particularly well written work in Latin.
9.25.2009 1:43pm
Fedya (www):
Prof. Volokh:

I don't like applying the word "correct" or "incorrect" to the "three times less" phrasing; I'd prefer something like "imprecise".

What I think [i]is[/i] incorrect is your use of argument by authority; my reading of your post is that you were the one arguing that the "three times less" phrasing is somehow correct because of Sir Isaac's use of it.

Dan Weber wrote:

Saying "we have 10 times fewer murders than we did last year" is perfectly clear, since everyone who doesn't have a stick up his klein jar will know you mean "we have one-tenth the murders we did last year."


True, but it would be more precise, and less open to misinterpretation in the general sense to say that murder is down by 90%. :-)
9.25.2009 1:45pm
Mike McDougal:

The formulation is confusing

No, it's not. You're a pedant with a lingering lust for your prescriptivist 7th grade English teacher.
9.25.2009 1:46pm
Soronel Haetir (mail):
Also on the topic of Newton, haven't his notations lost out? So he wouldn't even be able to describe modern mathematics correctly using the tools he had available.
9.25.2009 1:50pm
Snaphappy:
Paul's beaker is 9 degrees
Mary's beaker is 3 degrees
Jill's beaker is 1 degrees
Joe's beaker is -9 degrees
Jack's beaker is -15 degrees

The temperature of Mary's beaker is 6 degrees less than the temperature of Paul's beaker.

Which of the following is true?

A. The temperature of Jill's beaker is three times less than that. [i.e., 1/3 of Mary's]

B. The temperature of Joe's beaker is three times less than that. [i.e the difference between Paul and Joe's beaker (-18 degrees) is three times the difference between Paul and Mary's beaker (-6 degrees)]

C. The temperature of Jack's beaker is three times less than that. [i.e. 3x the delta between Paul and Mary, but starting from the temperature of Mary's beaker]



P.S. Isaac Newton was wrong about how to talk in English about mathematics, and I am right, because Newton stole the calculus.
9.25.2009 1:50pm
Arkady:


Fedya's argument is ____ times less likely to convince folks that Eugene's.

Scientific notation is allowed.
9.25.2009 1:50pm
Mike McDougal:
Snaphappy, I've never seen the "x times less" usage employed in the context of negative numbers. Why bother with it here?
9.25.2009 1:55pm
CDU (mail) (www):
People who use math in their work would never use the subject formulation.


I think Isaac Newton used some math in his work.
9.25.2009 2:01pm
Snaphappy:
The negative number isn't important. You could substitute 50 degrees and 48 degrees for the original number and 16, 42, and 40 degrees for the variations of "three times less than."

The answer to your question is that I originally started with a second number that was "three times less" than the first number. If you start out at 1/3, the second and third interpretations (3x the 2/3 delta, with different starting points) are always a negative number. I then realized that there was no need for the first number to be "three times less" than the first, but didn't change the numbers.

The reason I chose degrees rather than apples was that it would be better to say "fewer" rather than "less than" if it were apples.
9.25.2009 2:04pm
CDU (mail) (www):

Which of the following is true?

A. The temperature of Jill's beaker is three times less than that. [i.e., 1/3 of Mary's]

B. The temperature of Joe's beaker is three times less than that. [i.e the difference between Paul and Joe's beaker (-18 degrees) is three times the difference between Paul and Mary's beaker (-6 degrees)]

C. The temperature of Jack's beaker is three times less than that. [i.e. 3x the delta between Paul and Mary, but starting from the temperature of Mary's beaker]


None of them are correct. Unless you're using the Kelvin scale (which you obviously aren't, since you've got negative values), temperature data is interval data. When using interval data it doesn't make sense to use statements like "twice as much" or "three times less". In order for these sorts of statements to be meaningful, they need to be applied to ratio data: data that does not depend on a scale with an arbitrary zero point, like numbers of things, size, mass, etc.
9.25.2009 2:05pm
Snaphappy:
Whoops!

Paul: 50 degrees
Mary: 48 degrees

Mary is 2 degrees less than Paul.

Which is three times less than that?

A. Jill, 16 degrees (48/3)
B. Joe, 44 degrees (50 - 3*2)
C. Jack, 42 degrees (48 - 3*2)
9.25.2009 2:08pm
Snaphappy:
CDU: Okay then gallons. Now what say you?
9.25.2009 2:10pm
JJ799:
Snaphappy, the problem is that you're using the word "that" in a way where it can refer to several different things. This has nothing to do with the "times less than" usage: "that" is often ambiguous.
9.25.2009 2:24pm
Notmyleg (mail):
For people who argue that this is incorrect, I'm still confused as to why. Remember, this is about whether 10 times less is actually incorrect in standard English, not whether it is your preferred usage.

The mathematical claim (10 times less means something other than 1/10th in mathematics therefore it must mean that in plain English) would require that rules of English usaga be based on mathematics. While you may believe that is desirable, that doesn't mean it is how language actually works.

Perhaps that stronger claim is, "X times less means something else (or means nothing) if used in mathematics. Therefore, if it is used in English, it will be ambiguous which meaning is being employed." This, of course, is a simple claim of ambiguity. The problem with this is that it makes the common mistake of confusing potential ambiguity (this sentence can be understood in more than one way) with actual ambiguity (people are likely to understand this sentence in a way other than that which is intended, or in extreme cases, this meaning of this sentence is actually indeterminable). THe problem is, if you are attacking this for potential ambiguity, then nearly every sentence is going to be subject to criticism. The English language, and probably every language, has a great deal of built in potential ambiguity. This is eliminated through context and common usage (and other ways, like punctuation).

I would argue that X times less than is only potentially ambiguous. I would doubt that there are many native speakers who read that and actually think, "I don't know what this sentence means." I would argue that even those who are annoyed by it are able to immediately recognize the meaning, even if they find the usage disagreeable.

It is true that some people have argued against this usage (at least in the last fifty years), but this has been in use for at least ~300 years and is understood by virtually all native speakers. I think that places a heavy burden on the prescriptivists who claim it is not correct English.
9.25.2009 2:28pm
John Jenkins (mail):
Isn't "W is X times less than Y" suboptimal, not because it annoys people, but because it's less precise than it could be.

If what you mean is that W= Y/X, then why not say that expressly (e.g., W is one half of Y instead of W is two times less than Y)? The W=Y/X formulation is clearer and shorter, something that writers should generally strive for.

It really gets me when X is not a whole number (e.g., W is .70 times less than Y). No, Bruce, I don't have any problem understanding actual math, but this one always gets me. (Do they mean that W is 70% of Y? 70% less than Y?) Gets me every time.
9.25.2009 2:29pm
Notmyleg (mail):
ALso going to chime in to say that the temperature examples are particularly bad because the supposed "fix" doesn't work (which would be to say 1/3 as much instead of three times less). As noted above, this is because it only works on ratio data.

If it is changed to gallons, I agree, your use of "that" is ambiguous. I would no more know that "correct" answer if you phrased it in the "acceptable" form, "which is one third as much as that?"
9.25.2009 2:33pm
JJ799:
There should be no problem here. The logic behind the usage is that "more" and "less" are inverses, so if A is more than B, B is less than A. So if A is three times more than B, B should be three times less than A. It's completely natural and logical to reverse "X times more" and "X times less", and it fits nicely with the mathematical operation of an inverse.
9.25.2009 2:36pm
bbbeard (mail):
Finally, a request for people who want to argue the contrary: Please preface your comments with "Isaac Newton was wrong about how to talk in English about mathematics, and I am right, because ...."

Rubbish. Your quote is from Newton's Opticks, and in case you haven't heard, Newton didn't even get the physics right, much less the English (or the spelling!). He advocated a "corpuscular" theory of light, which we now know to be a woefully insufficient theory, albeit useful in limited circumstances. For an accessible treatment of the modern theory of light, I recommend "QED: The Strange Theory of Light and Matter" by Richard Feynman.

So, if Newton couldn't even get the physics right, why should we trust his English?

Looking at the page you linked, one finds the gem, "But in the Figure pt composed of the less Circles, the three less Circles ag, bh, ci, which answer to those three greater, do not extend into one another, nor are there any where mingled so much as any two of the three sorts of Rays by which those Circles are illuminated, and which in the Figure PT are all of them intermingled at BH." Dust thou thinkst it would be just fine for schoolchildren to learn to say "less Circles" instead of "smaller circles" -- because Sir Isaac did? Is it fine for them to learn punctuation and grammar from Sir Isaac, so that we may all capitalize important Nouns, and perhaps elide the terminal "and" in a sequence such as "ag, bh, ci"? Rubbish.

So: "Isaac Newton was wrong about how to talk in English about mathematics, and I am right, because ..."

(1) I was born in America in the 20th century,
(b) I got a "5" on my English AP exam, and
(iii) I was on the U.S. Mathematical Olympiad Team,
(IV) I have a doctorate in theoretical physics from MIT...

none of which Sir Isaac could claim, and none of which matters one whit to this argument!

So much for the Argument from Authority.

If you're comfortable with "3 times less", are you equally comfortable with "twice as less"? Eleanor Wilson Orr wrote a book with that title about a decade ago, explaining in great detail how defective grammar leads to defective mathematical understanding.

BBB
9.25.2009 2:37pm
Lior:
I think a better phrasing would be to consider the relation "X is A times larger than Y", to which no-one objects.

We would now like to express the opposite relation (by which I don't mean the negation, but the relation going the other way). Since the opposite to "X is larger than Y" is "Y is smaller than X", it is natural to say that "Y is ten times smaller than X" if and only if "X is ten times larger than Y".

Now the two statements are mathematically equivalent, but they are not equivalent rhetorically -- we usually perceive them as statements about the first among the two quantities mentioned. Similarly, "X is ten times less than Y" and "X is one-tenth of Y" are mathematically equivalent yet are not the same rhetorically.
9.25.2009 2:48pm
Notmyleg (mail):
I do have a problem with "!twice as less," but probably not the same problem as you. My problem is that it is, at best, an extremely non-standard usage. Someone reading it will probably understand what the person meant, but will also think the person probably doesn't speak English, because we would say "two times less." My problem has nothing to do with mathematical understanding, or being able to parse the sentence in a field outside of English, it has to do with the fact that it isn't something people say.

If, however, it had been a common usage for the past three hundred years, and everyone used and understood it, I probably wouldn't care. The book you point to identifies exactly the problem with the argument though. The statement that "defective grammar leads to..." begs the question. Maybe the book makes a better argument, but this seems to say "twice as less cannot be understood in mathematics, therefore using it in English will detract from your understanding of math, therefore it is ungrammatical in English." It may be ungrammatical, and using it may detract from understanding of math, but the two are in no way related.
9.25.2009 2:50pm
donaldk2 (mail):

I always dismissed it as a contemptible solecism created by and copied by ignorant writers. And, since "10 times less than" can't sensibly mean anything other than "one tenth as much," why not just say that?

"times" is part of the terminology of "increase"; "less" is part of "decrease." Putting them together is at best sloppy.

Must admit I knew nothing of this usage by pre-eminent scientists.

IF it was theirs. Maybe it must be credited to the translators.
9.25.2009 2:55pm
Snaphappy:
My problem is that the formulation "times less," when used as suggested, has a comparative term without anything to compare it to. There is no original "time" that is "less than" any other. I posited an original time that is 2 gallons less, and asked what's three times less than that? So let's try again, without the word "that":

Paul: 50 gallons
Mary: 48 gallons

Mary has 2 gallons less than Paul.

Who has three times less than Mary?

A. Ann, 16 gallons (48/3)
B. Bob, 44 gallons (50 - 3*2)
C. Cal, 42 gallons (48 - 3*2)

This is why you would never say "twice as less" to mean "half." Because one would rightly be confused as to what one was meant to double.
9.25.2009 3:01pm
Eugene Volokh (www):
Notmyleg: I entirely agree (and so would Horace). "Ten times less" reports over 1 million Google hits; "twice as less" reports fewer than 6000, of which most of the early hits simply refer to the book with that title. The former is common idiom, which reflects its clarity, reflects its compatibility with related English usage (and thus its natural fit into how English speakers speaks), and makes it quite unlikely that people will be confused. "Twice as less" is highly uncommon, lacks a direct analog that's a quick clue to meaning (we don't say "twice as more," and the analog to "twice as much" is "twice as little"). It is thus nearly certain to be jarring, and is probably somewhat more likely to be confusing.

The English language is full of idioms that don't correspond to the literal meanings of the individual words. My favorite example is "ice cream," which literally refers to iced cream, not cream made out of ice. That doesn't make the idioms wrong; it just means that you can't import the rules of a foreign means of communication -- Latin, mathematics, computer programming, and so on -- to understand them, and especially the rules of a means of communication that generally doesn't include idioms.
9.25.2009 3:01pm
Snaphappy:
Eugene: I think you have been asked this before, but do you think the number of google hits is really a measure of anything? You cite it an awful lot. "Pwned" has 2.6 million hits, for example. Should that make it into the OED?
9.25.2009 3:22pm
Lior:
Mathematics does include idioms; they are just not obvious to non-mathematicians. Suffice it to say that the following is grammatical: "Let R be a well ordering of the set X".

I think "twice as less" is rare, since "half" has been with us before written language. "One tenth" is a far more sophisticated concept and does not have such a long history.
9.25.2009 3:23pm
NotMyRealName:
This blog post is faintly ridiculous. Come on, the important issue is not whether "three times less than ten" is correct or incorrect; the important issue is whether "three times less than ten" is the best, clearest, least-confusing way to express the concept. I say that it is not. I say that it is a confusing way to write, and there are clearer alternatives.

Newton, Schnewton. Gimme a break. Spend some time reading how folks wrote about math at the time, and then look me in the eyes and tell me you honestly want to see everyone write that way today. Their writing style was often imprecise, confusing, and/or hard to read. I'd hardly point to that archaic style as the best model to follow today.
9.25.2009 3:24pm
SeanF:
I don't like "x times less," but I won't call it incorrect because, in English, common usage makes correctness.

The reason I don't like it, though, is this:

The difference between "three times more" and "four times more" is the same as the difference between "four times more" and "five times more."

However, the difference between "three times less" and "four times less" is not the same as the difference between "four times less" and "five times less".

That seems to suggest that "times" somehow means something different with "less", that one "time" is not the same as another "time" - and one "time" is never the same as the original number, which it always is when used with "more".
9.25.2009 3:44pm
Richard A. (mail):
What perturbs me is the variant on this enforced by the Miss Thistlebottom school of English usage that calls for "fewer" to be used in the formuation "few than one" rather than "less than one." The theory is that fewer should be used with discrete quantities, i.e. "I have fewer apples than he has." But you can't have fewer apples than one. At that point, you ain't got no apples and "less" is just fine.
9.25.2009 3:52pm
SeaDrive:
Does "half as less" mean the same as "twice as less"?


"Twice As Less" is a book, as it happens:
http://tinyurl.com/ya6vdap

I've read it, and the author has a little trouble parsing the meaning. IIRC, I was somewhat frustrated that she didn't talk to some of the students who used it to find out what they meant.

Anyway, this thread is all the evidence necessary to suggest it's a confusing usage.
9.25.2009 4:02pm
Henry T:
I'm not sure where kietharch's data comes from, since, as a mathematician, I can confirm that I see nothing wrong with "ten times less than." The claim that it's unclear is simply unjustified, and everything thus far posted to that effect has demonstrated only that there are particular ways it could be used which would be unclear, not addressing that the overwhelming majority of uses are clear.
9.25.2009 4:06pm
Jon Roland (mail) (www):
Bruce F. Webster:

"A times more than B" means (A/1) * B
"A times less than B" means (1/A) * B

Is essentially correct, but what is interesting is the origin of this usage, which is common to other languages than English. It arises from an era when most people were acquainted with multiplication, but not with division, and unfamiliar with the notion of "reciprocal". Most people today find that difficult to imagine, but just listen to the talk of children after they learn multiplication and before they learn division.

Incidentally, among mathematicians one does occasionally get the usage that translates into

A * [-∞, B)

where * == multiplication operator and
[ ...) == half-open interval, that includes the lower endpoint but not the upper. The above expression =

[-∞, A*B)
9.25.2009 4:10pm
rc:
EV: "Please preface your comments with "Isaac Newton was wrong...""

Oh snap!

But here's a counterexample for anyone who's interested. "Orders of magnitude less" An order of magnitude is a ten-base exponential multiplyer, so shifting orders of magnitude is effectively a timesing operation.

So is it wrong to say 'orders of magnitude less?' Cause if it's wrong, I don't wanna be right.
9.25.2009 4:21pm
SeaDrive:
rc:

As I'm sure you know, "order of magnitude less" means "the order of magnitude is less", as "order of magnitude" is the exponent of 10 in the scientific notation. It's the same as "ten times less". We allow it, but I think it's a colloquialism.

None of these things are as confusing or likely to be wrong as expressions including the word "percent."
9.25.2009 4:51pm
Jon Roland (mail) (www):
If n = the order of magnitude by which it is less, then that translates into

X *= 10^-n

which is computer notation for "X is replaced by its value multiplied by 10 to the -n power".

Hope that is clear. :)
9.25.2009 4:56pm
TNeloms:

NotMyRealName:

This blog post is faintly ridiculous. Come on, the important issue is not whether "three times less than ten" is correct or incorrect; the important issue is whether "three times less than ten" is the best, clearest, least-confusing way to express the concept. I say that it is not. I say that it is a confusing way to write, and there are clearer alternatives.



I disagree.

Dealing with fractions ("1/3 as much as before...") is less intuitive for many people than dealing with whole numbers ("3 times less than before...").

It's also much harder to say if you're dealing with a larger number, e.g., "three hundred fifty one times less than before..." versus "one three hundred fifty first as much as before..."
9.25.2009 5:19pm
Arkady:
@Eugene


Notmyleg: I entirely agree (and so would Horace). "Ten times less" reports over 1 million Google hits; "twice as less" reports fewer than 6000, of which most of the early hits simply refer to the book with that title.


However, "twice as few" gets 2,740,000 hits.

Finally, on our language, in general.


Do not be troubled by the fact that languages (2) and (8) consist only of orders. If you want to say that this shows them to be incomplete, ask yourself whether our language is complete;---whether it was so before the symbolism of chemistry and the notation of the infinitesimal calculus were incorporated in it; for these are, so to speak, suburbs of our language. (And how many houses or streets does it take before a town begins to be a town?) Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular streets and uniform houses.


Wittgenstein, Philosophical Investigations, para. 18.
9.25.2009 5:50pm
ohwilleke:
I was a mathematics major as an undergraduate. I've been clean and sober for seventeen years since then.

One clear way to think about the expression is that "times" in the phrase "times less" is a unit modifier.

You have five less soldiers less than a full brigade (i.e. 1495)

You have five score soldiers less than a full brigade (i.e. 1400)

You have five hundred soldiers less than a brigade (i.e. 1000)

You have five fold soldiers less than a full brigade (i.e. 300)

You have five times soldiers less than a full brigade (i.e. 300).

I also like the notion that "times" used without modification is really an abbreviation of "times more than". There is never a circumstance when "times" unmodified means something different than "times more" unmodified. And, this makes "times more than" and "times less than" inverse operations of each other.

In the same vein, without the "times" part the phrases "more than" and "less than" can be understood as inverse operations of each other rather than simply different operations than each other.

The historical dating is also interesting. The authorities cited and the way that the phrase is used suggest that the useage predates widespread use of the concept of negative numbers.
9.25.2009 5:55pm
Snaphappy:
So one last thing: I have always thought that the word "times" made a kind of sense in that it works the same, mathmatically, as it does when meant to mean "instances." So if I give you four marbles, three "times," I give you the same number as "three times four." If I give you four marbles, then do it "three times more," then you have received four marbles a total of four "times," that is, four times four.

It doesn't work the same way with "three times less." If I give you 9 marbles one time, it's not possible to give someone else 9 marbles "three times less." Giving someone else "three times less" doesn't make any sense because I only have you marbles one time.

But if I gave you one marble 9 times, and gave someone else one marble 3 times less, then the second person gets six marbles, not 3.
9.25.2009 5:58pm
Snaphappy:

There is never a circumstance when "times" unmodified means something different than "times more" unmodified.


I went to the water fountain three times today.

You would have me say that I went to the fountain three times more. More than what?
9.25.2009 6:01pm
ohwilleke:
Also, FWIW, while Newton's ideas were good, his expression of those ideas was frequently clunky, which has the side effect of dramatically slowing down mathematical thinking. His writings are hard for a modern mathematician to read. Doing calculus the way Newton did it is like trying to do arithmetic using Roman rather than Arabic numerals.

All modern mathematician carry out calculus using the notion methods of his co-inventer of calculus Gottfried Wilhelm Leibniz. Leibniz introduced several notations used to this day, for instance the integral sign ∫ representing an elongated S, from the Latin word summa and the d used for differentials, from the Latin word differentia. This ingenious and suggestive notation for the calculus is probably Leibniz's most enduring mathematical legacy.
9.25.2009 6:03pm
Fred the Fourth (mail):
BOR-ING.
Let's complain about improper uses of "quantum", "organic", and "parameter" instead. Oh, and since I'm OT already, how about "hone in on" instead of "home in on". And "factoid" to mean "little fact" instead of "something that looks like a fact, but isn't".

And back on topic: Since (as SeanF pointed out) one of the items being poorly communicated with this idiom is proportion, I want to register a complaint about journalists who seem to instinctively choose the least-informative representation whenever given a choice between an absolute number or increment, vs. a percentage or proportion.

It's Friday after lunch and I'm off my meds...who can focus on work?
9.25.2009 6:07pm
ohwilleke:
"So much for the Argument from Authority."

Generally, the Argument from Authority is not a logical basis for scientific proof. But, language is more like law than science. Sometimes "authority" is what makes a linguistic useage right. Useage dictionaries have used this principle for centuries.

The reason that most people in the world speak languages that are mutually intelligble by millions of people is that the general public has had an intuitive preference for dialects used by those in positions of authority. A dialect can be correct, while another can be incorrect, in a social context, precisely because of the relative authority of the speakers.

The speakers of Sanscrit has so much authority that their linguistic practices had a great deal of influence on which dialects of Dravidian were viewed as most correct, even though Dravidian languages aren't even derived from Sanscrit. Most modern languages arose from the dialect favored in royal courts. These days, television and radio personalities have supplanted to role of genuine power holders to some extent. One of the most influential style guides today, a book making calls on what is "correct," is published by the Associated Press.

As an exception that proves the rule, some of the distinctive elements of the upper class New England American dialects of English, like its non-pronunciation of the letter "h" at the beginning of a word, were specifically adopted after the American Revolution in defiance favored pronunciation in the English court at the time.
9.25.2009 6:15pm
Alex S. (mail):
I don't have a problem with it at all when the meaning is unambiguous.

If Mars has a diameter of 10,000 miles and you make it 3 times less it is clearly impossible that the diameter is now -20,000 miles and so he must mean reduce to 1/3 the original value.

However, where it tends to annoy me is in a formulation like this: "This year's drop in violent crime was three times less than last year's drop of 3%." I honestly could not say whether they mean crime had dropped 1% or 9% (let alone the discounted possibility they actually mean it rose 6%).

Plus there's the fact that I can't begin to predict whether the speaker is a pedant on this issue. So while I don't get worked up about it unless personally I try to keep it a phrasing that has minimal ambiguity (and won't run me afoul of usage nags).

So I'd say "If the Diameters of the Circles ... be made one-third that they were before" or "If the Diameters of the Circles ... be made 66% less than before" and "this year's drop in violent crime was 9% while last year's drop was 3%."

And if I heard Newton saying any of this, I'd try to touch his wig and giggle like a schoolgirl and not care at all about his phrasing. Then I'd tease him about all that alchemy stuff.
9.25.2009 6:22pm
Bruce Hayden (mail):
<blockquote>A times less than B</blockquote>I am still struggling with all of this. My natural tendency was to do the "A * \< B", which makes absolutely no sense. And, I don't think this mathematical naivety, since I do have a degree in mathematics, and quite a bit of engineering after that. Not the most experience here, by far, but also likely not the least. But, then, much of my engineering type experience was in software, and that I think colored how I interpreted this - of course, most compilers would kick it out as ambiguous.

And, if I tried to make sense, it would be multiplying A by some number that is less than B, or, probably more accurately, the set of numbers less than B (which, presumably, if both A\>0 and B\>0, and we are talking real numbers, means all the numbers less than A*B).

My thoughts are that if you want to do reciprocals, then call them that, and if you want to divide A by B, then call it that.

p.s. Interestingly, this blogger interface has real problems with less than signs in comments, as they are the start of a HTML command.
9.25.2009 6:54pm
NotMyRealName:
Some people seem to think that "three times less" is the clearest, least confusing usage. What's wrong with "one-third as much"? Or "the price was reduced by a factor of three" instead of "the price was three times less"? Or any of a number of other replacements. I submit that, in most cases, there will be an alternative that is clearer and has less potential for confusion.

ohwilleke has nailed the argument why referring to Newton is unconvincing.
9.25.2009 7:10pm
CheckEnclosed (mail):
Was this the use of a repost as a riposte?
9.25.2009 7:30pm
einhverfr (mail) (www):
From the update

People who use math in their work would never use the subject formulation. I thought it was limited to journalists.


Because Isaac Newton was a journalist, not a mathematician....
9.25.2009 7:43pm
einhverfr (mail) (www):
Snaphappy:

All of your statements about temperature are meaningless unless using a true temperature scale like Kelvin ;-)
9.25.2009 7:49pm
einhverfr (mail) (www):
notmyleg:

Remember, this is about whether 10 times less is actually incorrect in standard English, not whether it is your preferred usage.


Every single linguistic textbook which I have ever read (and I read as many of them for fun as I do court opinions, and I read enough court opinions for fun to hang out here....) starts off with the basic statement that natural languages are defined by use.

Technically a widespread construction cannot possibly incorrect in a natural language!
9.25.2009 7:52pm
Fedya (www):
Arkady wrote:

Fedya's argument is ____ times less likely to convince folks that Eugene's.


Zero, of course. :-)
9.25.2009 8:14pm
Harry Schell (mail):
Been around engineers too long.

"Ten times less" is nonsensical, even if people compensate for the failure and recognize the speaker is trying to say B is only 10% of A.

Noam Chomsky is a linguist, and one of the stupidest people I have ever read. He really thinks Marx wsa onto something, and has never changed his views despite the manifest failure of marxism as a political or economic system to provide the "liberation" or "abundance" or "justice" it promises. When an idea or set of them have a 100% failure rate...that would be, er...I dunno...but usually if one can think, one decides to stop trying it.

Progressives enamoured of marx are regressives, trying to sell bad failed ideas from the past as if they will work, this time.

Stepping back onto the thread, the linguistic books can cram it. Ten times less than something doesn't make sense mathematically, and so the comment is misleading, wrong or foolish.

No wonder kids can't do math, they have adults throwing foolish comparisons like this at them.
9.25.2009 8:15pm
rc:
'x times' is the preferred language of scale. Says me.

I think that "x times less" is a useful and understandable phrase to use when you're saying that less is actually more.

'x times less' is good language when you're expressing the factor by which the small is better than the large.
9.25.2009 8:52pm
rc:
Arkady: "Fedya's argument is ____ times less likely to convince folks that Eugene's."

Fedya: "Zero, of course. :-)"

I frown upon arguments that divide by zero.

Here's a picture of Fedya
9.25.2009 9:05pm
Jerry Mimsy (www):
I use this a lot when describing how to optimize images and other files for the web.

If the savings are small, I will usually say "the optimized file is now one-sixth the size of the original, and will download six times faster."

If the savings are larger (which is very common), though, that "seems wrong"; it seems clunky and likely to require multiple readings. So I use the "times smaller" phrasing; I'll say "the optimized file is now 80 times smaller than the original, and will download 80 times faster."

Isaac Newton was right about how to talk in English about mathematics, because... he agrees with me.
9.25.2009 9:16pm
einhverfr (mail) (www):
Harry Schell:

Noam Chomsky is a linguist, and one of the stupidest people I have ever read. He really thinks Marx wsa onto something, and has never changed his views despite the manifest failure of marxism as a political or economic system to provide the "liberation" or "abundance" or "justice" it promises. When an idea or set of them have a 100% failure rate...that would be, er...I dunno...but usually if one can think, one decides to stop trying it.


What does that have to do with linguistics principles which have remain largely unchanged for over a hundred years.

If you don't like Chomsky, try reading George Steiner (he can't stand Chomsky's linguistics and falls more in the Whorf/Sapir camp).
9.25.2009 9:41pm
D.O.:
As math teacher, let me ask this is 3 times less than -30 is
A) -10
B) -90
C) something else?
9.25.2009 10:31pm
Eugene Volokh (www):
"Nonsensical." Isaac Newton, talking nonsense. Herschel, talking nonsense. Locke, talking nonsense. Harry Schell knows better! Might there be anything wrong with this picture?
9.26.2009 1:11am
Eugene Volokh (www):
D.O.: My sense is that people would rarely use either "3 times less" or "3 times more" with negative numbers, precisely because for negative numbers, dividing by 3 yields a result that's greater than the original and multiplying by 3 yields a result that's less than the original. But that just shows that the English idiom "3 times less than X," as opposed to the mathematical formula "X/3," does not neatly fit all possible values of X.
9.26.2009 2:28am
D.O.:
Prof. Volokh, thank you for earnest reply. The question was a joke! I find myself sometimes saying "more negative" about a number, which is less than another negative number, which is not very common, but useful.
9.26.2009 4:52am
NateM (mail):
Prof. Volokh—

If you say "times more than" or "times less than," your audience is less likely to understand you, and more likely to have to struggle to understand you, than if you'd used some other formulation.

I think it's no accident that all your citations are from a very long time ago. I've been hanging out in technically-minded circles most of my life, and I can't remember ever hearing someone use "times more than" or "times less than." (Although, when a subject is implicit, "... times more" seems occasionally felicitous.)

Radical descriptivism, combined with a willingness to comb through centuries of English written by smart people, will lead you to some strange linguistic beliefs. If I added every Humean or Lockean formulation to my stock of English, my writing would look ridiculous. I have a responsibility to my listeners and readers, and that responsibility is incompatible with my helping myself to any locution any smart English user ever published.

P.S. — Of ~course~ I know some things that Newton didn't know about how to write scientific English... in 2009. How could Newton have known what English would be in 2009?
9.26.2009 11:13am
rc:
I don't think that this phrase is orbit-haltingly confusing.

'Times' is the preferred word to represent factor or scale. Its acceptability to express comparison is much better than percentages, and I think better than fractions on those occasions where 'less is more'- where you're expressing the smaller number as the better number.

So a thing is smaller, and better... by what factor? It's 'X times less than.'

It may be a bit cumbersome, but it's a bottom-line point-making phrase, not a math-handy calculation phrase.
9.26.2009 11:29am
einhverfr (mail) (www):
NateM:

I think "three times more than" and "three times less than" are perfectly understandable in their normal usages.

Next, I await the great debate over whether the following sentence is incorrect in English:

"Who did you meet yesterday?"
(Gotta love cases like these, pun intended)
9.26.2009 11:31am
NateM (mail):
einhverfr--

I might be quite wrong about whether the phrase is felicitous. It struck me as very odd, but it can be hard to retain a sense of usage when you know you're reading a blog post about usage.

As smart and reasonable as Prof. Volokh is, I just don't think extreme descriptivism is tenable. (David Foster Wallace's "Authority and American Usage" is excellent on this subject.) If "times more than" is acceptable, that is because the phrase is a part of correct contemporary English, not because Newton used it.
9.26.2009 11:57am
einhverfr (mail) (www):
NateM:

Extreme descriptivism is a practical necessity in fields like historical linguistics. The reason is that language usage changes over time and drifts relative to its roots. It is also important in comparative linguistics because various dialects may have different rules (see the habitual tense in African American Vernacular English for example).

I think that what you are arguing is "effective style" rather than "correct usage." This has to be separated or else we would all be reprimanded every time we say "So, who did you meet yesterday?" This is because the prescriptive approach would be to use the "objective" (merged dative and accusative) instead and say "So, whom did you meet yesterday?" But that just sounds wrong. Interestingly Edward Sapir in 1912 predicted the imminent death of "whom" in informal communications.

"Effective style" is prescriptive. This is what we REALLY want our schools to teach. However it is not the same thing as "correct usage" which is descriptive. IMO we should separate these issues.
9.26.2009 1:00pm
NateM (mail):
einhverfr--

Thanks for responding so helpfully.

I'm not sure I exactly see your point. Language definitely changes. Languages and dialects definitely have different rules. I don't see how any of that decides the question whether, given a language at a time, correct usage is to be determined descriptively or not. One can be a prescriptivist and hold that 2009 English uses "who" in the accusative/objective.

Using English and using English well are certainly two different things, but nothing you say shows that correct usage is descriptive. If I use Hume's English, I might fail not only to use English well but to use English--contemporary American English--at all.
9.26.2009 1:25pm
ralph:
I am an engineer. I used to write regulations for nuclear power plants. I would NEVER say "3 time less than", especially with regard to temperatures. I would say instead that the second temperature was " X degrees less" than the first temperature. I would say that the second pressure or the stress in a member was "one third" or "33 percent" of the first, NEVER 3 times less. This is a horrid construction, gauranteed to create confusion, and it should be absoulutely be BANNED, and the people who use it should be burned at the stake.
9.26.2009 1:42pm
ralph:
And, if any government lawyer had insisted on the "3 times less" construction, I would have died in a ditch to ensure that he/she did not get their way. And I would have won, in the end.
9.26.2009 1:46pm
einhverfr (mail) (www):
Nate M:

Using English and using English well are certainly two different things, but nothing you say shows that correct usage is descriptive.


So, do you begin business letters with "To who it may concern?" If we accept an objective (i.e. accusative/dative) "who" as a prescriptionist, then that should be the correct, accepted form. However it is not. The phrase we accept has "whom" in it.

Once again, I think this is a matter of effective style rather than correct usage. If I try to avoid the problem above by addressing my letter "To whoever it may concern" that certainly doesn't sound jarring but it isn't effective style either.

Similarly dialects which use double and triple negatives aren't necessarily wrong in their usage. We don't use them in formal communications because they don't sound very educated so once again it is a stylistic rather than a syntactic rule. On the other hand, "My aunt said he wanted to have us over for dinner" where "he" in context refers clearly to "my aunt" IS incorrect because it violated deeply held and pervasive gender checks built into the language, and it is not an accepted use of "he."

I think George Steiner ("After Babel") said it best, that language is fundamentally a simplified model of the world and that it necessarilly contains some ambiguity. I would add to that the notion that natural language is usually autochthonous and as such one doesn't have the same strictness in syntactic rules that one has with a computer language.

Why does this matter?

Consider the Ebonics debate. The debate centers largely around whether AAVE is "bad or lazy English" or whether it is an interesting dialect which is hence a part of our national heritage. To a prescriptionist, the former seems most appealing and to the descriptionist the latter seems most appealing. I certainly hold with the latter.

However, this doesn't mean that the concerns about what we teach our kids magically goes away just because we say "It's an interesting and important dialect." The worry (which is well justified by the way multiculturalism is handled elsewhere in our education system) that this will undermine rather than support critical education is a valid one if we continue with a prescriptionist framework exclusively.

However, a solution can be found in being more honest about what we want kids to learn in English class. Instead of teaching "correct English" we want kids to learn how to be able to write and speak in a certain dialect and style which we believe best prepares kids for economic success later in life.

Now, one other point relative to this forum. Legalese is a little different. It is far more prescriptive than modern English simply because of its hyperdefined nature and the effect of error. Similarly invented languages or dead languages are more prescriptive simply because they are tied to a specific static authority separate from the speakers.
9.26.2009 1:54pm
eyesay:
Some raised the issue in EV's post of three years ago that "times more" is itself ambiguous. If city A has three times more murders than city B, is that three times as many murders as city B, or four times as many? If you think it means three times as many, explain this: "City A has 300% more murders than city B" clearly means that city A has four times as many murders as city B. How can "three times more" mean anything different from "300% more"?

As someone who thinks that "three times more" means "three times as many," I guess the answer is: because with "percent more" expressions, by convention, no change or difference is 0% more, but with "times more" expressions, by convention, no change or difference is 1 time more; also by convention, these expressions are used only when there is non-zero change or difference, so one doesn't actually say "0% more" or "1 time more" to express no change or difference.

But, for the reason that "three times more" is inconsistent in meaning with "300% more," I think expressions of the form "three times more" should be avoided, and a fortiori, expressions of the form "three times less" should be avoided.

Also, "one-third as many" and "one-third less" mean very different things. If I hear that city A has one-third as many murders as city B, and city C has one-third less murders than city B, then I calculate the murder ratio of city A : city B : city C = 1 : 3 : 2. I assume that "three times less" is intended to mean "one-third as many" (or "one-third as much") and not "one-third less."
9.26.2009 2:09pm
PubliusFL:
Hmmm. 10 is one-tenth as much as 100. "Less than" is the inverse (multiplication by the reciprocal, as Laura put it) of "as much as." Therefore, 100 is one-tenth less than 10!
9.26.2009 3:24pm
NateM (mail):
einhverfr--

Thanks again for the response. You say a lot that's worth responding to, but quickly first: there's absolutely no reason that a non-descriptivist would have to hold that Ebonics is "bad or lazy English." She probably would have to hold that AAVE is different from some other English dialect.

For myself, I think it's very difficult to hold that AAVE is "bad English." AAVE is no less rule-governed than [pick another form of English], and dismissing it as "bad English" leaves one unable to account for AAVE mistakes as mistakes.

It's quite common for people to saddle prescriptivism (which might not be the best term for all anti-descriptivist views) with all sorts of crazy views that are no part of it. If prescriptivism committed me to the view that AAVE is bad English, I'd run away screaming from it [that is: from prescriptivism, not from AAVE].
9.26.2009 3:41pm
Bill Poser (mail) (www):
I have read a good deal of mathematical writing but nonetheless find this phrasing confusing and suspect that most other people do too. It isn't the clearest way to express the idea. The fact that Isaac Newton used the construction doesn't actually bear on whether it is clear to the modern reader. He wrote over three hundred years ago, is not known as a particularly clear writer, and for that matter, wrote most of his scientific work in Latin.
9.26.2009 3:51pm
rc:
I don't think we should apply the logic of inequality algebra to the English language. Very un-PC.
9.26.2009 3:59pm
Gunstar1 (mail):
Those that are saying that "times less than" is confusing or just wrong are people that are used to reading things from a certain field of study where the point of that writing is to convey an exact detailed meaning/understanding.

I hate to break it to the engineers and mathmaticians, but there are people in the world that do not see math in their heads when reading something. The general scale is all that is being referenced in saying things that way, not the exact to-the-decimal-point number.

My bet is that if you asked people who find reading engineering or math articles boring as hell would find absolutly nothing wrong with "times less than".

The ones that dont like it are the ones that read a sentence and automatically figure out the word problem in mathmatical form. In this case "times less than" does not conform to the basic word problem form and this is what bothers them about its usage.
9.26.2009 6:33pm
pgepps (www):
While math textbooks do tend to teach students a very wooden application of language like "times" and "less than," the language itself is not so hard to sort.

If you realize that the usual phrase is not "A times B equals C" (using "times" as an operator in an expression of the relationship of A and B to C) but that "times" refers to iterations or instances, you'll be fine.

So that "if I multiply B A times" would be an older form of our more compact expression, and "if I divide B A times" would also be quite sane. These yield the mathematical expressiong (A * B) and (B / A) which we gloss "A times B" and "B divided by A"; note the asymmetry of the translation, with division retaining its term, but multiplication losing its.

Hence the confusion of those who forget that "I can go to the store three times" is perfectly intelligible without any algebra in play.
9.26.2009 10:06pm
Aaron Denney (mail):
No, of course one would not write "one times (more/less) than x". But "1.1 times" or "0.9" times would come up, and they do confuse.
9.27.2009 12:03am
rc:
Aw, dang it. By posting snark and sarcasm on this issue, it's been on my mind enough to actually think about it.

And after a bit of thought, I find 'times less than' to be awkward and confusing.

People's posts on the internet actually caused me to change my mind. I apologize- it'll never happen again.
9.27.2009 10:15am

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