A reader writes, apropos my Slate piece,

Canada cannot have crime rate ten times lower than the crime rate in the U.S. It can have crime rate of one tenth the crime rate of the U.S.

I've heard this objection before, but I'm just not sure I understand its foundation. Indeed, if "X times lower" meant "lower by X times the original amount," "ten times lower" would make little sense (except if for some reason you said that -9 was ten times lower than 1). Or if you somehow just defined "times lower" as an error, it would be, by definition, an error.

What I don't grasp is what justification, besides the objector's own view of what the language should be, there is for this. This is, to my knowledge, a common part of normal English usage. It's not confusing. It's not illogical unless one defines "lower" in a pretty strange way.

Even if one is a prescriptivist, who argues that a statement is bad English if it doesn't conform with The Authorities, what Authorities actually condemn this? There might be some, and please let me know if there are, but I just don't know of any, and my quick and dirty search didn't come up with any.

Naturally, if you think that this usage is ugly or annoying, I can't really argue with that. And if enough people think that, one might want to avoid the usage simply to avoid annoying one's readers. But I took the "cannot" to mean "cannot, without violating the rules" rather than "cannot, without annoying me." Where are those rules set down?

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 ....