"Times Less Than"

A Facebook comment 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.

But somehow this logic was lost on, say, 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? My sense is that many people's objections rest on an assumption that English is, or ought to be, like math. 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.

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 …."

Comments

[Martinned]

Isaac Newton was definitely wrong about how to write and talk in 21st century English in many ways. More generally, all math texts from before the 19th century or so are pretty illegible, as far as I know.

But A times less than B means a multiplication by A^-1. I don't think that really confuses anyone.

[Michael P]

I probably wouldn't care much about this particular usage as long as people stopped writing "three times more [than X]" to mean 3X rather than 4X.

[Martinned]

Now that seems like a genuine ambiguity. I could read three times more as reasonably referring to either 3X or 4X. (E.g. by analogy to my previous it would be 3X.)

[Rossami]

No. Just as "10% more" is always 110% of the original, "three times more" is always 4x. If you want to say 3x, you just say "three times". As Dilan implies below, sometimes the translation from math to english (and they are most definitely two different languages) is ambiguous - but that translation rule is not.

[Martinned]

I agree with you that the ambiguity is less likely to arise when you say it with percentages, but I'm telling you that many people are likely to disagree about the meaning of "three times more". You don't have to like it, but it's true.

[Sue Holt]

I agree with commenters' discomfort with this phrasing. It's similar to what astronomers say about the early days of our moon, that it was "10 times closer" to Earth. I can't wrap my brain around that description -- first I'm told "10 times" which makes me think of a very big distance, and then "closer", so it's a smaller distance? This is verbal whiplash!

[Absaroka]

How about 1e-10 vs. 1e-20?

[Don Nico]

A is 5 times less than B simply means that A*5 approximately equals B. That is what Newton meant.

[MatthewSlyfield]

Which would necessarily mean that A approximately = B/5

[Drewski]

It troubles me because it's a bad habit. People who use that phrasing also tend to use it when they are trying to communicate an actual result, and frequently are so used to the construction that they literally cannot understand why it is unclear for that purpose. When it matters, language can and should be used to convey the appropriate level of precision.

This is analogous to the idea that laws should not be written ambiguously. People can still write ambiguously! Just not when they're writing laws. And if a lawmaker habitually writes in vague terms,we may worry that they'll do so when it matters.

It's not a personal crusade of mine or anything - I'd never mention it unless I needed to know the result or were proofreading technical writing. But it troubles me when I see it in real numerical contexts. It scans better when used metaphorically or hyperbolically.

Of course, as is the case with phrases such as, "I could care less," many people who use it are signaling that they're looking for a slapfight, sometimes because their point is weak but sometimes because they just love Internet slapfights.

Also, Isaac Newton was a lunatic. He was a genius (not least of all for his invention of the cat flap), but he was completely off the rails more often than not. For all we know, he may have believed that writing out fractional quantities of real materials numerically would interfere with the alchemical interactions.

[lucia_l]

>When it matters, language can and should be used to convey the appropriate level of precision.

When you really need to talk about math, you need to use formulas.
I tried to tell this go a copy editor at work once, and she insisted that (a) formulas confuse too many people and (b) it's always possible to state thing precisely in words.
She also hated what she saw as redundancy.

It may be true that it is always possible to state mathematical ideas in words, but it can get very, very wordy.

(a) "The ratio, B to A."
(b) "B times less than A".
(c) " B/A";

By her rules, writing "the ratio B to A, B/A." was redundant. So was "B times less than A, B/A". And just "B/A" was abhorrent because it had no words and one should always state things in words. But no, version (a) sets some people googling to be sure they haven't forgotten the word they learned in elementary school, (b) gets people arguing. Almost no one misunderstands "B/A".

Multiplication is less error prone because it commutes:

If you have to explain something that requires calculation, you are often better off just inserting a formula. It used to be difficult to typeset them, it's easy now. And in my view, a little redundancy is not always a terrible, horrible, very bad thing.

[Michael 11]

By her rules, writing “the ratio B to A, B/A.” was redundant. So was “B times less than A, B/A”.

The second one is wrong, not "redundant". B times less than A is A/B.

[lucia_l]

Your right... I should have written "A times less than B, B/A".

[Dilan Esper]

What's going on here? My sense is that many people's objections rest on an assumption that English is, or ought to be, like math.

I think that's at the root of all of these situations where people swear there are "rules" that do not in fact exist. Some people think that language is supposed to be a logical system.

But there's literally no reason why you should expect language to be logical. It is an evolved system of communication; like any evolutionary system, traits that are in some sense useful are selected over time. And that includes a lot of usages that ordinary people find useful and whose meanings they understand, even if they are not "logical".

[Rossami]

Very well argued. And more clearly than I could have done so.

[CJColucci]

Many so-called "rules," however seemingly logical, are nonsense considered as rules. Nevertheless, it is often a good idea to follow them as practices, and almost never a bad one. For example, the "rule" against splitting infinitives is nonsense, but there is still no positive case for the general practice of splitting them. Even the occasional sentence that sounds better with a split infinitive usually can be rewritten to sound even better than either the split or un-split version. (And the real problem with the Star Trek intro isn't that the last infinitive is split, but that the parallel structure of the sentence is thereby disrupted: to find...to seek out...to [boldly] go. I submit that "to go (pause) boldly (pause) where.." would retain the parallel structure without annoying people who believe in the "rule" against split infinitives or just object to the momentary disruption of the normal expectation that "to" will be followed immediately by a verb.)

[Dilan Esper]

To the contrary, I think "to boldly go where no man has gone before" is far better than not splitting the infinitive, "10 items or less" is far better than "fewer", etc. Breaking these "rules" clearly improves the language.

But more broadly, I would also argue that breaking these "rules" is also a good in itself. The people who put out these "rules" are pernicious. They are snobs who constantly criticize the way others write and talk so they can feel superior. They are putting forth a fundamentally wrong view of language that does not understand its nature. They need to lose.

And the way for them to lose is for us to loudly, repeatedly, intentionally break their "rules" and use correct English rather than what they wrongly think to be correct English. To isolate them, so they have no cultural purchase. To humiliate them. To have the best speakers of English out there repudiating them and repeatedly demonstrating that they are wrong.

[Á àß äẞç ãþÇđ âÞ¢Đæ ǎB€Ðëf ảhf]

Amen! What "to boldly go" gets right is how it scans, its rhythm. "To go boldly" plods along, something a bureaucrat would recite from a teleprompter.

Same with "10 items or less". Using "fewer" ruins the rhythm.

One of my peeves is abandoning "lit" in favor of "lighted". "A well-lit room" just sounds better, it maintains the rhythm in ways that "a well-lighted room" destroys.

[David Nieporent]

Same with “10 items or less”. Using “fewer” ruins the rhythm.

It's an instructional sign, not poetry. And if you were used to seeing "10 items or fewer" it would seem entirely natural.

[Á àß äẞç ãþÇđ âÞ¢Đæ ǎB€Ðëf ảhf]

No it would not seem natural. And neither assertion can be proved or disproved, so both assertions are meaningless.

Perhaps you just have no poetry in your soul. Perhaps you are an automaton. I don't know and don't care.

[Bwaaah]

Rhythm, cadence, meter...Yes!

[CJColucci]

Is it more important that they lose or that readers have their reasonable expectations satisfied?

[Dilan Esper]

The expectations you refer to are not reasonable. Anyone who is reading prose looking for split infinitives is being UNreasonable and should be treated as such.

[CJColucci]

Ninety-odd percent of the time, both in writing and in speech, people don't split infinitives, not because of some stupid rule, and not even because they're thinking about it, which they aren't, but because that's the natural way to talk or write in idiomatic English. Hence the expectation, eminently reasonable, that "to" will be followed immediately by a verb. When it isn't, there's a momentary hiccup. Not a huge deal; it is, after all, a hiccup and not a hemorrhage, but better to be avoided if you're more concerned about your reader than your umbrage at grammar snobs.

[Noscitur a sociis]

Yet another rule scrambling for a justification. The only people who have ever experienced even a “hiccup” at that construction are the people who have consciously trained themselves to look for split infinitives. And when those people read or hear it, they get to pat themselves on the back for spotting the putative solecism. So really, everybody wins.

[CJColucci]

If that's true, why don't people, most of whom have never heard of the "rule" against splitting infinitives and wouldn't care about it if they had, split infinitives all the time?

[Noscitur a sociis]

People do split infinitives all the time.

[Absaroka]

From the Font:

“In colloquial speech, the construction came to enjoy widespread use. Today, according to the American Heritage Book of English Usage, “people split infinitives all the time without giving it a thought.” In corpora of contemporary spoken English, some adverbs such as always and completely appear more often in the split position than the unsplit”

(their teddy bear example shows why)

[lucia_l]

Even the occasional sentence that sounds better with a split infinitive usually can be rewritten to sound even better than either the split or un-split version

Sounds better to whom? People who hate split infinitives? I think the split infinitive form often sounds much better.

And I agree with Dilan Esper that "to boldly go" sounds far better than the version you like. And there nothing wrong with breaking a parallel structure. Parallel structure in the Start Trek sentence would suggest all seeking, finding and going are equally important. Breaking that structure emphasize that the going is more important than the finding or seeking. You can find with a telescope. The main purpose of the Enterprise is to boldly go.

I don't care one bit if people who are annoyed at split infinitives are annoyed. They can learn to deal with their emotions as all adults should.

[Michael 11]

That's not relevant here; the construction already is logical, so it's pointless to observe that it doesn't have to be.

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."

This point of view is incompatible with the idea that saying "three times more" is fine, which was stipulated in the post.

But again, there is no problem in the logic: three times more than X refers to increasing ("more") X by a factor ("times") of three, and three times less than X refers to decreasing ("less") X by a factor ("times") of three.

[Stephen Lathrop]

I wonder if Herschel took account of the fact that perceptions of sensory stimuli do not scale identically to mathematical variations in the strength of the signals. For instance, experts in hearing perception insist that a mathematical doubling of sound intensity is nowhere near enough to get reported as twice as loud.

[John F. Carr]

When I was in school we were taught mechanical rules to assist in turning word problems into mathematical formulae. The rules were not sophisticated enough to distinguish "times" from "times less".

Also when I was in school one was not supposed to use a double negative, because two negatives make a positive. One was supposed to use "whom" as the objective case form of "who". And to address a woman properly one had to know if she was single, married, or a radical feminist.

[Á àß äẞç ãþÇđ âÞ¢Đæ ǎB€Ðëf ảhf]

Then there's the old joke about a grammar professor droning on about double negatives, how some languages allow it, some don't, but no language allows a double positive.

The wiseacre in the back responds, "Yeah, right."

[SRG2]

If something is a common idiomatic usage, arguments against the usage from logic or grammar are wrong.

[Paladin_44]

This is the type of arrant pedantry up with which I will not put.

[Á àß äẞç ãþÇđ âÞ¢Đæ ǎB€Ðëf ảhf]

Little girl asks her father to read a bedtime story to her. He goes downstairs, brings back a book, and she is dismayed, it's not the right book.

"Why did you bring this book that I don't want to be read to out of up for?"

[Don Nico]

How about a better choice of words:
"pedantry which I will not tolerate."

Constructions which "violate the rules" often occur because the writer or speaker is either lazy or has a limited vocabulary. English is an extremely rich language; use it and enjoy the diversity of nuance.

[Á àß äẞç ãþÇđ âÞ¢Đæ ǎB€Ðëf ảhf]

What makes that better?

The point of the chosen words is to make fun of the pedants. They do it so well that you took it literally.

[Absaroka]

Or they are, for example, a gifted orator like Churchill, more concerned with the flow of what they are saying than rules of grammar.

You don't take poetry and start diagramming sentences.

I'm less interested in whether Winnie split an infinitive or ended a sentence with a preposition, or Yogi Berra used a double negative, than whether or not their message arrived intact in the minds of their audience.

[MatthewSlyfield]

"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.)"

I wouldn't call it confusing, but it fells awkward to me.

[David Nieporent]

I would call it confusing. I'm not sure what a person who says it means by it.

[Barry Sullivan]

Isaac Newton WQAS wrong.

First of all, trying to shame people who disagree with you by a blatant "appeal to authority" (when you argue Isaac Newton did it -- makes no sense logically given that Einstein's Theory of General Relativity thoroughly trashed Newton's Theory of Universal Gravitation. Wrong is wrong no matter who is wrong. Just because Newton got a lot of stuff right does not make him omniscient.

Second, your construct of "A times Less Than (or Fewer Than) B" does not hold up in all cases -- so it is inherently ambiguous. What about the case of "One times less than B"? Under your construct does that mean zero? So, if "One times less than B" is zero, how could "Two times less than B" also not mean zero? How can you subdivide zero beyond itself?

By the way, the term you seek is "innumeracy." The point of using any language is to convey a thought with precision. Using "A times less than B" is innumerate.

[Á àß äẞç ãþÇđ âÞ¢Đæ ǎB€Ðëf ảhf]

Einstein’s relativity did not thoroughly trash Newton’s gravity. He merely improved it for marginal conditions.

You’ve been told a million times to stop exaggerating.

Secondly, your humor detection circuits have failed.

Thirdly, you need to apologize for your ugly typoe.

Fourthly, no one says "one times less". It doesn't make any sense no how no way.

[Don Nico]

" He merely improved it for marginal conditions."
Actually he extended it for strong field conditions." Here, "marginal" is incorrect.

[Á àß äẞç ãþÇđ âÞ¢Đæ ǎB€Ðëf ảhf]

Marginal is entirely correct. The vast majority of gravity calculations have no need for relativity's increased accuracy. The minority remaining is indeed marginal.

[BrianL.]

Try getting the GPS system to work accurately without relativity.

[Artifex]

What part of …

The vast majority of gravity calculations have no need for relativity’s increased accuracy.

… is this in response to ? Are you proposing that designing new GPS systems is what the common man does on a daily basis ?

Maybe I am simply misreading the concerns and there is a business opportunity here. Your household scales use straight Newtonian calculations and lack real relativistic correction. I would be perfectly happy to add some hardware and compute to provide a more accurate answer at a couple thousand dollars a pop. If you decide not to take me up on this, maybe you really do understand Random Alphabet guys point at some subconscious level even if you can’t articulate it.

[Squirrelloid]

"1 times less than B" isn't zero, it's not a valid construct. (I mean, I suppose you could say it's B/1 = B, but there's no reason to ever *say* that).

[Squirrelloid]

Multiplication and division are *the same operation* in principle, just like addition and subtractions are *the same operation* in principle. (Division by x is multiplication by 1/x, subtraction of x is addition of -x).

So if A less than B is B-A, then A times less than B should be B/A, by the same logic. I see no issue here (besides using english to describe what would better be described by an equation, but that’s a style preference).

[Martinned]

For the record, much of this is giving me flashbacks to the time when I once spent an inordinate amount of time examining ground rent increase clauses in leasehold deeds. Almost without exception the relevant lawyers took a reasonably clear formula and rendered it as a paragraph of complicated text, thus assuring that almost nobody would ever be able to understand it again. All with the goal of making it completely clear and unambiguous.

[CEF]

Interesting to use a quote from a book called "Opticks" to inform modern English usage. The title alone should give pause.
And, realistically, scientists are frequently *horrible* at communication. Why on earth would you use them as an example?

I have a bigger beef with "X times more" than with "X times less." If I have $10, and you have "5 times more" than I do, do you have $50 (5 times what-I-have) or $60 (5 times what-I-have more than what-I-have).

[Absaroka]

'"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 …."'

Classic!

[mydisplayname]

Oh, if Edwin Newman was still alive (https://www.npr.org/templates/story/story.php?storyId=129883697), he would chastise me for saying saying "I know someone who plays the piano." "Which piano," he might ask, due to my inclusion of the article "the."

[Nestor33]

Along these same lines: I was once asked chime in on a boisterous and spirited discussion among bar patrons, who asked "if it's twenty degrees in Boston, and it's twice as cold in Burlington, how cold is it in Burlington?"

[Absaroka]

Heh, that's a good one.

I'm not sure what people who object to '3 times less' would prefer - perhaps 'a third'?

If so, that doesn't work for temperature either: 'Burlington is half as cold' seems even worse that 'twice as cold', because if it's half as cold, presumably it's warmer 🙂

[lucia_l]

“if it’s twenty degrees in Boston, and it’s twice as cold in Burlington, how cold is it in Burlington?”

Well…. Let’s see. It’s the US. So you must be using F. But I think these calculations should be done in absolute. So, convert to R, divide the “coldth” by 2, then convert back to F. > (20+492.67)/2 – 492.67 [1] -236.335 It’s -236.3F in Burlington. I would stay away from Burlington.

The problem with “twice as cold” or even “half as cold” or hot or whatever, is that “cold” is not numerical or countable. You usually can’t even make sense of a statement like “the temperature is 90% of yesterday’s temperature”.

[Absaroka]

"I would stay away from Burlington."

Wear your long johns, anyway 🙂

My first thought was to base 'cold' at 32, so 20 is 12 'degrees of cold', so twice as cold would be 32-24=8 degreesF, which probably works intuitively for New Englanders. But then someone from Florida would argue that cold starts at 60, and people from Wyoming would be all '20 degrees? That's warm!'.

[lucia_l]

Absaroka
-236.3F is going to require some super warm long johns!

Relative temperatures (i.e. temperature differences) actually make more sense when discussing "cold" and "warm". For my calculation, I just picked the somewhat ridiculous absolute zero as my reference temperature. It has an aura of objectivity, but clearly gets a number no one from Vermont would call -236.3F only "twice as cold" as 8 F. I agree some Vermonter might mean what you suggested.

I tend to think of temperatures below 60 as "cool/cold" and above 80 as "warm hot". But other people have different ideas.

[Squirrelloid]

I mean, most people don't say it's cold until it's below what? 65? 60? 50? idk. Let's assume 60.

So a straight up literal interpretation would put it at -20 in Burlington.

However, people don't *perceive* coldness linearly. I'd interpret such a statement to mean the temperature in Burlington is in the ballpark of 0 degrees.

(All temperatures in F).

I mean, cold isn't an objective quality. (Temperature measures heat. Cold isn't a real thing, it's a perceptual thing). So you end up guessing at what would *feel* twice as cold.

As a rigorous question, yeah, its dumb. But it's not rigorous, and I think there is something there as a casual statement which is understandable, even if you can't turn it into a temperature measurement.

[Davy C]

It's clearly on a log scale. Every 10 degrees is twice as cold.

[NvEric]

Is this post to see how people reason on non-legal topics ?

Constructing words into sentences is for others to understand one's thoughts as much as it is for the writer too. Reading aloud also helps the writer, for it adds another way in, or means of, or avenue to, or pathway for, or conveyance in an alternate method of transmission not only for the receiver, but more importantly the writer too. Hearing is by a different process in the brain and can be insightful to fathom thoughts.

I know my constructs can be atypical, but the reader must not read in only one way as writers must not write in one way either. Why ? Because language can never be exacting enough for everyone to understand and more so over time. Reading words written by various writers over many time spans helps one to better understand the various ways in which language is used.

Deciphering each others words is how we understand each other. Please take the time to do this and we'll all be better off for it.

[Bwaaah]

A headline from the Washington Post this week was something like, “If inflation is coming down, why are prices still so high?”

All I could think was, “What a great question!”

(I didn't bother to read the article.)