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In english based math language it seems that

non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing)

decreasing $\Longleftrightarrow$ strict less ( strict decreasing)

Is that correct ? If so, how does it make sense ?

precision

I should note that even very good math teacher are making mistakes about this. Actually I asked this question after watching Boyd's video on convex optimization where even him is confused about this.... So I imagine many many people are, and there must be classes and tests about this absurd and buggy concept, which yields absolutely nothing interesting.

So I just wonder if I really am missing something, or if, yes, some people decided to create an abstraction that is leaky (not not increasing $\neq$ increasing ?) verbose (4 words, with special negation logic, instead of using the word 'strict' and keeping the usual well defined predicate logic rules)

absurdity of the concept

This notation is absurd for the following reason :

  • when dealing with element instead of functions we dont apply the same logic :

    we dont phrase $x < y$ as "$x$ is less than $y$" nor "$x\leq y$" as "$x$ is not-more than $y$". (If we did though, at least it would not be so harmful as not not-more would mean more)

  • you have to define functions using a not notation, $f$ is non-increasing function $\Longrightarrow$ if $x$ is not-less than $y$, say 0.3 feet and 2.5 inches, then $f(x)$ is not-more than $f(y)$

This also violates a very basic tenet in programming style 101, which is here for a reason : never define or use something with a negation, it is confusing.

  • To apply composition rules between functions, you better be buckled up with all the not. must be a fluff of cases

More profoundly, this violates a fundamental principle of logic which is that given some ambiguity, you should assume the most general case apply.

It is way worse than measuring things with non integral units. This is violating logical rules, and leaving a very basic concept obfuscated.

Community
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nicolas
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  • seems like an answer. – nicolas Mar 03 '12 at 08:53
  • OK; I've posted it as an answer. – joriki Mar 03 '12 at 08:56
  • Sorry for the deletion. Pierre asked the question "Is |x| not increasing?" on my answer. To nicolas: why do you think the absolute value function is, in your words, "not 'not-increasing'"? – anon Mar 03 '12 at 09:09
  • Dear @anon: Thanks! No problem. I had already reposted my comment below joriki's answer. – Pierre-Yves Gaillard Mar 03 '12 at 09:12
  • @anon are you saying the absolute value function is 'not-increasing' ? – nicolas Mar 03 '12 at 09:12
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    I don't think the language is absolutely standard, and you need to pay attention to the context and to the conventions announced by the lecturer or author. Note also that a differentiable function can be strictly increasing without having a positive derivative everywhere (zero is possible). – Mark Bennet Mar 03 '12 at 09:15
  • Dear nicolas: Out of curiosity: Which languages do you speak? – Pierre-Yves Gaillard Mar 03 '12 at 09:17
  • Perhaps one should say that $x^2$ is not monotonic on $\mathbb R$... – Pierre-Yves Gaillard Mar 03 '12 at 09:19
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    @Pierre-YvesGaillard french, perfect english, small amount of spanish, german, italian. But when I speak "math", consistency is not optional !! And I can not begin to imagine the torture the young students are submitted to with this non sense ! it is way worse than the imperial measure system. I really thought I had missed a point at first because I could not imagine someone as badly wired as to invent such a thing. And even if he existed, he would have been stopped by someone.. – nicolas Mar 03 '12 at 09:23
  • @MarkBennet indeed, the strictly positive with non strictly positive derivative can be confusing. but it is a fact of life we can't go against. such pitfall make it all the more compelling to not introduce another layer of obfuscation. – nicolas Mar 03 '12 at 09:28
  • @Pierre-YvesGaillard Those 'concepts' only apply to monotonic case. good call (I had forgotten that notion). from there we can introduce the logical NOT operator, NOT to be confused with the 'not' used in not-increasing for instance, and retain compositional predicate logic. – nicolas Mar 03 '12 at 09:34
  • Note that there are contexts in which "$x$ is not more than $y$" is not equivalent to $x\leq y$. For example, if $A$ is a set, we can put a "partial order" on the subsets of $A$, by saying that $x\leq y$ if and only if $x$ is a subset of $y$. If we do that, then "$x$ is not more than $y$" means "$y$ is not properly contained in $x$", but this does not yield "$x$ is a subset of $y$." For example, if $A={1,2}$, $x={1}$, and $y={2}$, then "$x\leq y$" is false, but "$x$ is not more than $y$" is true. – Arturo Magidin Mar 03 '12 at 21:43
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    Current usage is muddled. When writing, one way to avoid confusing anybody is to use "strictly increasing" and "non-decreasing" or "weakly increasing", and avoid the unadorned "increasing". There's no need for confusion. In French, there is a convention that most everybody follows, but there's no reason for an non-mathematician who speaks fluent French to know the convention, so I don't see that the situation is inherently better. – Stephen Mar 03 '12 at 23:07
  • @steve Using only "strictly increasing" and "weakly increasing" will be my personal rule to avoid spreading the disease. The convention that, unless specified otherwise, increasing, or any other word, do relate to weakest definition compatible with the said word seems only plain sound logic to me. If we allow ourselves to say that a word implies something other than the weakest concept compatible with it, then chaos ensues, as each person will claim it should be assumed that his own "stronger" version of the concept should holds implicitely. – nicolas Mar 04 '12 at 14:01
  • As a side question, is it a modern convention, or something from before ? What concerns me the most is younger people learning that. Weak minds will just be confused and drop off learning stuff. Those those with common sense will rightfully think that their teacher are a joke. – nicolas Mar 04 '12 at 14:06
  • I think this is as good for math as the imperial system is for physics : absolutely nonsensical – nicolas Jan 21 '14 at 18:30
  • I disagree that this is bad for math and I would say there is a very simple explanation for this system - we don't want to say that $0$ is both positive and negative and we also don't want to say that constant functions are both increasing functions and decreasing functions. Saying $0$ is both positive and negative might be excusable to some but certainly calling a constant function both increasing/weakly increasing and decreasing/weakly decreasing is undesirable. How can something be increasing and decreasing at the same time? Nonincreasing and nondecreasing avoid this issue entirely. – Jürgen Sukumaran Jun 22 '21 at 22:38
  • @TSF beside the confusing downside on one of the most basic construct in math (no pb with constant function being increasing and decreasing + pb on non non-decreasing etc.), it is logically inconsistent : If I give you a theorem with some degree of ambiguity, it should hold for the most general case of whatever I did not specify. – nicolas Jul 11 '21 at 20:00
  • It is logically consistent, nondecreasing and nonincreasing have specific meanings but it doesn't make it logically inconsistent. Language like non nondecreasing is virtually nonexistent and can be addressed still. For native english speakers it's clear that nonincreasing and nondecreasing are the more natural convention. If your boss says they're going to increase your pay and it stays constant is that okay? If the dr tells you to decrease your weight do you think they mean stay at a constant weight? The current convention reflects how the words increasing and decreasing are ACTUALLY used. – Jürgen Sukumaran Jul 11 '21 at 22:11
  • The way of assigning meaning (not the meaning itself) is inconsistent : If something is not specified, the most general case should be assumed. About the meaning, I doubt anybody was ever confused about a function being both concave and convex (the english books I have follow the logical way of giving a meaning to those words - though most are russian inspired). As for your examples, definitions in math have to be amenable to consistent abstraction. Has anybody actually said a flat table was convex ? Probably not. Is the definition of convex correct and amenable to consistent abstraction ? yes – nicolas Jul 19 '21 at 14:22

3 Answers3

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Personally I find this among the most awful terminology in existence. It starts with the ambiguity present in "increasing" and "decreasing" themselves: common sense would have that this means getting ever larger/smaller; yet (if I take Wikipedia as reference) both the terms monotonically increasing function and monotonically increasing sequence allow for (local) constancy. (It seems unlikely that the purpose of "monotonically" is to weaken the notion following it; rather it seems to indicate that a formally defined rather than colloquial notion is meant.) So if there is doubt about what a bare "increasing" meant, the proper remedy would be to always accompany it with a disambiguating "weakly" or "strictly"; this would settle the matter.

For some reason however many people seem to find that "nondecreasing" is preferable to "weakly increasing". I work a lot with integers partitions, which most authors introduce as nonincreasing sequences of integers (with finite sum). Clearly what is meant here is not the absence of "monotonic increase" between successive integers, since that would imply strict decrease. One might conclude that when using negative terminology, people implicitly revert to the colloquial rather than formal meaning of the base notion. For comparison, even here in France, where "négatif" is taken to include $0$ (as does "positif"), few people would be willing to interpret "entier non-négatif" as designating integers${}>0$.

However, even apart from the fact that negation does nothing to remove ambiguity from a notion, there are other drawbacks specific to this case:

  • Nonincreasing is not the negation of (strictly) increasing for sequences of length${}>2$, and should therefore be carefully distinguished from "not increasing". The sequence $0,1,-1,2,-2,3,-3,\ldots$ is all of "not increasing", "not decreasing" and "not constant"; however, it is neither of "nonincreasing" nor "nondecreasing", but it is "nonconstant". A nice mess.
  • In the presence of partial ordering, having "nonincreasing" mean "weakly decreasing" is even less justified; here weak decrease is stronger than the absence of strict increase even for sequences of length $2$. I think what is needed in such context is almost never "nonincreasing", even between successive elements. For instance a "plane partition" could be defined as a weakly decreasing sequence of partitions (for the containment-of-diagrams partial ordering); saying "nonincreasing" here would be utterly confusing.

If one must absolutely use negative terminology, then it would have been much better to use "nowhere increasing" rather than "nonincreasing" (and even then only for total orderings).

In conclusion: if you want to be precise, it is better to say what you mean rather than to say what you don't mean (or even to not say what you are nonmeaning).

Marc van Leeuwen
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8

There are two possibilities:
* increasing and strictly increasing
* nondecreasing and increasing
Someone who switched the terminology from one to the other perhaps did it thinking that the more common notion should have the shorter name. But not everyone has switched terminology, so now we have the two systems existing side-by-side, which is, indeed, confusing.

Similar situations:
* nonnegative and positive
* positive and strictly positive

also
* $A \subseteq B$ and $A \subset B$
* $A \subset B$ and $A \subsetneqq B$

GEdgar
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  • Dear GEdgar: $+1$! But the problem is that, if you use the "nondecreasing and increasing" possibility, then most of you functions will be neither decreasing nor nondecreasing. – Pierre-Yves Gaillard Mar 03 '12 at 13:12
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    Got it. there are 2 systems. As for myself, I definitely favor the one that preserves the (not (non (xxx)) = xxx. The promotors of the other system are not patriots. I suspect KGB. – nicolas Mar 03 '12 at 13:37
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    Wait. when one says "A is included in B" in english math, it might mean strict inclusion ? – nicolas Mar 03 '12 at 13:39
  • following the 'other' terminology, it would make "A weakly included in B" be phrased as "A non-including B", while the "non-including" itself is a non total order, I can imagine the weird sets of 'rules' that one has to devise... – nicolas Mar 03 '12 at 13:46
  • Is this terminology a recent thing, or a persistence of the past ? – nicolas Mar 03 '12 at 13:49
  • Dear @nicolas: This situation is the only instance I know where "not not x" does not mean "x". Example, for $x$ real, "$x$ is not non-negative" means "$x$ is negative" (as it should). – Pierre-Yves Gaillard Mar 03 '12 at 13:51
  • @Pierre-YvesGaillard. good to know, it is only here. I can move on. – nicolas Mar 03 '12 at 13:55
  • @Pierre-YvesGaillard I would not un-hold it against you – nicolas Mar 03 '12 at 14:02
  • Dear @nicolas: I hope what I said is right, but I'm far from being sure... (I mean, I'm far from being sure that there no other examples of this not-not thing.) (I’d say it’s a persistence of the past.) – Pierre-Yves Gaillard Mar 03 '12 at 14:04
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    A function is called "nonnegative" iff $f(x)\ge 0$ for all $x$; and is called "negative" iff $f(x)<0$ for all $x$. So, again, saying $f$ is not nonnegative is different than saying $f$ is negative. – GEdgar Mar 03 '12 at 17:11
  • @GEdgar and this is insane. that's not a consistent way to deal with ambiguity. The only consistent way to deal with ambiguity is to assume that the reality described by it is the weakest form. This is logic 101. – nicolas Mar 04 '12 at 14:09
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It might be easier to say something helpful if you explain why you think it doesn't make sense. It makes perfect sense to me -- "non-increasing" means it doesn't increase, i.e. doesn't become greater, i.e. stays equal or becomes less; "decreasing" means it becomes less -- those are the standard meanings of the words in everyday language, and it seems it's the other convention, the one that uses "decreasing" for "non-strictly decreasing" and "strictly decreasing" otherwise, that's in need of justification because it departs from the everyday usage of the words.

About your edited question: It sounds as if you're assuming that there's one standard convention and people who don't follow it are confused and making mistakes. In my experience there are two different conventions in use, one where "increasing" means "non-strictly increasing" and one where it means "strictly increasing". It seems to me that it's just a matter of taste whether you'd rather have shorter words ("non-strictly increasing" being a bit verbose) or whether you want to stay close to the everday usage of the words ("increasing" in everyday usage meaning "strictly increasing").

joriki
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    If the word 'strict' has any use, I can't think of any better use than not having to use 2 different words to designate the same concept.... – nicolas Mar 03 '12 at 08:57
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    Is "V" (i.e. $|x|$, that is, the absolute value function on $\mathbb R$) not increasing? (In French we don't have this problem.) – Pierre-Yves Gaillard Mar 03 '12 at 09:08
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    @Pierre: I'm not sure I understand the question. It's not increasing, but also not non-increasing. Your question made sense with respect to anon's answer because that said "not increasing", but I didn't use that expression, so I'm not sure what you're asking about. About French: One could either say you that don't have that problem or that you don't have that distinction :-) Anyway, these adjectives are usually not used predicatively ("the function is not increasing"), where they may seem ambiguous, but attributively ("a non-increasing function"). – joriki Mar 03 '12 at 09:26
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    @joriki re french, and I guess in many, many other languages, 'decreasing' relates to x <= y => f(x) <= f(y). and 'strict decreasing' to x < y => f(x) < f(y). this ecompasses all that zoology of special cases.. – nicolas Mar 03 '12 at 09:40
  • @joriki : what does 'inferior' means in english math ? is it < or <= ? – nicolas Mar 03 '12 at 09:41
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    Dear joriki: Thanks! I know that this is not a crucial issue, but I find confusing the fact that a function is in general neither increasing nor non-increasing. It would be easier if "non-increasing" meant "not increasing". In "French", we use "increasing" and "strictly increasing", and the meanings of "not increasing" and "not strictly increasing" are clear. (Same with decreasing.) – Pierre-Yves Gaillard Mar 03 '12 at 09:45
  • @nicolas: I've never seen "inferior" used in this context. About your previous comment: I believe this is also by far the more widespread convention in English; e.g. Wikipedia mentions only this convention. So I doubt whether this is a problem particular to English. The only time I've ever seen the other convention used is this question: http://math.stackexchange.com/questions/34796/injective-function-and-ultrafilters. – joriki Mar 03 '12 at 09:48
  • @joriki I mean, Are the name in english math for "x < y" and "x <= y" "non-superior" and "inferior" ? or inferior and strictly inferior ? – nicolas Mar 03 '12 at 09:54
  • @Pierre: My French is rather rudimentary, but I would have though you actually have the same distinction there; it might just seem clearer because the word order is different and the negations sound more different. Wouldn't "n'est pas croissante" mean "is not increasing" and "est non-croissante" mean "is non-increasing"? – joriki Mar 03 '12 at 09:56
  • @nicolas: I can only repeat, as I said, I've never seen "inferior" used in this context (nor "superior" or "non-superior"). – joriki Mar 03 '12 at 09:57
  • @joriki there must be some english expression for "x < y" and "x <= y".... – nicolas Mar 03 '12 at 10:00
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    @nicolas: There is: "less than" and "less than or equal". – joriki Mar 03 '12 at 10:01
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    Dear joriki: I don't think it has anything to do with languages, but rather with terminology tradition. For instance "positif" means $\ge0$, whereas "positive" (English) means $ > 0$. Also, you mention "everyday usage of the words", but in "everyday usage of the words", "not black" (say) means "of any color other than black". For instance, a red object is "not black". (Again, it's not so important.) – Pierre-Yves Gaillard Mar 03 '12 at 10:10
  • @joriki ok so it is not "less" and "non-more". another inconsistency within english-math. I mean it's better otherwise it can go nuts with all the subcases. – nicolas Mar 03 '12 at 10:15
  • @nicolas: I don't know what books you've been reading and I didn't watch the video you mentioned; I can only emphasize again that by the most prevalent convention in English is to use the terms "increasing", "strictly increasing", "decreasing" and "strictly decreasing". To disambiguate, one sometimes also uses "weakly increasing" or "weakly decreasing". People sometimes also use "non-increasing" and "non-decreasing", but these are not the most prevalent terms, so I don't see where the inconsistencies are that you keep emphasizing. – joriki Mar 03 '12 at 10:20
  • @joriki so the meaning of "increasing" depends on the person who says it ? (I mean, I understand, the 'non-increasing' stuff is flawed) – nicolas Mar 03 '12 at 10:24
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    @joriki anyway, at least there are some commonly known non ambiguous words for this notion. "weakly increasing" and "strictly increasing". I will use them. thank you. – nicolas Mar 03 '12 at 10:26
  • @nicolas: As I said above, there is a prevalent usage; it's the only one mentioned in the Wikipedia article and the only one I'd seen used until recently; but in the question I linked to above, someone used it differently and André in his answer then adopted that usage; and apparently you've also seen it used like that (else presumably you wouldn't have asked this question). So, yes, it seems to depend on who uses it, and it doesn't hurt to add "strictly" and "weakly" to disambiguate, but I believe the convention where "increasing" means "strictly increasing" is by far the more prevalent. – joriki Mar 03 '12 at 10:44
  • One of my teachers used to say, "If you get a salary increase then you expect to get more money next payday." We obviously used $f(x) < f(y)$ in his course. – Patrick Mar 03 '12 at 21:54
  • @Patrick I am horrified to hear that. that an ambiguous word might might refers to something other than the weakest form, unless otherwise specified, is just not logical. – nicolas Mar 04 '12 at 14:13
  • @nicolas: I think you need to work on that attitude. This is a matter of definition, preference and convenience, not of logic. Many of your comments are very harsh and judgemental; I don't see any basis here for anything other than personal preferences and find your insistence on labeling usages other than the one you prefer as bad/horrifying/obfuscating/weird rather inappropriate. – joriki Mar 04 '12 at 15:32
  • @joriki except that it is a matter of logic. the fact that you dont recognize it as such just underline the extent of the damages done. – nicolas Mar 04 '12 at 15:51
  • @joriki in the presence of a term, whatever it is, increasing or else, the only consistent way is to assume the weakest form. If I tell you I have a "licence", you should not assume I have a doctor's licence, nor a driving's licence, and you should fear I am james bond. nothing personal, just logic. – nicolas Mar 04 '12 at 16:15