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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.

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  • $\begingroup$ seems like an answer. $\endgroup$
    – nicolas
    Mar 3, 2012 at 8:53
  • $\begingroup$ OK; I've posted it as an answer. $\endgroup$
    – joriki
    Mar 3, 2012 at 8:56
  • $\begingroup$ 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'"? $\endgroup$
    – anon
    Mar 3, 2012 at 9:09
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    $\begingroup$ 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). $\endgroup$
    – Mark Bennet
    Mar 3, 2012 at 9:15
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    $\begingroup$ 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. $\endgroup$
    – Stephen
    Mar 3, 2012 at 23:07

3 Answers 3

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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).

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    $\begingroup$ Great answer!!! $\endgroup$
    – Pierre-Yves Gaillard
    Mar 3, 2012 at 14:07
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    $\begingroup$ I did not find this answer nonhelpful. $\endgroup$
    – Stephen
    Mar 3, 2012 at 23:12
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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 $A \subsetneqq B$

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  • $\begingroup$ 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. $\endgroup$
    – Pierre-Yves Gaillard
    Mar 3, 2012 at 13:12
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    $\begingroup$ 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. $\endgroup$
    – nicolas
    Mar 3, 2012 at 13:37
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    $\begingroup$ Wait. when one says "A is included in B" in english math, it might mean strict inclusion ? $\endgroup$
    – nicolas
    Mar 3, 2012 at 13:39
  • $\begingroup$ 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... $\endgroup$
    – nicolas
    Mar 3, 2012 at 13:46
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    $\begingroup$ 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. $\endgroup$
    – GEdgar
    Mar 3, 2012 at 17:11
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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").

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    $\begingroup$ 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.... $\endgroup$
    – nicolas
    Mar 3, 2012 at 8:57
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    $\begingroup$ Is "V" (i.e. $|x|$, that is, the absolute value function on $\mathbb R$) not increasing? (In French we don't have this problem.) $\endgroup$
    – Pierre-Yves Gaillard
    Mar 3, 2012 at 9:08
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    $\begingroup$ 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.) $\endgroup$
    – Pierre-Yves Gaillard
    Mar 3, 2012 at 9:45
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    $\begingroup$ @nicolas: There is: "less than" and "less than or equal". $\endgroup$
    – joriki
    Mar 3, 2012 at 10:01
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    $\begingroup$ 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.) $\endgroup$
    – Pierre-Yves Gaillard
    Mar 3, 2012 at 10:10

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