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 :
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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)
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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.
Best Answer
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:
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).