This is a single introductory sentence. You should not reason that far with it.
With that reasoning you could claim addition is impure, because time() + 1 changes over time. map is definitely a pure function. If you apply it to impure arguments, of course the result will be impure. The point in FP is to avoid using 'time', not 'map'.
That's a good argument - but it does not really resolve the definition problem. It seems that some people think that 'functional programming' means 'programming with mathematical functions' - while others think that it means 'programming with first-class functions'. I can take this or that definition - but if wikipedia uses one definition - this sounds like a good argument that more people use the 'mathematical functions' definition. I also think that 'mathematical functions' is a more useful definition - because it implies 'referential transparency' and probably other interesting things that are usually quoted as the advantage of using functional programming.
Let me repeat: you used an introductory sentence as a "definition" of FP and are overinterpreting it. If you read later you will see:
> Functional programming languages, especially purely functional ones...
So Wikipedia distinguishes functional programming and pure functional programming. In fact there is a separate article on pure functional programming, which says that destructive modifications are "excluded", not just "avoided".
Functional programming does not have clear boundaries. You can have advantages of functional programming even if you introduce _a bit_ of FP style in a very imperative program. Conversely, you can write in very imperative style in Haskell.
My point is that the definition of functional programming is this purity you mention not using first-class functions. Pure means that something has more of that thing you want and impure means it has less - so pure functional programming has more 'functional programming' then 'standard functional programming'. Sure completely pure functional programming is impractical - there are degrees to how purely functional are the languages we use in practice - but this is the point practical languages are more or less functional and how much are they functional depends on how much they depend on state manipulation not on using higher order functions or having first class functions.
Defining functional programming as programming with first-class functions is a consistent definition and could be used if not the fact that more people use the other definition (at least that is what the wikipedia definition and other online material makes me believe).
Is Prolog a functional language, as it avoids state manipulation but has no first-class functions? I think not. There are some characteristics of functional languages - including first class and higher order functions, relying on immutability, declarativity etc. Choosing just one of them to measure how much a language is "functional" is a mistake IMO.
With that reasoning you could claim addition is impure, because time() + 1 changes over time. map is definitely a pure function. If you apply it to impure arguments, of course the result will be impure. The point in FP is to avoid using 'time', not 'map'.