In general encodings do not enable operations on the encoded information and do not allow the information to be transformed into something different. Notations do enable such operations and transformations.
Agreed. Encoding maps symbols, but notation maps entire systems: symbols and operators/operations/mappings.
The difference between "notation" and "program code" is interesting, as "notation" tends to be used exclusively when humans do the interpretation (music, math, rally route marking, etc.), and "program code" when a machine does the interpretation. I can definitely see why both are considered "languages" (by some/many).
The difference between encoding and notation is very vague, of course. For example, Unicode is an encoding between visual glyphs and their numerical representation in digital systems. RPN is notation and not encoding because it defines not only the symbols but also the operations the symbols refer to.
I suggest encoding is contained within notation; with whether operations or mappings that affect other symbols are included, and whether it is self-contained or relies on other rules for full utility, being the separator between 'notation' and 'encoding'.
This is, of course, only my current understanding based on the observed effective use (in mostly technical and scientific contexts) and the dictionary definitions.
Why would anyone want to use prefix notation?
Why would anyone want to speak any other language but English?
Your question does not answer my question.
It is a rhetorical question, whose answer also answers your own question:
Because they find it useful.Just because you find something natural and obvious and comfortable, does not mean it is so for everyone else too.
For example, my native language is Finnish, which has plenty of grammatical cases; is typologically agglutinative; uses suffixal affixation; and just about everything is inflected. Something like "Can you see my car?" is just two words in Finnish, "Näetkö autoni?", and not because the expression is simpler: "Can you see your car?" is "Näetkö autosi?", and "Can they see my car?" is "Näkevätkö he autoni?". I'm pretty good (but nowhere near perfect) in Finnish grammar, even with possessive suffix ("auto
ni" instead of "minun auto" for "my car") which seems very difficult for even those who write Finnish as a job.
Nevertheless, gendered nouns in Swedish and German completely throw me in a loop. And no, it is not a matter of "not learning enough": I've got
years of Swedish studies completed with good grades, but still the only way to get the noun genders right, is for me to memorize every damn noun and their inflection and gender.
To expand a bit: As explained by Łukasiewicz, the principle of his (prefix) notation was to write the functors before the arguments to avoid brackets, and that he had employed his notation in his logical papers since 1929. The postfix notation, on the other hand, came decades later; in 1941 in Germany, gaining some traction in the 1950s and much more in 1960s.
To explore the reasons why others might find prefix notation easier than postfix notation, we'd need to delve into language, and especially the differences in word ordering. For example, although many do not realize it at all, English adheres strictly to adjective order, with opinion, size, age, shape, colour, origin, material, and purpose in that order. Saying 'red large old box' will feel unnatural or as if emphasizing one of the adjectives to native English speakers; they will always prefer to 'large old red box'.