I've been wondering what the origins of the verb 'to miss', as in a to have a longing for, come from. Is it anywhere similar to the origins of the verb 'to miss' as in to not hit?
Learn English – What are the roots on the verb ‘to miss’ come from
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The earliest reference I can find is from Hugh Wiley's first novel Wildcat, 1920:
The Wildcat, consuming a pork chop in the kitchen end of the mess hall, listened in. "Hot damn!" he exclaimed, "Grasty—was big words cooties, Honey Tone sho' would itch! Lissen at him go!"
A bit more about the book (from the Wiley link):
The Wildcat told the story of a black American drafted and sent overseas during World War I; several of Wiley's other early books, including The Prowler and Fo' Meals a Day (1927), were works depicting black life in comic and exaggerated manner, somewhat akin to minstrel show entertainment though perhaps a bit more subtle.
You seem to be asking about the origin of the term as used in category theory. The history of the term there is somewhat unclear, but it can at least be traced back a little ways:
The term is sometimes attributed to Mac Lane, but this seems to be inaccurate; however, the widespread use of the term is probably due to his influential "Categories for the Working Mathematician", replacing the remarkably terrible term "triple".
The frequent but unfortunate use of the word "triple" in this sense has achieved a maximum of needless confusion, what with the conflict with ordered triple, plus the use of associated terms such as "triple derived functors" for functors which are not three times derived from anything in the world. Hence the term monad.
Mac Lane's use of the term was apparently prompted by J. P. May:
The name "operad" is a word that I coined myself, spending a week thinking about nothing else. Besides having a nice ring to it, the name is meant to bring to mind both operations and monads. Incidentally, I persuaded MacLane to discard the term "triple" in favor of "monad" in his book "Categories for the working mathematician", which was being written about the same time. I was convinced that the notion of an operad was an important one, and I wanted the names to mesh.
Elsewhere, Ross Street attributes the term to Jean Bénabou:
Meanwhile Jean Bénabou had invented weak 2-categories, calling them bicategories. (...) He pointed out that a lax functor from the terminal category 1 to Cat was a category A equipped with a "standard construction" or "triple" (that is, a monoid in the monoidal category [A, A] of endofunctors of A where the tensor product is composition); he introduced the term monad for this concept.
The attribution to Bénabou is also mentioned here.
The motivation for the term is to suggest a relationship with monoids, as can be deduced from the construction given in the quote above, and the Greek root "monos" comes second-hand. The connection to philosophy in general, or Leibniz in particular, is often asserted but never to my knowledge supported in any way. More likely if anything would be a connection to the term "monad" used in non-standard analysis, also related to Leibniz, but I'm not sure what the conceptual link there would be. An anecdote from Michael Barr relates the first use of the term:
(...) The attendance consisted of practically everyone in the world who had any interest in categories, with the notable exception of Charles Ehresmann. (...) One day at lunch or dinner I happened to be sitting next to Jean Benabou and he turned to me and said something like "How about 'monad'?" I thought about and said it sounded pretty good to me. (Yes, I did.) So Jean proposed it to the general audience and there was general agreement.
The off-the-cuff nature of the suggestion, and immediate positive response from a large audience, suggests that there's probably no written record of the term being introduced formally. It's certainly possible that the word was borrowed from use in philosophy or elsewhere, but in any case there appears to be no connection more meaningful than the level of "cheap pun".
As far as I know, the only way you're going to get a better answer than that is by asking Bénabou himself.
Best Answer
If you miss your mother, you do not have what you want (her). If you miss your target, you do not have what you want (a hit) either. This use of the mental conception of "having something" has apparently shifted a bit from "keeping physically near" to "counting among one's successes" in the case of the verb to miss. You could say that "having someone near", "having something in possession", and "having success" are linked in some way.
You might also define both senses by using a word other than "having", in which case you'd need to describe some other but similar metaphoric shift in meaning; I believe this shift in to miss and "having" is all part of a vague complex of conceptions of "nearness" and "possession" and "likeness" that are intimately connected in our minds. This is probably because in Prehistoric times the concepts of physical dimensions, movement, and time developed first; our patterns of thinking about those concepts we later reused for (often abstract) phenomena that are essentially different but can still be dealt with in useful ways by means of these patterns (large chance: I see no "large" object?; the well runs dry: what is it that "runs"?; you have a point: I see nothing "pointy"?' etc.). See the Wikipedia article on Conceptual Metaphor.