Why is the “STAY” message sent in Interstellar?

Question

In the movie Interstellar, why does Cooper send the “STAY” message? When Cooper winds up in the tesseract (event Z), there are two related events that are in his past that he knows about: the reception of the coordinates of the NASA base (event A) and the goodbye scene in which Murph tells him that the “ghost” told her “STAY” (event B) which he ignores. There’s also the related event of Murph’s reception of data on Gargantua through the wristwatch (event C), but unlike the other two this is, I think, not a past event that Cooper knows about before actually causing it.

I get that cause and effect are time inverted here, in that event Z allows Cooper to cause events A, B and C, and that there’s a loop in that event Z leads to A and A leads to Z, with another loop in that B needs to be ineffective in order to allow Z to happen which in turn leads to B. I think I understand the A-C-Z events, it seems that Cooper realizes that by ensuring the A-Z loop he can also cause C which he hopes allows Murph to solve the gravity problem (which she does).

I have trouble understanding the B-Z events though. Before his realization, Cooper seems to at first want to prevent himself from leaving on the space journey, which is why he uses the books to spell out “STAY” (event B). But why spell out a message that he already knows he will / has ignore(d)? Should I assume he has simply forgotten about the message? Or that he is so overcome with emotion he’s acting irrationally? Is there another explanation or something I missed? I realize there’s a potential paradox if Cooper would not cause B, or would cause an alternate B with a less ambiguous message like “COOPER STAY WITH YOUR DAUGHTER AND GET THE FOLLOWING DATA TO DR BRAND”, but that still doesn’t quite help me understand it.

EDIT: there's a related question on “Why Cooper didn't simply not give the co-ordinates of NASA?”, but the answer given there does not (to me) really resolve the contradiction of Cooper sending a message he (as I understand it) already knows is ineffective.

Answer 1

Should I assume he has simply forgotten about the message? Or that he is so overcome with emotion he’s acting irrationally?

That's exactly what you should assume. When he arrives there he is indeed struck with emotion and doesn't really know how things work out there, let alone that anything he does he already did and he can't change the past. Also keep in mind that he did not seem to believe Murph when she told him that the "ghost" wants him to STAY. So when he comes there he doesn't yet realize he can send the quantum data to Murph and save them all. He assumes that he can change the past and that's what he tries to do. This is also addressed in the answers to this related question.

The first thing that's on his mind when seeing himself leaving his daughter on a supposedly dying planet is that he wants this guy to STAY with his daughter and not leave her alone. This is congruent with him trying to return to Earth and to his daughter after learning that Plan A is supposedly impossible earlier on Mann's planet, he doesn't care that Earth is doomed and simply wants to be with his daughter. And this is also what he now at least wants for the past Cooper to do, unaware of the fact that the past Cooper can't do anything else than the Tesseract Cooper already did anyway. It's also evident from the whole scene that he's reacting rather emotionally than with reason and careful thinking:

Murph: Just go. If you're leaving -- just leave now.
Cooper (to his earlier self): Don't go, you idiot! Don't leave your kids. You goddamn fool!...S...T...A...Y...Stay, you idiot! Tell him, Murph! Stay...Murph, tell him again! Don't let him leave!...Murph, don't let me leave...STAY!!

Only then after he is totally devastated that this idiot left his kids on a dying planet yet again, TARS comes up with the idea that this whole thing was actually constructed by "Them" in order to help us. Before that Cooper didn't really know what's up at all, let alone that he can actually transmit a message to save the Earth as a whole.

TARS: Cooper?
Cooper: You survived?
TARS: Somewhere. In their fifth dimension. They saved us.
Cooper: Who's 'They'? And why would they help us?
TARS: I don't know, but they constructed this three-dimensional space inside their fifth-dimensional reality to allow you to understand it.
Cooper: It isn't working!
TARS: Yes, it is. You've seen that time is represented here as a physical dimension -- you even worked out that you can exert a force across spacetime.
Cooper: Gravity. To send a message...Gravity crosses the dimensions -- including time -- And you have the quantum data, now...

And only when trying to figure out how to achieve this, he suddenly gets the whole picture, that he himself was it who sent him on this trip and that he can't change the past anyway as everything already happened. But he can change the future of Earth after he left and for this he is needed to give Murph the quantum data.

TARS: Cooper, they didn't bring us here to change the past.
Cooper (realizing): No, they didn't bring us here at all. We brought ourselves here...TARS, feed me the coordinates of NASA in binary...

So the answer why he tries to send a message that he already knows he ignored is, that he didn't know anymore that he already ignored it and, and even more so, he didn't yet realize that his past self can't ever do anything else than ignore it. He's emotionally struck, totally confused, clinging to hope and doesn't get how things work, let alone that the past is unchangeable. Not before TARS shows up and sparks some enlightening dialogue.

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I realize there’s a potential paradox if Cooper would not cause B, or would cause an alternate B

Just a little remark to this point. This is not possible at all. The movie employs and entirely consistent timeline, as we see when the movie shows that changing the past in the slightest way is impossible. Cooper just cannot "not cause B" or an "alternate B", since those events are already part of his causality chain and the movie shows that the past is unchangeable. If he would cause an alternate B, he would have experienced that alternate B in the past already and we would look at a completely different timeline for the whole movie. There is only one single timeline (though, one that contains a loop, apparently) ever possible, the one we see in the movie. This is also addressed in the answers to this related question.

Answer 2

It's a classic time-travel paradox. At this point in his adventure, Cooper is distraught that his mission is an apparent failure. To his viewpoint, he has missed his kids' lives for absolutely NOTHING. He has not realized yet that this "window through time" is the key to everything, and simply wants to change the past by making himself STAY on Earth with his family.

This attempt to change the past, however, is largely what kicks everything - including the mission - into existence to begin with. Basically, he accidentally started the very loop that he feels trapped by. It's when Cooper realizes this that he realizes he can also send messages back to his past self and/or Murph, and then sends the data. He also realizes that...

it was never aliens assisting mankind, but rather the future descendants of humanity itself. By providing the "time window" apparatus he used to start the loop, they were therefore ensuring their OWN survival in addition to Cooper's.

The obvious irony here is that this is the way things always happened. We just see the timeline from Cooper's point of view as he figures that out.