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When Marty decides to go back from 1958 to 1988 in Back to the Future, he decides to go back several minutes before the doctor is killed by the terrorists, to warn him.

But when he arrives late, he sees another copy of him re-enacting what had happened at that point of time.

How can this make any sense? Shouldn't the other copy be extinct? or at least not present at that place at that time?

For the sake of argument, if that copy's existence is justified, does that mean that he (the copy) will do the same actions done by the "real Marty" and perhaps take the same decisions to come back early and warn the doctor?

Wouldn't that cause an infinite cycle of Marty clones?

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Back to the Future characters operate on the basis of keeping the timeline mostly intact. Doc, in particular, is very very concerned with ensuring that future events do not affect past events, or it will cause a paradox. In this case, had Doc never befriended Marty, and Marty had never traveled back in time, Marty would never have gone back in time in the first place. It creates a Predestination paradox.

As Future-Marty never affects Past-Marty, Past-Marty will do everything exactly the same as before, changing himself into Future-Marty.

At any given point in that timeline, there are only two copies of Marty.

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    There are never two copies of Marty. The "original" Marty departed from the Twin Pine Mall, and returns to see "himself" at the Lone Pine Mall. Lone-Pine-Marty has lived a different life from Twin-Pine-Marty; he is not a copy. – T.J.L. May 30 '17 at 20:49
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To understand the answer, think about when Doc explains how the timelines work in Back to the Future II, and about how alternate realities can be created by making changes in the past. The whole story in BTTF is fluent to Marty, but at the end of the movie when he travels back to 1985 early to warn Doc about the terrorists he does not return to his original timeline because of the changes he had made in the past. He instead returns to 1985A where his Dad is now a famous writer and Biff cleans cars for a living.

In the 1985A timeline, Marty has still befriended Doc Brown and was helping him with the time travel experiment just as it happened in the original timeline. Marty gets there to witness Doc getting shot and himself going back to 1955. At that moment he is the only Marty that is in that reality. That 1985A reality will keep going and not loop to create "an infinite cycle of Marty clones". BTTF II starts in the alternate 1985A timeline.

It can be assumed that in the original BTTF timeline, Marty travels back to 1955 and never returns so the Marty in that timeline becomes a missing person and is never seen again.

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You can arguably see the whole trilogy as half the story. The film series is Marty 1’s story. Marty 2 would be the child of the more confident George and Elaine that Marty 1 created. Marty 2 would be different enough that his time travel back would've altered the past in diffent ways. Lets say George and Elaine 2 are a little bit wealthier, confident and strick with education. George 2 being a scfy writer could’ve tought Marty 2 more about time travel and he still would’ve looked up to Doc the same way as Marty 1. So Marty 2 could’ve went back in time, dodged Mr. Peabody’s tree (better driver having his own truck) and would have thought about the risk of running into his parents. He would lay low with Doc, travel back to 1985 not altercating with his parents. Now because he didn’t alter his parents path, George falls from the tree, Elaine starts dating him. George never knocks out Biff and everything returns to the original time line. Marty 2 however would’ve returned to 1985, watched Marty 1 travel back in time and be shocked to see his dad and mom not being confident and successful like in his timeline.

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The Marty you see heading off in the time machine at the end of the movie is the same Marty as the one we saw at the beginning.

I use the word "same" loosely here, because of course he has memories of growing up in a very different, more affluent family, whereas "our" Marty does not.

But it's not a copy of him, it is him. The question isn't really "what happens to that Marty?", so much as "what happens to those memories?". There isn't an infinite loop of Martys, there is just a single loop.

Being a movie about a very tricky subject (backward time travel), you basically have to let it lay down its ground rules, and judge it by how it obeys those rules. And in BTTF it's consistently held that people travelling through time always retain the memory of the version of events as they were when they left "before" (for want of a better word) the changes they made, whereas everyone in the changing timeline remember events "after" the changes.

In other words, Biff doesn't remember being George's boss, but Marty does. The Doc wouldn't either, but presumably he understands time travel enough to get the idea.

But really it's no different from asking, "What happened to the Marty who knew Clayton Ravine was actually called Eastwood Ravine?"

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