I find this episode confusing, which is one of the reasons I've delayed writing about it for so long. I'll give it my best shot.
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[1] After these things Jesus walked in Galilee: for he would not walk in Jewry, because the Jews sought to kill him.
"These things" -- feeding the 5,000, walking on water, the Bread of Life discourse -- all took place in Galilee, not "Jewry" (Judaea), but the crowd that witnessed these things apparently included "Jews" (Judaeans) who had followed him from Jerusalem after witnessing his miracles there. They would not try to kill him in Galilee, where he was among his own people and they themselves were strangers, but they would be bolder if he returned to Judaea.
Or perhaps "these things" refers to earlier miracles wrought in Jerusalem. Later, when Jesus does after all return to Jerusalem for the Feast of Tabernacles, he says to the hostile crowd, "I have done one work, and ye all marvel" (7:21), and what follows makes it clear that he is referring to his healing of a man at Bethesda on the Sabbath.
Although I don't know enough Greek to suggest this with any confidence, perhaps μετά is here being used in the sense of "with, among" rather than "after." Perhaps the correct reading is, "As he was doing these things (feeding the 5,000, walking on water, etc.), Jesus walked in Galilee, for (ever since the healing at Bethesda) he would not walk in Judaea, because the Judaeans sought to kill him (for breaking the Sabbath)."
Or perhaps this whole verse has been transposed from its proper place in the text? As I have noted before, John 5 ends with Jesus in Jerusalem (and the Jews there seeking to kill him), and then John 6 begins with his crossing the Sea of Galilee, which is nowhere near Jerusalem.
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[2] Now the Jew's feast of tabernacles was at hand.
[3] His brethren therefore said unto him, "Depart hence, and go into Judaea, that thy disciples also may see the works that thou doest. [4] For there is no man that doeth any thing in secret, and he himself seeketh to be known openly. If thou do these things, shew thyself to the world."
[5] For neither did his brethren believe in him.
The Feast of Tabernacles was (together with Passover and Pentecost) one of the Three Pilgrimage Festivals, when all Jews who were able were expected to make a pilgrimage to the Temple in Jerusalem.
The advice to "go into Judaea, that thy disciples also may see the works that thou doest" seems strange, since John 6 (which takes place in and around Galilee) is full of references to Jesus' disciples being with him. (Simon, Andrew, Philip, and Judas are mentioned by name.) It seems that his disciples "followed" him in a very literal sense, accompanying him in all his peregrinations. He must have had other disciples as well who stayed in Judaea, and his brothers must have meant "that they disciples there also may see" -- and I see that many modern translations insert that word.
As Mark tells the story, when Jesus began preaching, working miracles, and drawing crowds, his friends and family "went out to lay hold on him: for they said, He is beside himself" (Mark 3:21). Not believing that Jesus was who he said he was, they very understandably wanted to stop him from embarrassing the family by making a public spectacle of himself. In the Fourth Gospel's account, we have the opposite situation: Jesus is lying low in Galilee, and his unbelieving relatives are urging him to make more of a public spectacle -- to "go into Judaea" and "shew thyself unto the world."
What could their motive have been in urging this? I can think of three possibilities:
1. They just wanted to get rid of him. They lived in Galilee, and their brother was embarrassing them, so they wanted to get him out of Galilee. I don't think this is a plausible reading. First, they must have known that the Judaeans wanted to kill him, and it seems unlikely that they would have sent their own brother to his death. Second, they were encouraging him to go to Judaea during the feast, at a time when all the Jews went to Judaea -- "shew thyself unto the world," they said. If it was embarrassing to have Jesus raising a ruckus in Galilee, how much more embarrassing to have him do so before all Jewry -- including all their friends and neighbors, who would come back to Galilee full of talk about Jesus! Finally, they themselves also intended to go to Judaea for the feast; they were urging Jesus to go with them. If their motive was to save face by hushing up their brother's scandalous claims about himself, they would have said the opposite of what the Gospel records them saying; they might have said something like, "I know you're supposed to go to Jerusalem for the feast, but the Judaeans want to kill you. I think you'd better just stay here in Galilee and keep a low profile, and we'll go to the feast without you."
2. They were calling his bluff, or trying to use reverse psychology. They were trying to cure Jesus of his "delusions" by pushing them to their natural conclusion and forcing him to realize that he didn't actually believe what he was saying about himself. "You're the Messiah, are you? Well, in that case, you should have no problem waltzing into Jerusalem and announcing that fact to the Pharisees and the Roman governor, right? What's that, you're afraid they might kill you? You don't actually believe you're the invincible Messiah after all? Yeah, that's what we thought." I can't rule out this reading, but it doesn't sit well with me.
3. They believed, at least tentatively, that Jesus was the Messiah and were genuinely troubled by his reluctance to go to Jerusalem. They were urging him to do what they thought the Messiah ought to do, and to prove thereby that he was in fact the Messiah. In this reading, "neither did his brethren believe in him" might be better translated as "even his brethren had doubts," or "even his brethren lacked confidence in him." They didn't disbelieve him in the sense of thinking he was an impostor or mentally ill, but they couldn't understand why he was hanging around in some Galilean backwater instead of going to Jerusalem to announce his identity and claim the throne of David.
It is this third reading that seems most likely to me. Although we never hear anything else about Jesus' brothers in the Fourth Gospel, his brother James is later referred to as an apostle (Gal. 1:19); and Jude, the author of the epistle, may also have been Jesus' brother (see Jude 1:1, Mark 6:3); so it appears that his brothers were not out-and-out unbelievers but merely struggled with doubt at times, when Jesus' behavior seemed inconsistent with what the Messianic prophecies had led them to expect.
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[6] Then Jesus said unto them, "My time is not yet come: but your time is alway ready. [7] The world cannot hate you; but me it hateth, because I testify of it, that the works thereof are evil. [8] Go ye up unto this feast: I go not up yet unto this feast: for my time is not yet full come."
[9] When he had said these words unto them, he abode still in Galilee. [10] But when his brethren were gone up, then went he also up unto the feast, not openly, but as it were in secret.
This is the second time in this Gospel we see Jesus refusing to do as requested, saying it is not his time yet, and then almost immediately doing it after all. The first instance of this in in John 2; when Jesus' mother informs him that there is no wine, he replies, "Woman, what have I to do with thee? mine hour is not yet come" (v. 4) -- a reply which his mother ignores, telling the servants to do whatever Jesus says, and Jesus proceeds to provide wine miraculously. Did he work this miracle despite the time not being right, because his mother had asked him to? Or was his idea of "the right time" so specific that hours or even minutes after his mother's request the time was right?
As we read later in this chapter (v. 14), Jesus finally appeared "about the midst of the feast" -- meaning around the fourth day of this seven-day festival. Is that because his time was "full come" by then? What difference could it possibly make whether he showed up for the festival on time or three days late?
My best guess is that the delay had to do with the fact that Judaeans sought to kill him for healing on the Sabbath. The first and last days of the seven-day Feast of Tabernacles were Sabbath-like observances on which all work was forbidden, but work was permitted on the other five days. If Jesus, with his reputation as a healer, had arrived in Jerusalem on the first day of the feast, there would doubtless have been people there in need of healing, and he would have to heal them or else refuse to do so. Refusing to heal on the first day of the feast would look like a concession that the Pharisees were right, that it had been wrong to heal on a day when work was forbidden by the Law of Moses. On the other hand, healing on that day, brazenly repeating the offense for which "the Jews sought to kill him," could have put his life in danger -- and, as Bruce Charlton has recently discussed, it was important for Jesus to die at the right time, staying alive until his mission had been accomplished. Jesus may have judged that he was relatively safe during the feast -- in the Synoptics, those who plot his death say, "Not on the feast day, lest there be an uproar of the people" (Mark 14:2, Matt. 26:5) -- but openly healing on a mandatory day of rest might have been pushing his luck a little too far.
Against this interpretation, we have the fact that Jesus did again heal on the Sabbath during this visit to Jerusalem (John 9:14). This was after he had already been in Jerusalem for several days, though, publicly saying provocative things and defending his earlier Sabbath healing. That he was able to get away with this was taken as evidence that he was the Messiah: "Then said some of them of Jerusalem, 'Is not this he, whom they seek to kill? But, lo, he speaketh boldly, and they say nothing unto him. Do the rulers know indeed that this is the very Christ?'" (vv. 25-26). When, days later, Jesus finally provoked the Jews to the point that they took up stones to stone him (John 10:31), he "escaped out of their hand" (10:39). This escape was probably possible in part because in the preceding days he had impressed the crowd and built up enough good will that a substantial portion of the citizenry were on his side. If he had appeared on the first day of the feast and immediately started flagrantly breaking the Law of Moses, the reception might have been rather different.
This is all speculative, of course, but it is the only interpretation that makes sense to me. I think Jesus was constantly walking the razor's edge, pondering precisely how provocative he could afford to be and going that far and no further, at least until his mission was accomplished and he could submit to death.
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[11] Then the Jews sought him at the feast, and said, "Where is he?"
[12] And there was much murmuring among the people concerning him: for some said, "He is a good man": others said, "Nay; but he deceiveth the people."
[13] Howbeit no man spake openly of him for fear of the Jews.
Here is more evidence that it was important for Jesus to make his appearance at precisely the right time and in the right way. Everyone was anticipating his arrival and wondering what he would say and do. He delayed his arrival, letting the anticipation build up, and then appeared in the Temple preaching from the Bible -- an appearance calculated to defuse the accusation that he flouted the authority of Moses.