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Nobody suggested they were not written by Paul, just saying that those don't carry first hand account weight as evidence of Jesus and that there is no way to corroborate if the stories he was told, were true.
As for his mental health, he used to talk about a "thorn in the flesh" that afflicted him, so it could very well be a case of mental illness. As for the possible symptoms of bipolar illness, he could talk for hours non stop even with people not listening to him (pressured speech/talkativeness): "Paul spoke to the people and, because he intended to leave the next day, kept on talking until midnight" Acts 20:7 "And there was a young man named Eutychus sitting on the window sill, sinking into a deep sleep; and as Paul kept on talking, he was overcome by sleep and fell down from the third floor and was picked up dead." Acts 20:9. He also had a sense of grandiosity (Inflated Self-Esteem or
Grandiosity): "I do not think I am in the least inferior to those "super-apostles." 2 Corinthians 11:5. Decreased need for sleep (The diagnostic criteria indicate that during manic episodes there may be a reduced need for sleep): "I have labored and toiled and have often gone without sleep;" 2 Corinthians 11:27. Goal directed activity, Pychomotor Agitation: "Therefore I do not run like someone running aimlessly; I do not fight like a boxer beating the air. No, I strike a blow to my body and make it my slave so that after I have preached to others, I myself will not be disqualified for the prize." 1 Corinthians 9:26. DSM-5 Excessive involvement in activities that have a high potential for painful consequences: "I have worked much harder, been in prison more frequently, been flogged more severely, and been exposed to death again and again." 2 Corinthians 11:23.
As for Tacitus, he was born in 56 AD so it seems he is just writing about christians and their beliefs or borrowing from court records by the 2nd century, no eye witness testimony is supplied in his work. We would know so much more if christians had not decided to destroy the actual records when they came to power. And going by what american theologian Robert E. Van Voorst has to say:
"Does this "Testimonium Taciteum" therefore provide definitive evidence of the existence of Jesus? If we could be certain that Tacitus's account was based on nonChristian sources, the answer would be yes; but as we have seen, such independent knowledge is unverifiable. As R. T. France concludes, while the evidence from Tacitus corroborates the New Testament accounts of the death of Jesus, "by itself it cannot prove that events happened as Tacitus had been informed," or even the existence of Jesus."
https://epdf.mx/jesus-outside-the-n...duction-to-the-ancient-evidence-studying.html