I, Joe Hinman, defending proposition, Resolved: That the external (not in Bible) evidence is strong enough to warrant belief in Jesus' historicity.
I do not have to prove factually that Jesus actually did exist, only that belief in his historicity is warranted. This question has nothing to do with miracles. A man could exist and be thought to have worked miracles even if he did not. So miracles in no way limited Jesus' actual existence. I was an atheist, and when I was I thought the Jesus myth thing was groundless. This is not a Christian vs Atheist issue. It's really a struggle over how to understand historical facts.
I have five arguments. I will summarize them briefly and link to longer pages with material for documentation. The arguments are:
I. The Talmud
IV. Josephus (mainly the brother passage)
(in second post)
I am going to ask the reader to read a couple of very long pages and also just supply information you may want to see in addition. This page is just sort of a portal. If you want to just read the summary I've distilled the major points. Or if you want more evidence see the major link over the name for the long pages.
The Talmud is the written record of Hebrew oral tradition. It gets back realistically to mid first century and began to be written down around 200 or so AD. We know Jesus was in the Talmud and that is a fact admitted by Rabbis. Some references use his name (Yeshua) some use code words such as "such a one" or "Panthera". The reason codes are used, is that the commentators censored the works and removed overt reverences to Jesus (although they missed some) to prevent Christians from inflicting persecution. We have many of the out takes in various libraries such as Cambridge. These readings continued to be produced privately not officially. Encyclopidia Hebreica:
"Beginning with the Basle edition of the Talmud (1578–80), those passages in which Jesus was mentioned, as well as other statements alluding to Christianity, were deleted from most editions of the Babylonian Talmud by the Christian censors or even by internal Jewish censorship. These deletions were later collected in special compilations and in manuscripts."  Instone-Brewer and b. Sanh. 43a A scholar at Tyndale House, Cambridge, argues that one of the Talmudic outtakes from Munich actually records the charge sheets against Jesus for his execution.  See the link over "Talmud" for more details.
There are anomalies about these works. The codes given to Jesus allow complete confusion and thus plausible deniability as to who is meant. Other anomalies include the information that Jesus was the son of an adulteress. Also that he was charged with witchcraft and other things not in the Gospels. These are polemics and can be ignored. The point is he is always taken as a historical figure. It is claimed they have information about his genealogy. Perhaps this is also polemical but he is never treated as a myth. Celsus claimed that he got his information on Jesus from the Jews and what he says coincides with some these figures in the Talmud who are supposed to be Jesus stand-ins. 
Papias was the student of the Apostle John. There is a famous quotation by him, which creates confusions to which John he really knew. For our purposes, however, that issue can be mooted: he said both Johns were disciples of Jesus, whether the Apostle or "the elder". Please read the material. Mythyers will raise the issue that this is fifth hand and we don't have the writings of anyone who actually knew Jesus. That is a false standard: real historians don't demand writing from people who actually knew the subject before accepting their existence if it can be established in some way. Besides it's not fifth but second. Jesus is first, Apostle John is second. Although the saying is generated to Irenaeus the student of Polycarp and Papias. Irenaeus quotes Papias saying he knew elder John and/or Apostle John but Irenaeus also says Papias and Polycarp and Ignatius (another major Apostolic father) all studied together with John. Ireneaeus does report Polycarp speaking to Papias recalling to mind when they studied with John together. Below is the most famous passage of Papias, where he names the elder John separately from John Apostle.
I shall not hesitate to set down for you along with my interpretations all things which I learned from the elders with care and recorded with care, being well assured of their truth. For unlike most men, I took pleasure not in those that have much to say but in those that preach the truth, not in those that record strange precepts but in those who record such precepts as were given to the faith by the Lord and are derived from truth itself. Besides if ever any man came who had been a follower of the elders, I would inquire about the sayings of the elders; what Andrew said, or Peter or Philip or Thomas, or James, or John or Matthew, or any other of the Lord's disciples; and what Aristion says, and John the Elder, who are disciples of the Lord. For I did not consider that I got so much from the content of books as from the utterances of living and abiding voices...
Knew the Apostle John and studied with him. He speaks of where the apostle sat when they studied together. We have those sayings reported by Irenaeus who heard Polycarp's discourse. They are in the fragmentary lost writings, Eusebian fragment Letter to Florianus.
I saw thee in Lower Asia with Polycarp, distinguishing thyself in the royal court,3 and endeavoring to gain his approbation.Problems: He begins that discourse saying "when I was a boy." He's saying that to Florianus. He would have been a teenager. The fragment is part of "the lost writings" that come to us from Eusebius. Mythers will make a big deal out of the alleged Pious fraud quote. Eusebius never said the pious fraud quote. That was actually Gibbon the atheist historian who made up the pious fraud quote. Eusebius was actually a good historian and was careful about his sources.
For I have a more vivid recollection of what occurred at that time than of recent events (inasmuch as the experiences of childhood, keeping pace with the growth of the soul, become incorporated with it); so that I can even describe the place where the blessed Polycarp used to sit and discourse -- his going out, too, and his coming in -- his general mode of life and personal appearance, together with the discourses which he delivered to the people; also how he would speak of his familiar intercourse with John, and with the rest of those who had seen the Lord; and how he would call their words to remembrance. Whatsoever things he had heard from them respecting the Lord, both with regard to His miracles and His teaching, Polycarp having thus received [information] from the eyewitnesses of the Word of life, would recount them all in harmony with the Scriptures. These things, through God's mercy which was upon me, I then listened to attentively, and treasured them up not on paper, but in my heart; and I am continually, by God's grace, revolving these things accurately in my mind.
The passage was known in Eusebius' form: J.B. Lightfoot, Eusebius of Caesarea, (article. pp.308-348), Dictionary of Christian Biography: Literature, Sects and Doctrines, ed. by William Smith and Henry Wace, Volume II (EABA-HERMOCRATES). This excerpt pp.324-5. Here Lightfoot disproves the notion that Eusebius made up the Testimonium Flavianum.
This treatment may be regarded as too great a sacrifice to edification. It may discredit his conception of history; but it leaves no imputation on his honesty. Nor again can the special charges against his honour as a narrator be sustained. There is no ground whatever for the surmise that Eusebius forged or interpolated the passage from Josephus relating to our Lord quoted in H. E. i 11, though Heinichen (iii. p. 623 sq., Melet. ii.) is disposed to entertain the charge. Inasmuch as this passage is contained in all our extant MSS, and there is sufficient evidence that other interpolations (though not this) were introduced into the text of Josephus long before his time (see Orig. c. Cels. i. 47, Delarue’s note), no suspicion can justly attach to Eusebius himself. Another interpolation in the Jewish historian, which he quotes elsewhere (ii. 23), was certainly known to Origen (l. c.). Doubtless also the omission of the owl in the account of Herod Agrippa’s death (H. E. ii. 10) was already in some texts of Josephus (Ant. xix. 8, 2).Lightfoot one of the great historians of 19th century,
Pious Fraud Quotation Itself A Fraud
Roger Pearse, an experienced and mature scholar demonstrates that this rumor about Eusebius goes back to a quotation by Gibbon, and Eusebius never said anything like it:
"Some very odd statements are in circulation about Eusebius Pamphilus the Historian. Recently someone quoted one of them at me, as a put-down. I had the opportunity to check the statements fairly easily, and the results are interesting, if discouraging for those looking for data on the internet. Since then I have come across other variants, and added these also.
Note that the Greek text is rendered using the Scholars Press SPIonic font, free from here.
*'I have repeated whatever may rebound to the glory, and suppressed all that could tend to the disgrace of our religion'
*'It will sometimes be necessary to use falsehood for the benefit of those who need such a mode of treatment.'"
Roger Peirce goes on in a long page to disect and disprove this whole thesis, and to show that it was the 18th century historian Gibbon who said this about Eusebius, and not Eusebius himself.
IV. Josephus (Brother passage) 
I will make only scant mention of the Testimonium in this debate; if I did we would never get to the end of it. The brother passage does not have the kind of doubt, or attack, or charges of forgery the TF does. The only Myther answers I've heard on it suck.
But the younger Ananus who, as we said, received the high priesthood, was of a bold disposition and exceptionally daring; he followed the party of the Sadducees, who are severe in judgment above all the Jews, as we have already shown. As therefore Ananus was of such a disposition, he thought he had now a good opportunity, as Festus was now dead, and Albinus was still on the road; so he assembled a council of judges, and brought before it the brother of Jesus the so-called Christ, whose name was James, together with some others, and having accused them as law-breakers, he delivered them over to be stoned.The leading Josephus scholars, Prof. Louis Feldman (Yeshiva University) and Steve Mason (York University), state:
"That indeed, Josephus did say something about Jesus is indicated, above all, by the passage - the authenticity of which has been almost universally acknowledged - about James, who is termed (A XX, 200) the brother of "the aforementioned Christ" Mason says:
Josephus refers to James by referencing Jesus as though he's mentioned Jesus or the reader should know who he is. Jewish scholar Paul Winters states: "if . . . Josephus referred to James as being 'the brother of Jesus who is called Christ,' without much ado, we have to assume that in an earlier passage he had already told his readers about Jesus himself."
Nevertheless, since most of those who know the evidence agree that he said something about Jesus, one is probably entitled to cite him as independent evidence that Jesus actually lived, if such evidence were needed. But that much is already given in Josephus' reference to James (Ant. 20.200) and most historians agree that Jesus existence is the only adequate explanation of the many independent traditions among the NT writings. 
Encyclopaedia Hebraica] in , "Jesus" Jewish Virtual Library,
|American-Israeli Cooperative Enterprise, 2012, On line Resoirce URL|
they draw upon: cf. R.N.N. Rabbinowicz, Ma'amar al Hadpasat ha-Talmud (1952), 28n.26.
"The Munich Talmud on Sanhedrin and Jesus' Trial" Rodsh Pina roject on line
Documents of the Christian Church, edited by Henry Bettonson, Oxford:Oxford University press 1963, 27
 Origin quoting Celsus, On the True Doctrine, translated by R. Joseph Hoffman, Oxford University Press, 1987, 59 Polycorp, "Letter to Florinus," Christian Classics Ethereal Library, on line URL:
http://www.ccel.org/ccel/schaff/anf01.ix.viii.ii.html accessed 6/17/16
 Chris Price, "Did Josephus Refer to Jesus?," Bede's Library, ed. Dr. James Hannam, blog
http://www.bede.org.uk/Josephus.htm accessed 6/17/16
My own Josephus pages on the brother passage are inadequate. I've linked to a work by a member of the Christian CADRE. Not a scholar but the article is edited by Dr. Hannam Historian from Cambridge.
 Feldman, Louis H, Introduction In Feldman, Louis H. & Hata, Gohey "Josephus, Judaism, and Christianity", page 56)
 Steve Mason, Josephus and the New Testament, page 174 ff
 Paul Winter, "Josephus on Jesus and James," in History of the Jewish People in the Age of Jesus Christ, ed. Emil Schurer, Edinburgh, 1973, 432