The original Christians were called the Nazorians. They kept the Torah and, per Epiphanius, they believed Jesus died in the time of Alexander Jannaeus (103-76 BC). The Babylonian Talmud knows of no other form of Christianity. Other Christians believed Jesus died in the 40s AD rather than the 30s AD. If Jesus existed, how could different Christians believe he lived in different centuries? This bit of evidence is more likely for mythicism than historicism.
We have very little reliable information for Socrates, yet we do have both favorable and hostile eye witnesses for him, whereas we have no eye witnesses for Jesus. Why do we have no contemporary records for Jesus? The most likely reasons are he didn’t exist, he was incredibly insignificant, or there was a conspiracy to destroy such records.
There were numerous Jewish and pagan historians who discussed 1st century Judea, but none mention Jesus. (Most of these writings weren’t preserved, but surely Christians or their critics would have cited any passages that mentioned Jesus.) If medieval Christians didn’t preserve a work, or decided to destroy it, we don’t have it. So the evidence we have is skewed in favor of Christianity. Yet even with this biased view, we still have no references to Christianity from the 1st century outside of Christian literature. Even within Christian literature, suspiciously little survives.
Why did no one preserve the family records of Jesus that would have existed if he did? Tax records, deeds, contracts, court records, church records, census data, and letters all would have existed. It’s hard to believe early Christians wouldn’t care about preserving any of it. Christians discarded almost all of their own records from the first and second centuries. This lack of records would seem to favor mythicism, but in order to avoid being biased in his own favor, Carrier decides this lack of records favors neither historicity nor mythicism.
Possibly our earliest non-canonical source, 1 Clement tells us nothing about the historical Jesus. Clement does tell us Paul was recently killed in Spain (see also Romans 15:24-28) which contradicts the account in Acts. Clement knows of Jesus only through the Old Testament and visions. When he quotes Jesus, he’s usually quoting scripture. He has no knowledge of the Gospels. In fact, the Gospels seem to expand upon sayings found in Clement. Clement is similar to both Hebrews and Ephesians, so either all three are quoting from a now lost scripture or they’re quoting from each other. 1 Clement is more likely on mythicism than historicity.
Writing sometime between 110 to 160 AD, Ignatius is the next earliest datable Christian writing outside the New Testament. (The Epistle of Barnabas was likely written around 130-132 AD and thus is not independent of the Gospels.) Ignatius believed Jesus was a historical figure, but he argued with Docetists, Christians who either believed Jesus was only seen in visions or who believed his body was an illusion.
Ignatius gives no evidence of Jesus existing historically; he just argues that he did using a now lost gospel as evidence. So by the year 100, no Christian had any evidence that Jesus existed besides the Gospels. Ignatius believed Jesus was kept hidden until his death when he appeared as a star brighter than the sun and abolished magic from the world (To the Ephesians 19). This isn’t in any Gospel we know of, but it is consistent with 1 Corinthians 2, Acts 26:13, Philo’s On the Creation 31, The Ascension of Isaiah, Justin Martyr, and Psalm 24. Overall, Ignatius’s writings are more likely on mythicism.
Papias of Hierapolis (130-150 AD) is unreliable and favors neither theory. Later Christians didn’t even bother to preserve his works so we only have quotations and summaries found in other authors.
The writings of Hegesippus (about 180 AD) are likewise not preserved. His writing comes too late to bear on the question of Jesus’s existence and it’s obviously fiction anyway. However, the section describing the martyrdom of James the Just describes Jesus as a celestial figure who hasn’t come to earth yet, which is slightly more likely on mythicism.
Josephus was a Jewish historian who wrote Jewish Antiquities about 93 AD. There are two passages that mention Jesus, but both are almost certainly interpolations. The Testimonium Flavianum is based on Luke’s Emmaus Narrative. It doesn’t make sense in context. It doesn’t match Josephus’s style of writing. No Christians mention the passage until Eusebius in the fourth century (not even Origin who often quotes from Josephus). It also doesn’t make sense that a book written for a gentile audience wouldn’t go into more detail to give a better explanation. Only Christians could make sense of the passage as is. The passage before describes a massacre, the passage after begins with “about the same time also another terrible thing threw the Jews into disorder.” Also Josephus usually writes at length and this passage is uncharacteristically short.
Another passage in Jewish Antiquities mentions James the brother of Jesus. Origin also doesn’t quote this passage even though he would have desperately wanted to quote a passage like this when arguing with Celsus and his other critics. Again, Eusebius is the first to notice it. The Jesus in this passage is most likely Jesus ben Damneus who is mentioned again a bit later. If not, Josephus would have provided more explanation since the passage as is would be confusing to non-Christians. Luke used Josephus as a source, but doesn’t mention this passage in Acts as we’d expect. In the end Josephus supports neither historicity or mythicism.
There is a passage in Tacitus about Nero persecuting Christians around 116 AD, however this was also likely added by a later Christian author. It’s likely this passage was originally about the followers of the Jewish instigator Chrestus who were scapegoated for burning Rome (see Suetonius). This passage isn’t mentioned until the fourth century (it’s not even mentioned by second century Christians when they discuss Nero persecuting Christians which we’d expect if the passage existed at the time). However, to avoid being biased, Carrier assumes this is a genuine passage.
If Tacitus really wrote about Jesus, where would he have gotten this information? He wouldn’t have gotten it from government records since the city libraries had burned down twice since the event and his report does not contain the distinctive information we’d expect from such a source. He likely didn’t get the information from other historians either. He most likely got the information from his friend Pliny, who would have gotten the information from Christians, and thus the information would not be independent and thus doesn’t count for or against either theory.
Thallus is sometimes used as evidence for the historicity of Jesus, but Thallus never actually mentioned Jesus (per quotations of him in Eusebius). People just think he refers to Jesus because Julius Africanus misquotes him.
Suetonius likewise never mentions Jesus. He discusses a Jewish instigator named Chrestus (a common name at the time). People think it’s a misspelling of Christus, but there’s no reason to think this. First of all, the incident is about Jews, not Christians per Suetonius, Dio Cassius, and even Acts 18:2. Jesus was also not alive or in Rome at the time the incident occurred, and Paul’s letter to the Romans doesn’t mention it as we’d expect. A second passage in Suetonius mentions Christians being persecuted under Nero. This was probably a reference to the Chrestians originally, but even if not, it doesn’t tell us anything about the historical Jesus, so it favors neither theory.
In Justin Martyr’s Dialogue with Trypho, he has a fictional debate with a Jew who accuses him of inventing a Christ. 2 Peter 1-2 also criticizes fellow Christians who believe the story of Jesus was a myth. There are other hints of this in 1-2 Timothy and 1-2 John.
It’s more effective in the long run for a religion to be thought literally true. If the religion is revelation-based, anybody could have a revelation at any time and the religion couldn’t effectively be organized. Christianity started out as a revelation-based religion per Galatians 1 where Paul insists that revelations are superior to traditions. We can’t expect there to be any surviving refutations since Christians wouldn’t have preserved them. After all, we don’t have refutations of the miraculous feedings, resurrection of the dead, and other fantastic elements, so why would we expect there to be any refutations of Jesus existing? That we have no refutations thus counts for neither theory.