As the "Jesus mythicists" will happily tell you (and I should emphasise that such individuals lie very much outside the boundaries of mainstream scholarship) this leaves us with only the gospels and other early Christian writings to work with. Because such works are the products of a particular theological mindset and for the most part lack even the pretence of historical objectivity, this leaves us with virtually no incontestable evidence for Jesus Christ or the early years of the movement which bore his name. For the mythicists, this lack of evidence proves decisive: if there is no unambiguous evidence for Jesus, then epistemological prudence must push us to the position that either Jesus did not exist, or - at best - that we cannot say he existed with any confidence at all. While those claiming such a position are undoubtedly correct about the paucity of quality evidence available to us (and you would do well to keep this in mind every time Jesus is discussed in this thread), I think the conclusion they have reached is a little extreme and ultimately ends up raising more problems than it solves.
In the first place, our demands for hard, incontrovertible evidence cannot be as strict in the study of ancient history as they are in the study of modern history. The reason, simply, is that - barring the occasional chance archaeological find - hard evidence for the events of the ancient world usually haven't been preserved down to the modern day. Even events that have been meticulously documented by quality historians - the Peloponnesian War by Thucydides, for example - must be treated with a skeptical eye, leaving us with comparatively little that can be said with absolute certainty about events of even this magnitude. The reason such accounts must be treated skeptically is because the standards of modern historiography - with the need for meticulous sourcing and the drive for objectivity - just weren't an active concern for ancient historiographers. Thucydides, for example, would invent speeches out of whole cloth for dramatic effect (we know this because he tells us). Herodotus - the father of history - believed strongly in the influence of the divine over the progress of history, and would attribute the incidence of many events to the intervention of the gods. In the Roman world, Tacitus was as much a moralist as a historian, happy to engage in rumour and innuendo if it would better serve his ends.
And so on and so forth: no matter where you look, no matter how important an ancient event was, we usually only know about it today through the lens of an ideologically compromised ancient historian or two... and that's if we're lucky. If paucity of evidence were enough to make dubious the people and events of ancient history, we could probably compress everything we know with absolute certainty about the ancient world into a single book. If we can dismiss the existence of Jesus on such grounds - a figure whose public life played out in perhaps the space of a year or less, in an obscure, undocumented part of the world in front of perhaps a few dozen followers - we should probably dismiss the existence of Pythagoras, Socrates, Hannibal and whole host of other ancient figures as well.
With respect to Jesus, it is clear that virtually everything we know about his life is to be found in the gospels. These books obviously cannot be read with naive credulity, as though they were written with the aim of faithfully transcribing actual historical events, but that is not to say that they do not contain nuggets of reliable history that can be mined from the text if we would only use the appropriate methodological tools. Just as we can't dismiss everything Herodotus has to say because he believes in divine intervention, or everything Thucydides has to say because he has a penchant for making things up, we shouldn't dismiss everything the gospels have to say simply because their construction was heavily influenced by the theology of their authors. I won't go into detail about the kind of historical-critical tools we can use to distinguish fact from fiction in the gospels (here is something I wrote earlier if you want such detail), but it can be admitted that we aren't left with much we can say about the historical Jesus with any certainty once these methods have been applied. In my opinion, we can say that Jesus was an itinerant prophet, preaching an eschatological message in Palestine in the first century. He was likely born in Galilee, was likely a disciple of John the Baptist, and he likely ended his ministry in Jerusalem. Here he attracted notoriety (perhaps due to his sacking of the Temple), was apprehended by the Romans (possibly with the assistance of the Jewish authorities) and sentenced to crucifixion by Pontius Pilate. For me, that is about all that I would assert about the life of Jesus with any confidence: everything else I have to say about the man comes with a big asterisk next to it.
On the other hand, it's still something. These facts go a long way to explaining the shape and nature of early Christianity as preserved in the writings of Paul and others. And this is the important fact I want to emphasise to the mythicists, or those who find their arguments compelling: it's all very well and good to assert that there is no direct evidence for Jesus, but it cannot be denied that there was a movement which existed in his name barely two decades after the putative date of his death. If you wish to deny the existence of Jesus, then it is surely incumbent upon you offer some coherent explanation for how this body of belief and literature could have possibly grown up in his name in such a short space of time. This is not an easy prospect, and I'm yet to encounter any compelling alternative theories to the one that Jesus of Nazareth existed as a flesh-and-blood human being.
Now, at this point most mythicists will play the usual denialist game of throwing up their hands and saying "hey man, just asking questions!", but I think this tactic is a little intellecually dishonest. It's easy to attack, with simply untethered skepticism, a historical theory that is working with necessarily fragmentary evidence, but it's much more difficult to posit alternative theories for the evidence available that are more probable and less convoluted than the original theory. The parallels here between Jesus mythicism and other denialist movements like those of the "climate change skepticism" or 9/11 truthers are pretty easy to identify, much as it may gall the mythicists. All denialist movements are ultimately ideologically motivated, and all involve the highly selective use (and criticism) of the available evidence. More importantly, as valid as denialist criticisms of the prevailing theory may occasionally be, their ability to present an alternative theory - which better explains the evidence, with a minimum of superfluous pluralities - is generally laughable. The mythicist case is no different.
Basically, regardless of the details, the rejection of a historical Jesus necessitates the positing of some other historical origin for the early Christian movement. This, for most mythicists, necessitates the claim that Christianity was actually created by Paul, who by this logic believed in a purely heavenly Jesus, and that the flesh-and-blood Jesus of history was a mere literary contrivance of the four subsequent gospel authors. Now I can't detail all the problems with this assumption, because we'd be here all day, but a few difficulties off the top of my head would include:
- What was Paul's motivation for creating these
beliefs? Where did he get the idea of a heavenly messiah, beaten and
crucified in heaven to atone for our sins, when such beliefs had no
precendents in either Jewish or Hellenic thought?
- Why does the most natural reading of Paul appear to strongly
suggest that Jesus was someone who walked on the face of the Earth, with
not a single unambiguous indication anywhere in his letters of a belief
a pre-resurrection heavenly Jesus?
- What happened to such beliefs as time passed? Why does not a
single Christian in the first two centuries of Christianity - or any
time since then - profess belief in a Jesus who never walked the face of
the Earth? How did Paul's theology get so thoroughly garbled and
misunderstood so quickly?
- Where did the gospel authors get their historical details
concerning the life of Jesus from? Why would they have been motivated to
situate a heavenly redeemer on the Earth if they didn't believe that to
be the case? More to the point, if the gospel authors were merely
inventing historical details to furnish the theology started by Paul,
why (with the possible exception of Luke) do they show so little
awareness of Pauline thought?
- Why did none of the early critics of Christianity - who left almost no area of the faith immune from criticism - make no mention of the idea that Jesus never existed? Surely this would have been a useful polemic for them to use if it had ever existed in the cultural millieu of the time?
Now note that these are entirely contrived problems, unique to the theory of Jesus mythicism. The only inherent difficulty with the historical Jesus theory I can find concerns the lack of solid evidence, but the mythicist theory also has this problem (as I said, not a shred of incontestible evidence that a single Christian ever believed in the purely heavenly Jesus of the mythicist theory!) in addition to the problems listed above. So really, I can only ask which seems more plausible: the idea that there was an itinerant prophet called Jesus - the man that the early Christians wrote about - or the contrived and convoluted jumble of illogic found above? Even if we presume that the evidence for both claims is equal (something that I would dispute), which explanation is the most parsimonious, requiring the smallest number of pluralities and presumptions to explain the data? Without pre-empting your answer, I think it's telling that the mythicist explanation requires so many more leaps in logic or unfounded presumptions than the supposedly tenuous theory it seeks to replace. So, when someone asks me why I believe that there was a historical Jesus, this is the answer I give them: it's simply by far the most probable explanation for the available evidence that we have. Perhaps the day will come when a better alternative explanation offers itself, but until then the "Jesus as historical figure" theory is the only one to explain the data without resorting to fantastic assumptions or contorted chains of logic.