Basically, in my mind, the way that history seems to have made it's way into the books, not much of the early stuff is all that reliable. Even if we could translate it perfectly (which we can't in most cases) the simple fact that the people from those times believed in radical fantasies cannot be ignored. They made up stuff to explain phenomenons and even some stuff that they did know perfectly well about, but it just probably felt better a certain way on paper. Just like if you were to recall an 'epic moment' in your lifetime, you're likely to touch the facts... I mean, we look at religions alone, people argue that Jesus must have certainly existed, because of his documentation. Although there are very good criticisms. (I will not get into these, I don't feel like arguing over this anymore). Anyhow, if you're going to make a basis of something happening or existing, and saying that somehow maybe there was a small grain of truth in it, then why not Thor? Maybe there was a Viking with a hammer who smacked down some people during a thunderstorm, who knows? Why not start pulling all these ridiculous things out of our asses? Seriously, history has a few easy to pick out events, natural disasters (we can prove these happened with science), wars (there are usually many accounts, and sometimes even remains)?
So why do we lend any credit to ancient mythology? I really do want to know why any historian would try to prove something on a basis of 'half-truths' when there is no good way to prove it definitely (with science).
Often I find historians are on a quest to 'prove' something they think happened, isn't this against the core concept of trying to find out the actual truth?
This is a good question and I have been thinking a little about some related issues lately.
Allow me to work backwards and first take issue with your second last sentence "...when there is no good way to prove it definitely (with science)". This in my view is an unfortunate misunderstanding of the discipline of History. History is not Science.
Science as a modern discipline is concerned with explaining observable phenomena on the assumption of naturalistic causes, with procedures such as repeatable experimentation. Science, in our society, has experienced enormous success in the last 200 years. Which has led to the unfortunate situation where many people assume that the scientific method is the only method of knowledge. This is problematic. It's trying to fit everything into a single epistemology. The scientific method is not establishable by the scientific method.
History is not a science. Science is neither the origins of the discipline of history, nor is it a model for how to do history. One cannot run experiments on a large scale to replicate historical events. History is not open to repeat-testability. History emerged as a humanistic discipline aimed at answering the questions, "What happened?" and "Why did it happen?" and its product is meaningful narratives that connect the what- of history why- of history.
This leads into my second point: standards of evidence and what to do with them. Sometimes I encounter views like this following caricature: "Unless we have multiple eye-witness accounts written down simultaneously with the event, we can't even be sure that it ever occured at all." This is a bad way to do ancient history. It's setting up your standard of evidence and then blanketly asserting that nothing can be known. I suppose one could follow this methodology, but they would not be doing very much 'history'. They would mainly be writing books about how nothing can be known.
Instead, a far better approach is to see what evidences there are first, then work out what to do with them. My area involves a lot of documentary evidence from the 1-5th century. That's what there is to work with. So a responsible method is to assess those texts, work out what they say, what they purport to be, what they are trying to achieve, their reliability, and then produce an account that most plausibly accounts for those evidences. Indeed, that is what I would characterise as the task of writing history of ancients: what is the most plausible account for the evidences that we have.
When you come to things like ancient mythology, very rarely these days do you find historians trying to make arguments about whether Thor was a historical figure who got turned into a god. Instead the kinds of things you tend to encounter are studies of, "what role did mythology play in society", "what ideas of the culture did these mythological accounts express", "how did mythology function within the society"; mythology is not utilised as a source for trying to construct speculative histories. You say you don't want to discuss the Jesus question, but the comparison is not very compelling. The types of texts that discuss Jesus are not mythological-type texts. Whether one concludes that he existed or not aside, the question itself is worth investigating, and it goes back to what I said above: given the documents that exist, what is the most plausible account of their origin? This is entirely comparable to other founding figures of religions.
To conclude, let me say that as far as History is concerned with constructing plausible narratives of what happened and why, ancient history practices restraint and caution by tempering its claims according to the strength of the data. The stronger the evidence, the stronger our conclusions may be. In some areas of ancient history that may involve being speculative, which deserves the acknowledgement that some theories are speculative.
Like CommodoreCoCo I'd like you to clarify when and where a bit more. I'm also a tad confused since you start off with asking about ancient history but in the end you ask about ancient mythology - these aren't the same thing and ancient mythology is never taken seriously as historical truth. They are, however, used to understand the cultures they stem from.
Often I find historians are on a quest to 'prove' something they think happened, isn't this against the core concept of trying to find out the actual truth?
While it can appear to be that way, it's not really how it works. Often you work from an hypothesis which you research the probability of and while you certainly hope your hypothesis is true, it would be truly bad academics to work in a way that proves it while leaving out other facts - which would make it, like you say, "a quest to 'prove' something". Such studies are not well recieved by the historical community and they can often be ridiculous.
This is certainly an interesting question worth discussing. Can you first clarify what time you mean by "ancient" and in what part of the world? The responses about 5th century BC Greek texts, 1st century BC Roman texts, and 5th century AD Mayan texts will be very different.