Let me start by proposing that history is not a mere pile of facts that we prove or disprove one by one and add to the pile. There are many things that may or may not have happened, but we can still learn lots from them.

A great example of this is in the book by Alfred Fabian Young, The Shoemaker and the Tea Party: Memory and the American Revolution (Boston, Mass.: Beacon Press, 1999). Young looks at the memories and tales told by George Robert Twelves Hewes about his role in the event that came to be known as "the Boston Tea Party." To pick one example of many in the book, Hewes claims that he worked shoulder to shoulder dumping tea overboard with the great Samuel Adams and John Hancock. While history seems to indicate that John Hancock was NOT actually involved in the physical dumping, it is unclear about where Adams was. Now, Young could have given up writing the book because he couldn't establish Hewes' memories as fact, but instead he uses the stories to illustrate how the rhetoric and ideology of the revolution infused Hewes' memory -- and to show what resonated with his eagerly listening audiences in the 1820s as he told his tales. Young reveals a story about the promise of egalitarian democracy and freedom from class bondages in Hewes' memories; and in that story, the question of the 'facts' of the Boston Tea Party is almost totally irrelevant.

"Ideology did not set Georges Hewes apart from Samuel Adams or John Hancock. The difference lies in what the Revolution did to him as a person. His experiences transformed him, giving him a sense of citizenship and personal worth. Adams and Hancock began with both. [...] John Hancock and George Hewes breaking open the same chest at the Tea Partty remained for Hewes a symbol of a moment of equality."

I'll end by eloquently copy-pasting something I posted in another thread recently:

Basically, I think every historian comes to their own philosophy/epistemology/ontology of the past (how can we know what we know; what does exist and in what forms). However, there are certain broad movements in theory that basically set a standard for the field, upon which contemporary practitioners usually practice variations. (E.G. Post-modernism) If you are interested in reading scholars' various takes on the topic, two short and simple reads (if outdated!)that are good to start your journey with are:

• Marc Bloch, The Historian’s Craft (New York: Knopf, 1953)
• Edward Hallett Carr, What Is History? (New York: Knopf, 1962).

Heavier reads include:

• Peter Novick, That Noble Dream: The “Objectivity Question” and the American Historical Profession (Cambridge University Press, 1988).
• Gabrielle M. Spiegel, “The Task of the Historian,” The American Historical Review 114, no. 1 (2009): 1–15, doi:10.1086/ahr.114.1.1.

I can give a perspective from archaeology, because we deal with this all the time also. We have lots of "facts" in that I can see the remains of a building or hold a pot in my hands, but what those mean is very much an interpretive effort and so establishing facts based on those material things is a hugely difficult task.

The first thing they teach you in archaeology is that context is everything. A single wall doesn't really tell you much. A single fork doesn't tell you much. It is the association of an object in relation to all the other objects around it that give it meaning and allow us to interpret it.

It is really just an extension of what you do in your everyday life. How do you know which doors you should or should not walk through? You use contextual clues about whether the door is open or closed, locked or unlocked, clear or opaque, where in a building it is, signs hung around it, and what rooms it has access to. That it is a door really doesn't tell you much about it unless you have that contextual information.

We establish a lot of archaeological "facts" based on this contextual information as it reoccurs in multiple instances. A basic archaeological fact would be a chronology of pottery types. So, for instance, two or three pottery types might have been made for two or three hundred years before being replaced with two or three different types of pottery. We know this as a "fact" because we only ever find those first three types at archaeological sites in context together, and the second three in context together, but never a mixing of the two. This is very simplified, and it pretty much always messier than the example I'm giving, but that relates the idea at least.

Of course, it isn't so easy, and there is a huge amount of literature discussing how to interpret the archaeological record and establish what we do and do not know about an archaeological site. For instance, how do we know that a pot we found in a room was actually used by people living in that room or if it was put there by someone else when the room was abandoned. That is a very basic archaeological question that is fairly easy to answer (but perhaps more complicated than you would think), but these are the sort of "taphonomic" issues we have to deal with all the time. Basically, how did the artifacts we are looking at end up where they ended up. This is really important because it totally changes the way we interpret them. For instance, if I were to go into my kitchen right now I would find pots and pans in the cupboards and under the sink. The odds that I go into the kitchen and find them being used on the stove are much smaller than finding them in storage. If I didn't think critically about the situation and the contextual information, I might come to the reasonable conclusion that the person who uses that kitchen cooks their meals under the sink. Of course that is preposterous, but that is why archaeologists spend a lot of time talking about how to interpret material remains without knowing what they are a priori.

Then there is the question of interpreting these artifacts and places once we have a pretty good idea of how they were used. In other words, why do we care that you cook on the stove and not under the sink? The tricky part here is that, unlike historical documents, archaeologists have a very hard time talking about specific events. I can't really tell you much about how one person used this particular cooking pot, but I can probably tell you about how a whole population of people used these very similar cooking pots for similar activities.

For the most part, archaeology can't tell you much about individuals, but it can tell you a lot about populations and what people are doing in aggregate. We can say something about an individual artifact, but I don't think any archaeologist is going to claim there are many instances were one artifact can establish some historical "fact". It is only by looking at large numbers of artifacts and comparing with each other that we can say with some certainty that our interpretation is a "fact". For example, if a particular type of building with images of a particular deity show up in the same places in multiple villages, we can say with pretty good certainty that people in that region worshiped this deity in that kind of religious building. We can even potentially say that about villages we haven't excavated, because our sample is so large. Of course there are always exceptions, but the more evidence we have and the more similar it all is to each other, the more sure we can be of how "factual" something is.

We also have another criteria for establishing an archaeological fact in addition to having a large sample. Generally, we like to see multiple lines of evidence to really solidly establish something as fact. Basically, if we reach the same interpretation from different kinds of evidence, that strengthens the reliability of that interpretation. So, maybe we can say that it is a fact that people in region #1 were trading with people in region #2. We might be able to say that because we find two different styles of pottery in both regions and we have figured out that one kind is made in region #1 and the other kind is made in region #2. That is pretty compelling, but what if we find that shell jewelry made from a particular type of marine shell is found in both regions, but only the people in region #1 are actually by the sea and can access that shell. That is also evidence for trade between the two regions, but both together really strengthen the whole argument. I could say that potentially the people in Region #2 were going down to the coast to get shell for themselves and not trading with Region #1, but if we add in the ceramic evidence it supports trading between the two.

Basically, we can never really know something as "fact" in archaeology or history because we can't directly observe what happened, but we can say that certain things are more likely than others. I'm not going to argue that pyramids couldn't have been built by aliens, but given everything else we know (the multiple lines of evidence and the context here), it is incredibly unlikely that is the case. The most reasonable and most well supported answer is what we generally consider archaeological fact, even if there is always some ambiguity about what exactly happened.

Edit: Going to echo the "history isn't a pile of facts". Same with archaeology. Yeah, we spend a lot of time just establishing what people did and when, but the interesting part comes in interpreting the why.

I think one important point to make is that historians rarely speak in absolutes. I am a great fan of the "In Our Time" radio show/podcast, which chooses one topic, per episode, from the fields of history, science, religion, etc, and invites a panel of experts to discuss it. Many of these experts are pioneers in their field, and almost without exception they are professors/lecturers/researchers. The only term heard more frequently than the host's indroductory "hello" is "the sources tell us...". Even with archaeological corroboration, there aren't many things that are treated as "facts"