Herodotus has an interesting, if somewhat strange story:
Now the Egyptians, before the reign of their king Psammetichus, believed themselves to be the most ancient of mankind. Since Psammetichus, however, made an attempt to discover who were actually the primitive race, they have been of opinion that while they surpass all other nations, the Phrygians surpass them in antiquity. This king, finding it impossible to make out by dint of inquiry what men were the most ancient, contrived the following method of discovery:- He took two children of the common sort, and gave them over to a herdsman to bring up at his folds, strictly charging him to let no one utter a word in their presence, but to keep them in a sequestered cottage, and from time to time introduce goats to their apartment, see that they got their fill of milk, and in all other respects look after them. His object herein was to know, after the indistinct babblings of infancy were over, what word they would first articulate. It happened as he had anticipated. The herdsman obeyed his orders for two years, and at the end of that time, on his one day opening the door of their room and going in, the children both ran up to him with outstretched arms, and distinctly said "Becos." When this first happened the herdsman took no notice; but afterwards when he observed, on coming often to see after them, that the word was constantly in their mouths, he informed his lord, and by his command brought the children into his presence. Psammetichus then himself heard them say the word, upon which he proceeded to make inquiry what people there was who called anything "bekos," and hereupon he learnt that "bekos" was the Phrygian name for bread. In consideration of this circumstance the Egyptians yielded their claims, and admitted the greater antiquity of the Phrygians.
There's always some worry about Herodotus's reliability, in particular when dealing with civilizations at or beyond the margins of Greek familiarity, and in particular with Egypt (there is debate of whether he did, in fact, visit that land personally, although he claims to have done so). He certainly doesn't mind picking out the most outrageous tidbits to share from time to time. Thus, it might be dangerous to assume that this story is word-for-word true, or even more so to assume that the pharaoh's results were at all useful. It is still an interesting view into the concept that there was some worry over the prestige of whose civilization was "oldest," though, as well as a wonderful insight into a so-close-but-yet-so-far attempt at scientific/experimental thinking.
So I'm only gonna be talking about the Sumerians here. The only way that we are going to have any knowledge of what the first civilizations knew of their past is through written records. Writing first appears in Sumer in around 3000 BC, this is during the Uruk period (4000-3001 BC). At this time it was very much proto-writing and did not develop into a full written language until the Early Dynastic Period (2800-2500 BC)
It is not until the end of the Early Dynastic Period (2300 BC) that we see the emergence of the Sumerian King List which documents the kings of Sumer and gives us an insight into what the Sumerians may have believed about their past.
The King List, a collection of several sources, details the rulers of the cities of Sumer from the first antediluvian rulers to the last dynasty of Isin. The first king was Alulim, first king of Eridu. He ruled for 28,800 years. Now at first glance that may seem an unreasonably long rule and that is because it is. Alulim, the first king after the kingship descended from heaven created by the god Enki is clearly almost entirely mythical. This list of antediluvian kings ends with Ubara-tutu and the coming of the great flood that wipes the world clean.
The list resumes with Jushur the first of the dynasty of Kish. He ruled for 1200 years. 20 kings and 15255 years later we have En-me-barage-si who is the first king we have archaeological evidence for. He is dated from around 2600 BC from two pieces of alabaster vases found at Nippur which bear his name. Though he ruled for 900 years, a rather long time, it can be surmised that he is indeed real. He is mentioned also in the Epic of Gilgamesh alongside Gilgamesh himself giving credence to the thought that Gilgamesh is a historical figure. The King List continues into the time of rulers that can easily be verified such as Sargon of Akkad who ruled for 40 years and founded the Akkadian Empire. Thus we see the transition of the mythical into the semi-mythical and then verifiable history. But more on that later.
The Sumerian creation myth is important to note in this discussion. It recounts that the gods Enki among them created the first “black-headed people” (the Sumerians) and settled them in the land giving them the kingship and thus the first cities were created. A large part of the story is missing but at some point the gods decide not to save mankind from a flood which strikes destroying man and cites. Later the world is presumably repopulated.
In addition there is the “Debate between Summer and Winter” a creation myth from the mid 3nd millennium. This details the creation of the land and seasons by Enlil. In it he is seen to irrigate the land “guaranteeing the spring floods at the quay” and to begin the agricultural tradition of the land “making flax grow and barley proliferate.”
Finally and most interestingly for this topic is the “Debate between Sheep and Grain” another creation myth written in the mid-3rd century. The myth details a time in which sheep and grain were unknown to the land. The people “went about with naked limbs in the Land. Like sheep they ate grass with their mouths and drank water from the ditches.” The myth ends with the virtues of grain being extolled “from sunrise to sunset may the name of Grain be praised. People should submit to the yoke of grain.”
Therefore we can see that early history of Mesopotamia, the Ubaid period and before, is in Sumerian text seen in a divine light. The land was created by the gods as were the people and they were given cities and kingship. Enlil gave the people the summer and the winter, he gave them wheat and irrigation as Enki gave them kingship. Only in the “Debate between Sheep and Grain” is there indicated any knowledge of a time before sedentary agriculture. This myth clashes with that of the “Debate between Summer and Winter” though it is part of the same tradition indicating the lack of a unified view of their past. It seems that the Sumerians saw their past as part of a very real mythical tradition. Their kings begin as mythical figures and progress towards the non-mythical. The mythical and the non-mythical are closely linked in the Sumerian view of themselves and their past
I would conclude that the Sumerians did believe themselves to be not just the first civilization but the first people, it is part of their creation myth. In addition there was no knowledge in the way we would think of a hunter-gatherer life preceding their urban civilization. If there is any hint it exists as another facet of the extensive and contradictory creation myth of the peoples of Sumer.
I recall Carl Sagan mentioning that the Library of Alexandria held something like ~10,000 years of recorded Babylonian history, all/most of which was destroyed when the library was burned to the ground. Does anyone know anything more about that? It's very interesting, and very depressing, considering the likely volume and value that was lost.
No Plato's Timaeus about Atlantis?