What reason do we have to believe it did happen? Why should we believe the fanciful tale in the Bible?
Archaeologists haven't produced enough evidence to back the story as written.
For one: the numbers involved in the Exodus are absolutely absurd and yet we have no trace of these people, despite them being millions strong
As for conquest (which is the second part of the tandem): the amount of places that were actually destroyed in the correct period as claimed by Joshua are very limited (I know Bruce Halpern claims one or two fit the period but Dever and Finkelstein provide many examples that weren't and almost none that do). Most places were not or may not have even been occupied at the time. The later writers used their own context and projected it back, and it didn't always work out or match the archaeological record.
Scholars nowadays think that Israelites were not some horde that swept in, separate from Canaanites. They were either Canaanites or Canaanites plus some nomadic groups that settled down in what we consider "Israel's" territory. It's actually hard to distinguish them from canaanites early on, despite all their claims to special status and 500 years of Egyptian living.
Moreover: in the appropriate time period Egypt still had control over Canaan. Enough slaves to populate a nation ran from Egypt through Egyptian controlled forts to Egyptian territory and no one notice or it isn't mentioned? Some have suggested that this Egyptian control of Canaan would be a good seed for a narrative of Egyptian dominance being ended by Yahweh without needing a horde of Hebrew slaves actually be in Egypt.
Moreover, it was not necessary to have lived in Egypt to suffer from Egyptian oppression. In a series of landmark essays, Ronald Hendel has advanced the thesis that memories of late Bronze Age (fifteenth to twelfth century) Egyptian colonial rule in Canaan survived in the Exodus story.28 The Exodus myth could have drawn its key motifs from an anticolonial, “nativist” resistance movement against the colonizing power,29 gaining all the more in forcefulness and emotional impact from the mid-eighth century onward, as colonial oppression resumed under the Assyrians and continued unabated under the Babylonians, Persians, Seleucids, and Romans.
Assmann, The Invention of Religion
The version of the story we have seems to be deliberate nation-making propaganda from Judahite and post-Exilic powers.
The truth is that virtually every archaeologist who has investigated the story of the Exodus, with very few exceptions, agrees that the way the Bible describes the Exodus is not the way it happened, if it happened at all....When we look for convergences, how do the various books of the Hebrew Bible fare as a history? A story on this very topic appeared in The New York Times on July i9, 2000, entitled “The Bible, as History, Flunks New Archaeological Tests.” But does it? Perhaps the books of Exodus and Numbers do, because as we have seen their accounts of escape from Egypt, of wandering in the wilderness, and of massive conquests in Transjordan are overwhelmingly contradicted by the archaeological evidence. That may make many uncomfortable, but it is a fact, one from which no openminded person can escape. There is little real history in these books, although there may be some vague memories of actual events, as I shall argue presently. For example, we may suppose that a historical figure like the biblical Moses did exist and was recognized as a leader among a group of Semitic slaves in Egypt, someone who indeed seemed to be a miracleworker, and perhaps the mediator of knowledge about the new deity Yahweh
William Dever, Who were the Israelites and where did they come from.
And Dever himself notes that he's more charitable than others might be.
He also said:
But of course that doesn’t mean that we agree about everything.) The conquest model is not subscribed to by most biblical scholars today—certainly no one in the mainstream of scholarship—and that’s been true for some time. Moreover, there isn’t a single reputable professional archaeologist in the world who espouses the conquest model in Israel, Europe or America. We don’t need to say any more about the conquest model. That’s that
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Such questions about historical reality end in a blind alley. The biblical stories contradict each other, while extrabiblical sources are in short supply. The “historical Moses” has vanished into thin air and a verifiable exodus cannot be reconstructed from the stories.
Assmann This time.
Putting aside the possibility of divinely inspired miracles, one can hardly accept the idea of a flight of a large group of slaves from Egypt through the heavily guarded border fortifications into the desert and then into Canaan in the time of such a formidable Egyptian presence. Any group escaping Egypt against the will of the pharaoh would have easily been tracked down not only by an Egyptian army chasing it from the delta but also by the Egyptian soldiers in the forts in northern Sinai and in Canaan.
...
The conclusion—that the Exodus did not happen at the time and in the manner Kadesh-barnea for thirty eight of the forty years of the wanderings. The general location of this place is clear from the description of the southern border of the land of Israel in Numbers 34 . It has been identified by archaeologists with the large and well-watered oasis of Ein el-Qudeirat in eastern Sinai, on the border between modern Israel and Egypt. The name Kadesh was probably preserved over the centuries in the name of a nearby smaller spring called Ein Qadis. A small mound with the remains of a Late Iron Age fort stands at the center of this oasis. Yet repeated excavations and surveys throughout the entire area have not provided even the slightest evidence for activity in the Late Bronze Age, not even a single sherd left by a tiny fleeing band of frightened refugees.
...
The pattern should have become clear by now. Sites mentioned in the Exodus narrative are real. A few were well known and apparently occupied in much earlier periods and much later periods—after the kingdom of Judah was established, when the text of the biblical narrative was set down in writing for the first time. Unfortunately for those seeking a historical Exodus, they were unoccupied precisely at the time they reportedly played a role in the events of the wandering of the children of Israel in the wilderness.
Israel Finkelstein.