Is the Hebrew Bible the History of Israel or of Judaism? 

Here, I outline a basic argument that Julius Wellhausen proposed in 1878 that the Hebrew Bible (or Law) reveals a history of Judaism during the Babylonian and Assyrian period. The argument here denies that the Hebrew Bible testifies to ancient Israel's history but was written much later than the books imply.
While I can see why Wellhausen might make such arguments, I think other explanations are plausible; and ultimately, I would argue that the Hebrew Bible is a work of theology which, yes, exists in real history and testifies to real history but aims primarily to tell us God and his mighty works.
In other words, due to the highly selective nature of the Hebrew Bible, Wellhausen might have been right that it can be tough to create a history of Israel. After all, the Bible narrates the whole of creation until the Captivity! All of this in a short number of pages!
No, the Bible is a theological book that includes history to make its theological point. But it cannot be a history book as we think of it.
Oh, and I am a hopelessly conservative. So I affirm Mosaic authorship of the Pentateuch. Have fun with that :).