NO EXPLICIT CLAIMS OF MOSAIC AUTHORSHIP
Jews and Christians believe that the Torah we have today represents the original words of God as dictated to Moses, who wrote them down. What we find when we examine the different versions of the Torah is that there is no claim within the Torah itself that Moses wrote all 5 books. To the contrary, there are strong indications that they were written by someone other than Moses.
For example, there is a consistent use of the third person narrative throughout the Book of Exodus:
Then He said, “I am the God of your father, the God of Abraham, the God of Isaac and the God of Jacob.” At this, Moses hid his face, because he was afraid to look at God. [Exodus 3:6]
God said to Moses, “I am who I am. This is what you are to say to the Israelites: ‘I am has sent me to you.’” [Exodus 3:14]
Clearly, a third person such as a scribe or chronicler wrote these accounts, and not Moses himself. If Moses had been the writer then we would see the first person narrative being used.
Another issue is that of anachronisms. These are details that do not fit in with the supposed time of writing. For example, here the death of Moses happens before the book of Deuteronomy ends:
Moses was a hundred and twenty years old when he died, yet his eyes were not weak nor his strength gone. [Deuteronomy 34:7]
Since then, no prophet has risen in Israel like Moses, whom the Lord knew face to face. [Deuteronomy 34:10]