Why is there a controversy over who wrote Deuteronomy?

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Moses with the tablets of the Ten Commandments...
Moses with the tablets of the Ten Commandments, painting by Rembrandt (1659) (Photo credit: Wikipedia)
By its own testimony (Deut. 1:1, 5; 31:22), Deuteronomy is the work of Moses. Mosaic authorship is affirmed many times elsewhere in the Old Testament (e.g. 2 Kin. 14:6), in ancient Jewish sources (e.g., Josephus), and in the New Testament. This view was almost universally held until the rise of rationalistic criticism in modern times.

Critics correctly point out that the last chapter could not have been written by Moses. It is widely agreed that ch. 34 is an addendum, perhaps appended by Joshua. In the same way, the Book of Joshua ends with the death of Joshua, this record clearly having been supplied by the author of the Book of Judges, who appended verses from Judges to the end of Joshua (Judg. 2:7–9; cf. Josh. 24:29–31). 

 Likewise, the first verses of Ezra are copied and appended to the last chapter of Chronicles (Chronicles ends in the middle of a sentence). This way of linking a subsequent book to the preceding one (or variations of this practice) was common in antiquity and was intended to show the proper sequence of scrolls or clay tablets. It is probable that Joshua added the note on Moses' death and Israel's acceptance of Joshua in order to link his own book to Moses' great production. Such obvious additions do not, however, negate the general authorship of Moses.

More controversially, some critics have alleged that the language of Deut. 1:1, 5 indicates that the writer of the book must have been located on the west side of the Jordan River, i.e., in Canaan (the Hebrew phrase here translated "beyond the Jordan" is often translated "the other side of the Jordan" or "this side of the Jordan"). 
Such a description, they argue, belies the credibility of Deuteronomy as a work of Moses, since Moses never crossed the Jordan. This argument assumes that the Hebrew phrase in question must always refer to the region east of the Jordan. It is evident, however, that the precise meaning of the phrase must be determined by the context, and that it can refer either to Transjordan (the region to the east of the Jordan and the Dead Sea, Deut. 1:1–5; 3:8; 4:41, 47, 49) or to Canaan itself (Deut. 3:20, 25; 11:30; Josh. 9:1, 10). Here it clearly means the region east of the Jordan, as the geographical descriptions indicate (Deut. 1:1, 5).
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