Where was the Canaanite city of Arad? Now referred to as "Tel Arad" (in contradistinction to the modern Israeli city of Arad), it was located about 20 miles west of the Dead Sea. The location of Hormah is not so clear, but it was apparently located near Arad. Similarly, the location of Atharim is not positively known, but it was presumably a roadway in southern Canaan (one scholar suggests that it was a primary route between Arad and Kadesh-Barnea).
The obvious problem is that Numbers describes Hormah as being conquered by Israel before the Israelites crossed over into Canaan! Thus, Joshua 12:7,14:
And these are the kings of the country which Joshua and the children of Israel smote on this side Jordan on the west... The king of Hormah, one; the king of Arad, one...
and Judges 1:17:
And Judah went with Simeon his brother, and they smote the Canaanites that inhabited Zephath, and utterly destroyed it. And the name of the city was called Hormah.
How could the king of Arad war against Israel when the latter were still on the eastern side of the Jordan?? Ramban discusses this chronological problem at considerable length. (Chavel translation used here.) He states that the king of Arad heard of the coming of Bnai Yisrael so he travelled by way of Atharim to the plains of Moab to fight against Israel, that Israel took a vow to dedicate everything that they had if Arad was conquered, and that the vow was finally fulfilled after the death of Joshua at which time they named the city Hormah (="Utter Destruction"). Ramban continues that even though the vow was fulfilled later, Scripture "completed the account here, just like it did in the section regarding the falling of manna". Thus in Exodus 13:35 "And the children of Israel did eat the manna forty years, until they came to a land inhabited; they did eat the manna, until they came unto the borders of the land of Canaan".
Ramban mentions another example in Numbers 34:17: "These are the names of the men that shall take possession of the land for you: Eleazar the priest, and Joshua the son of Nun." This passage, he says, "constitutes a prophecy that these men will live and function at that time, for it is impossible to say that God would specify for them men about whom there was a doubt as to whether they would still be living; for otherwise He should have commanded Joshua at the time of the division of the land. Since the command was given to Moses, we see that the Torah speaks of future events."
Ramban continues with another explanation that also relies on a prophetic understanding of the verses, suggesting that the king and his people were destroyed during the time of Moses and that they named the place of battle Hormah. Later, during the time of Joshua, they called the cities Hormah - because they were utterly destroyed - and they then fulfilled their vow.
I find Ramban's explanation unconvincing, but not primarily because of his reliance on prophecy to explain the chronological problem. Note that he states that the king of Arad heard of the coming of Bnai Yisrael so he [the king] travelled by way of Atharim to the plains of Moab. He thus engages in a twisting of the translation so that the king of Arad travels from Canaan to Moab - rather than have Bnai Yisrael traveling to Canaan - to better fit his exegesis. Now Ramban was certainly more expert than I in Hebrew, but this does seem like a stretch: וַיִּשְׁמַע הַכְּנַעֲנִי מֶלֶךְ-עֲרָד, יֹשֵׁב הַנֶּגֶב, כִּי בָּא יִשְׂרָאֵל, דֶּרֶךְ הָאֲתָרִים;
But my intention is not to argue with Ramban on his translation. It is to simply state that his approach demonstrates one of the fundamental differences between the traditional approach and a scholarly one. The former accepts the notion of prophecy, the latter does not. Thus a traditionalist has no problem explaining an anachronism - such as is found here - as being indicative of a prophetic statement. A Biblical scholar will simply say that it is an error in chronology and thus proof of a later authorship or compilation.
Because an obscure or problematic passage can almost always be made to fit a subsequent historical event after the fact, claims of prophecy are difficult - if not impossible - to falsify. Fans of Nostramadus are guilty of using this to their advantage, as are Christian missionaries and - of course - JOFs (Jewish Orthodox Fundamentalists). Torah code proponents (mostly a subgroup of OrthoFundies) likewise rely on carefully crafted "unfalsifiable code prophecies."
This is one area in which I don't see any room for debate between the assertions of the traditional and scholarly approaches to Bible. One accepts a priori the notion of prophecy, the other denies its existence altogether.