The evidence reveals that the temple of the moon god was active even in the Christian area. Evidence gathered from both North and South Arabia demonstrate that moon god worship was clearly active even in Muhammad's day and was still the dominant cult.
According to numerous inscriptions, while the name of the moon god was "sin", his title was "al-ilah", meaning "the deity [the god]", further meaning that he was the chief or high god among the gods. As Coon points out, "The God Il or Ilah was originally a phase of the Moon God." [Carleton S. Coon, Southern Arabia (Washington, D.C.: Smithsonian, 1944) page 399]
The moon god was called al-ilah, the god, which was shortened to allah in pre-Islamic times. The pagan Arabs even used allah in the names they gave to their children. For example, both Muhammad's father and uncle had allah as part of their names. The fact that they were given such names by their parents proves that allah was the title for the moon god even in Muhammad's day. Professor Coon says, "Similarly, under Mohammed's tutelage, the relatively anonymous Ilah, became Al-Ilah, The God, or Allah, the Supreme Being." [Carleton S. Coon, Southern Arabia, page 399]
This fact answers the questions. "Why is allah never defined in the quran?" "Why did Muhammad assume that the pagan Arabs already knew who allah was?" Muhammad was raised in the religion of the moon god allah. But he went one step further than his fellow pagan Arabs. While they believed that allah [the moon god] was the GREATEST of all gods and the supreme deity in a pantheon of deities, Muhammad decided that allah wasn't the greatest of all gods, but the only god. In effect he said, "Look, you already believe that the moon god allah is the greatest of all gods. All I want you to do is to accept the idea that he is the only god. I am not taking away the allah you already worship. I am only taking away his wife and kids [the three daughters of allah] and all the other gods."
This is seen from the fact that the first point of the Muslim creed isn't "allah is great" but "allah is the greatest"---he is the greatest among the gods. Why would Muhammad say that allah is the "greatest" except in a polytheistic context? The Arabic word is used to contrast the greater from the lesser. That this is true is seen from the fact that the pagan Arabs NEVER accused Muhammad of preaching a DIFFERENT allah than the one they already worshipped. This "allah" was the moon god according to the archeological evidence.
Muhammad thus attempted to have it both ways. To the pagans, he said that he still believed in the moon god allah. To the Jews and the Christians he said that allah was their God, too. But both the Jews and the Christians knew better and rejected his moon god allah as a false god. Al-Kindi, one of the early Christian apologists against Islam, pointed out that Islam and its god allah didn't come from the Bible but from the paganism of the Sabeans. They didn't worship YHWH of the Bible but the moon god and his daughters, al-lat, al-uzza, and manat [Three Early Christian-Muslim Debates (ed. by N.A. Newman [Hatfield, PA: I.B.R.I., 1994], pages 357,413,426)]
Dr. Newman concludes his study of the early Christian-Muslim debates by stating, "Islam proved itself to be...a separate and antagonistic religion which had sprung up from idolatry." [Three Early Christian-Muslim Debates (ed. by N.A. Newman [Hatfield, PA: I.B.R.I., 1994], page 719]. Islamic scholar Caesar Farah concluded, "There is no reason, therefore, to accept the idea that Allah passed to the Muslims from the Christians and Jews." [Caesar Farah, Islam: Beliefs and Observations (New York: Barrons, 1987), page 28].
The Arabs worshipped the moon god as a supreme deity. But this WASN'T biblical monotheism. While the moon god was greater than all other gods and goddesses, this was still a polytheistic pantheon of deities. Now that the actual idols of the moon god are found, it's impossible to avoid the fact that allah was a pagan god in pre-Islamic times.
Is it any wonder then that the symbol of Islam is the crescent moon? That a crescent moon sits on top of their mosques and minarets? That a crescent moon is found on the flags of Islamic nations? That the Muslims fast during the month which begins and ends with the appearance of the cresent moon in the sky? Also, the idol of the moon god allah was at Mecca, the pagans faced toward Mecca and prayed to the moon god allah, Muslims today still does this today, praying fives times a day toward Mecca. These are a dead giveaway that allah is a moon god. These are visible evidence that they are still worshipping the moon god today.
The pagan Arabs worshipped the moon god allah by praying toward Mecca several times a day [Encyclopedia of Islam (ed. Eliade) pages 303. International Standard Bible Encyclopedia, page 1:219] making a pilgrimage to Mecca; running around the temple of the moon god called the Kabah; kissing the black stone; killing an animal in sacrifice to the moon god; throwing stones at the devil; fasting for the month that begins and ends with the crescent moon; giving alms to the poor; etc.