One of the reasons I have rejected Christianity is that I studied the Bible. That's right. I studied the Bible. As I did so, I didn't just read works published by Zondervan or InterVarsity Press. I read the works by Christian scholars from a wide variety of scholarly sources. I didn't read atheist works about the Bible so much as I mainly read scholarly Christian literature. What Christian scholars wrote led me to reject Christianity. For those of you who read my "Letter to Dr. James Strauss" in my book, you know some of the books I read, and almost every book I mentioned (and there were plenty I didn't) was written by a Christian, or someone within the Christian tradition.
Now as we approach the Christmas season, Let's see what Christian scholarship says about the infancy birth narratives. Raymond Brown is the author of the massive 752 page book titled, Birth of the Messiah (updated 1999), and "Gospel Infancy Narrative Research from 1976 to 1986" (CBQ 48: 469–83, 661–80). Here are four points about the contents of Matthew and Luke that Brown mentioned in the Anchor Bible Dictionary (1996) ["Infancy Narratives in the NT Gospels"] (emphasis is mine):
(1) [Brown discusses the agreements between Matthew and Luke’s gospels, but those are obvious and not part of my point]
(2) Matthew and Luke disagree on the following significant points. In chap. 1, the Lucan story of John the Baptist (annunciation to Zechariah by Gabriel, birth, naming, growth) is absent from Matthew. According to Matthew, Jesus’ family live at Bethlehem at the time of the conception and have a house there (2:11); in Luke, they live at Nazareth. In Matthew, Joseph is the chief figure receiving the annunciation, while in Luke, Mary is the chief figure throughout. The Lucan visitation of Mary to Elizabeth and the Magnificat and Benedictus canticles are absent from Matthew. At the time of the annunciation, Mary is detectably pregnant in Matthew, while the annunciation takes place before conception in Luke. In chap. 2 in each gospel, the basic birth and postbirth stories are totally different to the point that the two are not plausibly reconcilable. Matthew describes the star, the magi coming to Herod at Jerusalem and to the family house at Bethlehem, the magi’s avoidance of Herod’s plot, the flight to Egypt, Herod’s slaughter of Bethlehem children, the return from Egypt, and the going to Nazareth for fear of Archelaus. Luke describes the census, birth at a stable(?) in Bethlehem because there was no room at the inn, angels revealing the birth to shepherds, the purification of Mary and the presentation of Jesus in the temple, the roles of Simeon and Anna, and a peaceful return of the family to Nazareth.
(3) None of the significant information found in the infancy narrative of either gospel is attested clearly elsewhere in the NT.In particular, the following items are found only in the infancy narratives. (a) The virginal conception of Jesus, although a minority of scholars have sought to find it implicitly in Gal 4:4 (which lacks reference to a male role), or in Mark 6:3 (son of Mary, not of Joseph), or in John 1:13 (“He who was born . . . not of the will of man”—a very minor textual reading attested in no Gk ms). (b) Jesus’ birth at Bethlehem, although some scholars find it implicitly in John 7:42 by irony. (c) Herodian knowledge of Jesus’ birth and the claim that he was a king. Rather, in Matt 14:1–2, Herod’s son seems to know nothing of Jesus. (d) Wide knowledge of Jesus’ birth, since all Jerusalem was startled (Matt 2:3), and the children of Bethlehem were killed in search of him. Rather, in Matt 13:54–55, no one seems to know of marvelous origins for Jesus. (e) John the Baptist was a relative of Jesus and recognized him before birth (Luke 1:41, 44). Rather, later John the Baptist seems to have no previous knowledge of Jesus and to be puzzled by him (Luke 7:19; John 1:33).
(4) None of the events that might have been “public” find attestation in contemporary history. (a) There is no convincing astronomical evidence identifiable with a star that rose in the East, moved westward, and came to rest over Bethlehem. In Matthew’s story this would have happened before the death of Herod the Great (4 b.c. or [Martin 1980] 1 b.c.). There have been attempts to identify the star with the supernova recorded by the Chinese records in March/April 5 b.c., or with a comet (Halley’s in 12–11 b.c.), or with a planetary conjunction (Jupiter and Saturn in 7 b.c.; Jupiter and Venus in 3 b.c. [Martin 1980]). (b) Even though the Jewish historian Josephus amply documents the brutality in the final years of Herod the Great, neither he nor any other record mentions a massacre of children at Bethlehem. Macrobius’ frequently cited pun (Sat. 2.4.11) on Herod’s ferocity toward his sons is not applicable to the Bethlehem massacre. (c) A census of the whole world (Roman provinces?) under Caesar Augustus never happened, although there were three Augustan censuses of Roman citizens. It is not unlikely that Luke 2:1 should be taken as a free description of Augustus’ empire-cataloguing tendencies. (d) Luke’s implication that Quirinius was governor of Syria and conducted a “first census” (2:2) before Herod’s death (1:5) has no confirmation. Quirinius became legate of Syria in a.d. 6 and at that time conducted a census of Judea, which was coming under direct Roman administration because Archelaus had been deposed (Brown 1977: 547–56; Benoit DBSup 9: 704–15). (e) Although this item differs somewhat from the immediately preceding one, Luke’s idea that the two parents were purified (“their purification according to the Law of Moses”: 2:22) is not supported by a study of Jewish law, whence the attempts of early textual copyists and of modern scholars to substitute “her” for “their” or to interpret the “their” to refer to other than the parents.
A review of the implication of nos. 1–4 explains why the historicity of the infancy narratives has been questioned by so many scholars, even by those who do not a priori rule out the miraculous. Despite efforts stemming from preconceptions of biblical inerrancy or of Marian piety, it is exceedingly doubtful that both accounts can be considered historical. If only one is thought to be historical, the choice usually falls on Luke, sometimes with the contention that “Those who were from the beginning eyewitnesses and ministers of the word” (Luke 1:2) includes Mary who was present at the beginning of Jesus’ life. See Fitzmyer Luke I–IX AB, 294, 298, for the more plausible interpretation that it refers to the disciples-apostles who were eyewitnesses from the beginning of Jesus’ public life (Acts 1:21–22) and were engaged in a preaching ministry of the Word. There is no NT or early Christian claim that Mary was the source of the infancy material, and inaccuracies about the census and purification may mean that Luke’s infancy account cannot be judged globally as more historical than that of Matthew.
[For Richard Carrier's assessment of the date of the Nativity in Luke see here.]