... continued from yesterday.
The scholarly story of Matthew is not as romantic as the one we hear more often. Normally, we hear of how Matthew was a tax collector that Jesus called to be a disciple. Then later Matthew, under the inspiration of the Holy Spirit, wrote down his reminiscences of Jesus' earthly ministry. We get the impression that Saint Matthew sat down one day (or two) and the Holy Spirit inspired him to write down his memories of his time with Jesus two or three decades earlier.
God's inspiration of the Gospel of Matthew was probably a little more complicated and less romantic. The book of Matthew itself is anonymous. Nowhere in its pages does it tell us who its author is. There is an early tradition by a man named Papias who wrote this: "Matthew collected the sayings [of Jesus] in the Aramaic language, and everyone translated them as best they could." 
It is not clear that Papias was talking about the Gospel of Matthew as we have it. Nevertheless, at some point early on, the heading, "The Gospel according to Matthew," started circulating as the title of this document. It is the only name we have offered from the ancient world for the name of its author. My sense is that these sort of traditions usually preserved some kernel of truth but also that they also usually got a little mangled and romanticized.
Here is the scholarly story. The Gospel of Mark was written first, probably in the late 60s or early 70s of the first century, probably primarily to a Gentile Christian audience. Then a Christian Jew, sometime after the destruction of Jerusalem, perhaps in the 70s, took the Gospel of Mark and created the Gospel of Matthew out of it.
[insert text box on the Synoptic Question]
I should probably stop and remind you that God can inspire a person to use sources just as much as he can dictate to you. In fact, on the romantic model, Matthew was remembering things. That is, his memory was a source, helped by the Holy Spirit. Certainly 1 and 2 Kings mention sources like the Annals of the Kings of Judah, and Jude quotes a book called 1 Enoch.
So what did the author of Matthew add to Mark to make the Gospel of Matthew? Basically, Matthew adds five large sermons scattered throughout. Then he added the Virgin Birth and resurrection details on the front and the end. If you take Mark and drop these sermons into it, add the introduction and conclusion, then you have Matthew.
Now the author of Matthew had to have a source for these sermons. One possibility is of course that the source of these sermons was a collection of Jesus' sayings that in some way went back to the disciple Matthew. While this is mostly speculation, it would be typical of how traditions often worked in the ancient world. The Gospel of Matthew would get its name from one of its major sources but the precise story would have grown a little. 
The fact that there are five sermons has not been lost on interpreters of Matthew, since the cornerstone of the Jewish Scriptures is the five books of the Law, the Pentateuch. Could it be that the author of Matthew was thinking of the Law when he added these five sermons?  After all, Matthew arguably presents Jesus as the authoritative interpreter of the Law, and as we will see, the way it tells Jesus' birth story makes us think of Moses.
Was it the disciple Matthew who put this all together? On the whole it seems more likely that material from Matthew was a source for the Gospel than that the actual disciple Matthew put the Gospel together into its current form. It's certainly possible, just not the most likely conclusion. For example, why would an eyewitness (Matthew) draw on a non-eyewitness (Mark) as his primary source? And although Greek was certainly spoken in Palestine, the only evidence we have of Matthew suggests he preferred Aramaic (or Hebrew).
Nevertheless, I will refer to the author of the Gospel as Matthew. If this sort of issue is annoying, it's important to realize that half the things that upset us are actually traditions we have learned about the Bible rather than things in the Bible itself. The Gospel of Matthew does not tell us who its author was--the titles were surely added later. But since we grow up forming our faith around the Bible as we know it--or the version in which we read it--we can feel like our faith is being challenged when in fact it is only our traditions about faith that are in question...
 My translation. Found in Eusebius, Ecclesiastical History, 3.39.16. Three elements of this translation are highly debated. Did Matthew collect sayings or stories? Was it in Aramaic or Hebrew? Did people translate or interpret them?
 One problem with this speculation is that the sermons of Matthew are not exactly "translation Greek." Some of it could be, although it is virtually impossible to trace it without wild speculation. Assuming the Gospel of Matthew used a written source, it would likely have been a Greek source.
 B. W. Bacon is often credited with first developing this idea. ***