Down yonder, a reader writes:
Has there yet been proof one way or another as to whether St. Matthew
the Apostle is the Matthew who wrote the Gospel of Matthew? Ray Brown
gives credence to the Apostle via Tradition but some other Matthew by
Content. B16 apparently goes with Tradition. What sayest thou??
I haven’t looked up what Ray Brown said on the subject, so I can’t really comment on that. As to B16, he uses traditional language regarding biblical authorship without intending this language to be a comment one way or another on what critical scholarship would say on the subject. A careful reading of his words in the audience on Matthew shows that he is not trying to settle the question of the authorship of Matthew. Indeed, he is trying to avoid doing so, saying:
We recall, finally, that the tradition of the early Church agrees with
attributing the authorship of the first Gospel to Matthew. This was
already the case beginning with Papias, bishop of Gerapolis in Phrygia,
about the year 130.He wrote: "Matthew took up the Lord’s words in Hebrew, and each one
interpreted them as he could" (in Eusebius of Caesarea, "Hist. eccl.",
III, 39, 16). The historian Eusebius adds this detail: "Matthew, who
previously had preached to the Jews, when he decided to go also to
other peoples, wrote in their maternal tongue the Gospel he was
proclaiming: In this way he tried to substitute in writing what they,
whom he was leaving, lost with his departure" (Ibid., III, 24, 6).We no longer have the Gospel written by Matthew in Hebrew or
Aramaic, but in the Greek Gospel that has come down to us we still
continue to hear, in a certain sense, the persuasive voice of the
publican Matthew who, in becoming an apostle, continues proclaiming to
us the saving mercy of God.
This is a very neutral and ambiguous statement that can mean a number of things, from the (minimalist) idea that the gospel’s treatment of the figure of Matthew allows us in a sense to hear his voice due to learning from his example to the (maximalist) idea that the Greek gospel is just a translation of Matthew’s Hebrew or Aramaic original or anywhere in the middle (e.g., the Greek gospel of Matthew builds on the Hebrew or Aramaic one as one of its sources). B16 is thus trying to honor the traditional authorship of the gospel without committing to it.
When it comes to what I would say on the question, I consider the matter less decisive than the authorship of the Pauline epistles since there is no attribution of authorship contained within the document itself, but I give very significant weight to the authorship tradition for this document in the early Church and would say that there is significant evidence (not the same as proof) that Matthew was the author.
Some have argued this based on internal considerations. For example, W. Graham Scroggie (who deals with the authorship of the gospels extensively, albeit from a Dispensationalist perspective) argues that Matthew shows a preoccupation with money (as one would expect from a tax collector) the same way Luke shows a preoccupation with healing (as one would expect from a physician).
Even more than Scroggie’s book (which is good for raw data, though I don’t always like what he does with analysis), I’d recommend the introduction to the New Testament by Donald Guthrie, who also deals extensively with authorship issues.
I do think that there are indications that Matthew was not the product of a "community" (though it certainly responds to the needs of a community, and specifically a Jewish one) and that the authorial vision of a single individual is responsible for it. Regardless of what sources this individual may have used, the book exhibits far too much literary architecture and organization to be a patchwork document assembled without a single authorial vision.
As evidence for this fact, I would point both to large-scale structures in the work, like the fact that the sayings of Jesus that are scattered in Mark and Luke tend to be gathered into collected discourses on distinct topics that are then organized chiastically as follows:
1. The Sermon on the Mount: Jesus on the Law (ch. 5-7)
2. The Evangelization Discourse: The Church’s Outer Function (ch. 10)
3. The Kingdom Parables (ch. 13)
4. The Discipline Discourse: The Church’s Inner Functions (ch. 18)
5. The Olivet Discourse: Jesus on the Prophets (ch. 23-25)
If you look within these discourses, one also finds at times intense literary structuring. The Kingdom Parables as presented in Matthew (as opposed to in Mark 4) are themselves structured as a chiast, and the Sermon on the Mount is built around recurring phrases ("Blessed are the . . . " "You have heard, but I say . . . " "When you X, do not Y . . . " etc.) so that it exhibits clear internal structure, and in the Olivet Discourse Matthew organizes material that Luke has scattered all over the place.
The result is that Matthew’s gospel, while following the general synoptic account of the ministry and passion of Christ, has been organized in such a way that the teachings are gathered up into distinct discourses that are both internally structured and arranged in an overall structure that spans the whole book.
Of particular note is the fact that Jesus is presented in the Sermon on the Mount as the new Lawgiver and in the Olivet Discourse as the fulfillment of the Prophet, so the gospel is bookended with major discourses that correspond to the Law and the Prophets.
We also have subtle recapitulations that bespeak the hand of a single author, such as the way Jesus recapitulates the Old Testament history of Israel and Moses in the first four chapters of the book before he assumes the role of Lawgiver in chapters 5-7.
All of this speaks to the authorial vision of a single individual, and that leaves us with the question: Who was that individual? We have no better, more reliable guidance on this point than the voice of tradition, and I see no reason to ascribe the gospel to anybody other than Matthew.
One reason for this–aside from the early date and weight of the tradition itself–is that it is singularly unlikely that Matthew’s name would become attached to the gospel if he were not the author. In fact, if one were looking to fictitiously attach one of the apostles’ names to a gospel in order to give it more authority, Matthew is the last person whose name you’d want to use (except Judas Iscariot, who offed himself before the gospels were written).
Why’s that?
Because not only was Matthew a non-major apostle (we know very little about him), he also is the apostle who would have had the hardest time with a Jewish audience, given that he was a tax-collector for the Romans. He was a former collaborator with the hated enemy, who became even more hated after the two Jewish Wars and the destruction of Israel and Jerusalem.
Yet Matthew’s gospel, as is clear, is the one most written for a Jewish audience. There could scarcely be anything more offputting for such an audience than having the gospel story told to them by a tax-collector/collaborator, and thus "Matthew" strikes me as a name unlikely to become attached to the most Jewish of all the gospels if he were not the author.

