Can we trust a source such as Papias on the question of whether our Gospel of Matthew was written by the disciple Matthew and that our Gospel of Mark was written by Mark, the companion of the disciple Peter?
It is interesting that Papias tells a story that is recorded in our Matthew but tells it so completely differently that it appears he doesn’t know Matthew’s version. And so when he says Matthew wrote Matthew, is he referring to *our* Matthew, or to some other book? (Recall, the Gospel he refers to is a collection of Jesus’ sayings in Hebrew; the Gospel of Matthew that *we* have is a narrative, not a collection of sayings, and was written in Greek.) If he *is* referring to our Matthew, why doesn’t he see it as an authoritative account?
Here’s the conflicting story. It involves the death of Judas. And it’s quite a story! Here is my translation of it from my edition, The Apostolic Fathers (Loeb Classical Library, vol. 1; 2004).
But Judas went about in this world as a great model of impiety. He became so bloated in the flesh that he could not pass through a place that was easily wide enough for a wagon – not even his swollen head could fit. They say that his eyelids swelled to such an extent that he could not see the light at all; and a doctor could not see his eyes even with an optical device, so deeply sunken they were in the surrounding flesh. And his genitalia appeared more disgusting and greater than all formlessness, and he bore through them from his whole body flowing pus and worms, and to his shame shame, he emitted pus and worms that flowed through his entire body.
And they say that after he suffered numerous torments and punishments, he died on his own land, and that land has been, until now, desolate and uninhabited because of the stench. Indeed, even to this day no one can pass by the place without holding their nose. This was how great an outpouring he made from his flesh on the ground.” [Apollinaris of Laodicea]
You gotta love it. But, well, what does one make of it? Matthew’s Gospel – the one we have in the New Testament – also describes the death of Judas. But it is not like this at all. According to Matthew, Judas hanged himself (Matt. 27:5). If Papias saw Matthew’s Gospel as an eyewitness authority to the life of Jesus and those around him, why didn’t he accept its version of Judas’s death?
Another alternative is that when Papias describes a Gospel written by Matthew, he isn’t actually referring to the Matthew that we now have. Recall: Papias says two things about the “Matthew” he is familiar with: it consists only of sayings of Jesus and it was composed in Hebrew. Neither is true of our Matthew, which does have sayings of Jesus, but is mainly composed of stories about Jesus. Moreover, it was not composed in Hebrew but in Greek. 
It is possible, of course, that like other early Christian scholars, Papias thought Matthew was originally composed in Hebrew when it was not. But it is also possible that these later writers thought Matthew was written in Hebrew because they knew about Papias’s comment and thought he was referring to our Gospel. But he appears not to be: Matthew is not simply a collection of Jesus’ sayings; and in the only place that Papias’s comments overlap with (our) Matthew’s account (the death of Judas), he doesn’t appear to know (our) Matthew.
If Papias was not talking about our Matthew, was he talking about our Mark? As Papias’s quotation about Mark that I cited yesterday indicates, he considered “his” Mark to be problematic because of its disorderly arrangement: that’s why he says that the preaching of Peter was not given “in order.” But that somewhat negative remark in itself is odd, because he doesn’t make the same comment about Matthew, even though the narrative outline of our Matthew is pretty much the same as our Mark – with additional materials added in.
Apart from that, Papias indicates that Mark’s Gospel gives an exhaustive account of everything Peter preached and that it gives it without changing a thing. The reality is that there is no way that anyone could think that the Gospel of Mark in our Bibles today gives a full account of Peter’s knowledge of Jesus. Our Gospel of Mark takes about two hours to read. Are we to think that after spending months (years?) with Jesus, Peter had no more than two hours’ worth of memories?
Of course it may be that Papias is exaggerating for effect. But even so, since he does not appear to be referring to the book we call Matthew, why should we think that he is referring to the book we call Mark? And that, therefore (as Papias indicates) Mark’s Gospel is actually a transcription of Peter’s version of what Jesus said and did?
Despite repeated attempts over the centuries by readers to show that Mark’s Gospel is “Peter’s perspective,” the reality is that if you simply read it without any preconceptions, there is nothing about the book that would make you think, “Oh, this is how Peter saw it all.” Quite the contrary – not only does Peter come off as a bumbling, foot-in-the-mouth, and unfaithful follower of Jesus in Mark (see Mark 8:27-32; 9:5-6; 14:27-31), but there are all sorts of stories – the vast majority – that have nothing to do with Peter or that betray anything like a Petrine voice.
There is, though, a still further and even more compelling reason for doubting that we can trust Papias on the authorship of the Gospels. It is that that we cannot really trust him on much of anything. That may sound harsh, but remember that even the early Christians did not appreciate his work very much and the one comment we have about him personally from an educated church father is that he was remarkably unintelligent.
It is striking that some modern authors want to latch on to Papias for his claims that Matthew and Mark wrote Gospels, assuming, as Bauckham does, that he must be historically accurate, when they completely overlook the other things that Papias says, things that even these authors admit are not and cannot be accurate. If Papias is not reliable about anything else he says, why does anyone think he is reliable about our Gospels of Matthew and Mark? The reason is obvious. It is because readers want him to be accurate about Matthew and Mark, even though they know that otherwise you can’t rely on him for a second.
Does anyone think that Judas really bloated up larger than a house, emitted worms from his genitals, and then burst on his own land, creating a stench that lasted a century? No, not really. But it’s one of the two Gospel traditions that Papias narrates. Here is the only other one. This is the only saying of Jesus that is preserved from the writing of Papias. Papias claims that it comes from those who knew the elders who knew what the disciple John the Son of Zebedee said that Jesus taught:
Thus the elders who saw John, the disciple of the Lord, remembered hearing him say how the Lord used to teach about those times, saying:
The days are coming when vines will come forth, each with ten thousand boughs; and on a single bough will be ten thousand branches. And indeed, on a single branch will be ten thousand shoots and on every shoot ten thousand clusters; and in ever cluster will be ten thousand grapes, and every grape, when pressed, will yield twenty-five measures of wine. And when any of the saints grabs hold of a cluster, another will cry out, ‘I am better, take me, bless the lord through me.” (Eusebius, Church History, 3.39.1)
Really? Jesus taught that? Does anyone really think so? No one I know. Does Papias think Jesus said this? Yes, he absolutely does. Here is what Papias himself says about the traditions of Jesus he records in his five-volume book, in Bauckham’s own translation:
I will not hesitate to set down for you along with my interpretations everything I carefully learned from the elders and carefully remembered, guaranteeing their truth.”
So, can we rest assured about the truth of what Papias says, since he can provide guarantees based on his careful memory? It doesn’t look like it. The only traditions about Jesus we have from his pen are clearly not accurate. Why should we think that what he says about Matthew and Mark are accurate? My hunch is that the only reason readers have done so is because they would like him to be accurate when he says things they agree with, even when they know he is not accurate when he says things they disagree with.
However one evaluates the overall trustworthiness of Papias, in my view he does not provide us with clear evidence that the books that eventually became the first two Gospels of the New Testament were called Matthew and Mark in his time.
 That is obvious If Matthew was based in large part on the Gospel of Mark, as is almost everywhere conceded. Matthew agrees with the Greek text of Mark verbatim throughout his account. The only way that would be possible is if he was copying the Greek text into his Greek text.