Even if Papias learned about the two writings he is referring to from the Apostle John, son of Zebedee—which is highly unlikely—his descriptions of Peter’s sermons and Matthew’s oracles are not at all like the gospels we have today. He says the following:

- And the presbyter said this. Mark having become the interpreter of Peter, wrote down accurately whatsoever he remembered.* It was not, however, in exact order that he related the sayings or deeds of Christ. For he neither heard the Lord nor accompanied Him. But afterwards, as I said, he accompanied Peter, who accommodated his instructions to the necessities [of his hearers], but with no intention of giving a regular narrative of the Lord’s sayings. Wherefore Mark made no mistake in thus writing some things as he remembered them. For of one thing he took special care, not to omit anything he had heard, and not to put anything fictitious into the statements. [This is what is related by Papias regarding Mark; but with regard to Matthew he has made the following statements]: Matthew put together the oracles [of the Lord] in the Hebrew language, and each one interpreted them as best he could.
Since Mark appears to be a narrative and the account is organized, how could Papias be referring to the gospel we currently have? Matthew also provides scant evidence to support the translation’s use of Greek. Matthew, p. 7: Herman N. Ridderbos writes:
This means, however, that we can no longer accept the traditional view of Matthew’s authorship. At least two things forbid us to do so. First, the tradition maintains that Matthew authored an Aramaic writing, while the standpoint I have adopted does not allow us to regard our Greek text as a translation of an Aramaic original. Second, it is extremely doubtful that an eyewitness like the apostle Matthew would have made such extensive use of material as a comparison of the two Gospels indicates. Mark, after all, did not even belong to the circle of the apostles. Indeed Matthew’s Gospel surpasses those of the other synoptic writers neither in vividness of presentation nor in detail, as we would expect in an eyewitness report, yet neither Mark nor Luke had been among those who had followed Jesus from the beginning of His public ministry.
It’s possible that Papias is referring to original material that was included in the gospels; however, the way he describes it makes it nearly impossible to connect it to the gospels we currently have.
https://ehrmanblog.org/the-writings-of-papias-guest-post-by-steven-carlson
After quoting a well-known quote from Papias, Carlson says:
The length and detail of this passage make it virtually irresistible for critics to bypass the layers of embedded discourse and treat this comment about the Gospels of Mark and Mathew as if they were a self-contained block of a tradition. It is not. The elder’s comment about Mark was presumably uttered not out of the blue but within some larger discourse context. This context is lost to us. Indeed, what the elder said is not by any means intact, but extracted, edited, and embedded by Papias into a different context of his own creation. Furthermore, Papias’s presentation of these remarks also does not come down to us intact, but only as preserved by Eusebius—and Eusebius’s agenda is different from Papias’s. Eusebius too extracted, edited, and embedded this statement into a context of his own making. We have to be cautious in interpreting it. As one scholar put it, “Papias says only what Eusebius wants him to say.” As a result, the most famous statement in antiquity about the origins of Mark and Matthew is a joint production of three different people, living at three different times, with three different purposes: the elder, Papias, and Eusebius. All of them have contributed to this passage in their different ways, and all of them had different purposes for discussing their writings.
The description that Papias gives does not match GMark:
It does not correspond to any aspect that Papias discusses. Not a memoir. It is not a verbatim transcript. It opposes Petrine. It describes numerous scenes in which Peter was not present (such as the temptations, where no one else is present at all). The creator of Imprint’s Gospel commits various geological errors, (for example, setting Lebanon South of Galilee) that a local of the district wouldn’t make.
Additionally, Mark’s Gospel contains a significant amount of narrative material that is constructed from passages from the Old Testament as well as material that is blatantly fictive or implausible. It also contains sophisticated Greek literary techniques and structures that could not arise from natural speech.
But let’s continue:
Actually, I think another possible conclusion was that Papias was referring to secondary derivative works that probably circulated in the early second century rather than our canonical gospels or source materials. The many unusual details in Papias’ descriptions, in my opinion, would be explained by this hypothesis. We already know that Matthew was probably written as a derivative of Mark between 80 and 90, and the Gospel of Peter was probably written between 100 and 125 as another expanded version of Mark. The idea behind this is that Papias was familiar with two other works around this time (ca. 100-125). a collection of Petrine sermons that may have been based on Mark (now known as Didascalia Petrou or Kerygma Petrou, i.e. the Teaching of Peter or the Preaching of Peter) and a Matthew version that had a lot of additions and changes (now known as the Gospel according to the Hebrews), which was translated into Aramaic. These probably circulated with overt authorial attribution, in contrast to Mark and Matthew, who used the first person in the Gospel of Peter.
Since Heracleon (ca. 150-160) and possibly Aristides (ca. 125) used DP/KP, it probably dates back to the first half of the second century. Only a few fragments of DP/KP remain, so we have no idea if it claimed to have been written by Mark or if it contained significant passages from Mark’s gospel. However, the few fragments we do have contain specific Markan language, such as Fr. 3b, which is identical to Mark 3:28 but differs from Matthew 12:31 and Luke 12:10 in that it states that “As many things as any of you did in ignorance, not knowing God clearly, if he should recognize them and repent, all his sins shall be forgiven him ( ),” However, the author probably also knew John and Hebrews.) Even though DP/KP did not have a direct connection to Mark’s gospel, it could have been the work that Papias mentioned. Clement of Alexandria quotes a post-resurrection commissioning of the apostles, and Origen also claims a resurrection appearance story in DP/KP, which presents itself as a collection of Peter’s speeches. This story is the same as the one that Ignatius of Antioch used in his epistle to the Smyrnaeans (circa 110), which was written at the same time as Papias. Mark is said by Papias to have adapted Peter’s “teachings” (), which is exactly what DP/KP claims to be.
The Gospel of the Hebrews is frequently dated to the first half of the second century. According to Jerome, it was likely based on Matthew. A fragment from this work, which was quoted by Epiphanius, suggests that it used the first person in a manner similar to that of the Gospel of Peter. For example, Jesus asks the author, “you also, Matthew, did I call while you were sitting at the tax table, and you followed me?” It merged with other Jewish-Christian gospels from the third and fourth centuries, such as the Gospel of the Nazoreans and the Gospel of the Ebionites, according to various church fathers. Eusebius claims that Hegesippus (circa 200) cited both the Hebrew Gospel and a Syriac (Aramaic) version of the Gospel. Eusebius added that Papias told the story of the woman who was accused of a lot of sins (cf. John 8), which he claimed was also included in Hebrews’ Gospel. Even more intriguingly, Jerome claimed that the Gospel according to the Hebrews was the source of the resurrection appearance story in Ignatius’ Smyrnaeans, which Origen attributed to the KP. Ignatius also knew a gospel that was very similar to Matthew’s because it mentions the star at Jesus’ nativity (Ephesians 19:2) and the resurrection of the saints (Magnesians 9:2; cf. 52-53 of Matthew 27).
Therefore, it’s possible that the first generation of gospels was formally anonymous, and later, in the early second century, a flurry of gospels claimed to have been written by the apostles followed. The attributions to the canonical gospels () were motivated by this. Additionally, it’s possible that Papias was not talking about the gospels that we are familiar with; By the time Papias wrote, they may not have yet acquired their ascriptions. As a record of Peter’s speeches, DP/KP resembles Mark’s work very much. There may have been both a Greek and an Aramaic translation of the Hebrew Gospel. Then, Papias would have assumed that the Aramaic was the original and that the Greek translation “interpreted” it differently. According to Papias, our canonical Matthew and the Greek Gospel according to the Hebrews were interpreted as two different versions of the Aramaic book. The fact that Papias mentioned a story from the Gospel according to the Hebrews makes it even more likely that Matthew wrote this. See the Novum Testamentum article “The ‘Gospel According to the Hebrews’ in the Apostolic Fathers” by Pier Franco Beatrice for a similar perspective on this.
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- Papias composed his treatises 130 CE or later. B.F. Westcott had it at 1450-150 CE, and B.W. Bacon at 140 CE.

It seems like Papias is not referring to gMatthew at all.

Papias has a limited understanding (according to Eusebius).

Papias is the most unreliable.

Papias had a remark that gMatthew was originally written in Hebrew then to Greek, which is a mistake.
