What evidence do we have that the four gospels were written by Matthew, Mark, Luke, and John? I hear a lot of people saying they were anonymous, and we have no proof they were written by the people whose names are given to them.
To claim there is no evidence would require setting terms on admissible evidence so strict that it rejects all proof.
Matthew, also known as Levi, was a tax collector before he was called by Christ (Matthew 9:9; Mark 2:14). All early copies of Matthew are titled "The Gospel of Matthew." Papyrus fragment P4, dated to the late second century, is the earliest copy that we have which contains the title attributing it to Matthew. However, this isn't the only evidence. Early Christian writers all state that Matthew was the author of this Gospel.
"Matthew published his Gospel among the Hebrews in their own language, while Peter and Paul were preaching and founding the church in Rome. After their departure Mark, the disciple and interpreter of Peter, also transmitted to us in writing those things which Peter had preached; and Luke, the attendant of Paul, recorded in a book the Gospel which Paul had declared. Afterwards John, the disciple of the Lord, who also reclined on his bosom, published his Gospel, while staying at Ephesus in Asia." [Ireanaeus, Against Heresies, 3.1.1]
There are also clues recorded within the gospels regarding their authors. One is that events that might give praise to the author are recorded without mentioning the author's name. Comparing those events to the other gospels then tell us who was left unnamed. For example, in Matthew 9:10 there is a reference to Jesus dining "in the house" without directly saying whose house this took place in. But the use of "the" seems to indicate that it was a house important to the author. Comparing this to a parallel account in Luke 5:29 we learn that a great feast was held in Matthew's house. Matthew avoided putting the spotlight on himself in his own account of Jesus' life.
The original title for the book was "According to Mark." All early Christian writers state that Mark is the author of this gospel. We already quoted Ireanaeus' testimony, so here is another:
"Again, in the same books [the Hypotyposeis], Clement gives the tradition of the earliest presbyters, as to the order of the Gospels, in the following manner: "The Gospels containing the genealogies, he says, were written first. The Gospel according to Mark had this occasion. As Peter had preached the Word publicly at Rome, and declared the Gospel by the Spirit, many who were present requested that Mark, who had followed him for a long time and remembered his sayings, should write them out. And having composed the Gospel he gave it to those who had requested it. When Peter learned of this, he neither directly forbade nor encouraged it. But, last of all, John, perceiving that the external facts had been made plain in the Gospel, being urged by his friends, and inspired by the Spirit, composed a spiritual Gospel." This is the account of Clement." [Eusebius, Ecclesiastical History, 6.14.5-7]
Given that the Gospel writers tended to avoid naming themselves in their own books, it is suspected that Mark was the young man in Mark 14:51-52. A detail that is mentioned in no other gospel. Because of the unique citation, we can't cross-reference it to prove who was the unnamed young man.
The books of Luke and Acts are a two-volume work, being addressed to the same person, and written in the same style. The sections of Acts where "we" is used indicate that the author was a part of the group traveling with Paul. This tells us that the author was a close associate of Paul. Only two associates of Paul are not named in Acts: Titus and Luke. That Luke was a close companion of Paul is seen in Colossians 4:14 and Philemon 24. He was with Paul at the end of Paul's life (II Timothy 4:11). The earliest manuscript of Luke that includes the title, Papyrus Bodmor XIV, dated A.D. 200, has the title: "The Gospel According to Luke." Once again, the early Christian writers all attribute the book to Luke.
The author calls himself the one "whom Jesus loved" (John 13:23; 20:2; 21:7). This puts the author in the inner circle of those close to Jesus: Peter, James, and John. He is not Peter (John 20:2-6) and James was killed early on (Acts 12:1-2). This leaves us with John.
A subtle hint in John 1:6. When multiple people of the same name are mentioned, usually a qualifier is given to specify which one (Matthew 3:1; Luke 3:2). But since John never mentions himself by name, the only other John in the account is left unqualified.
Albert Barnes states: "There is no doubt that it was written by John. This is abundantly confirmed by the ancient fathers, and was not questioned by Celsus, Porphyry, or Julian, the acutest enemies of revelation in the early ages."
We have indirect internal evidence for most of the books, but we have strong historical evidence for all the books recorded by multiple authors. We also have early manuscript evidence as well. Therefore, the claim that there is no evidence is false. A few, because of their own agendas, might turn a blind eye to the evidence, but this doesn't mean the evidence is non-existent.