The early church is unanimous in attributing the four gospels to Matthew, Mark, Luke, and John but they are also unanimous that Matthew wrote first. However Mark probably wrote first since most of Mark is in Matthew. So my question is if they’re wrong on who wrote first then why should we believe that their right that Matthew, Mark, Luke, and John wrote the gospels?
The bottom line is this: the fact that certain early Christians believed that Mark wrote Mark or that Matthew was written first is not proof that these things are true. Noone knows with absolute certainty who wrote Matthew, and noone knows for sure if Mark was written before Matthew. We have two kinds of “evidence,” both for authorship and date of writing. The “evidence” (quotations used on purpose) is the statements of the very early Christians and human reasoning from the content of the books. My faith in the reliability and inspiration of the four gospels is not based on the identity of the author or the date of authorship. These are somewhat important questions, but not essential ones.
So, let me apply my own reasoning here. Which is it more likely that the early Christians might get wrong: the relative date of the writing or the identity of the author. My human reasoning tells me that I am far more likely to forget WHEN a book was written than who wrote the book. I am not going to be mistaken about who wrote the book The Screwtape Letters (it was C. S. Lewis), but I am fairly likely not to know when he wrote the book (without looking it up, I would guess somewhere between 1940 and 1965). Using my reasoning, which is not a terrifically powerful argument, I believe it is more likely that second century Christians such as Irenaeus would get the relative date of authorship wrong than the identity of the author. Therefore, I believe it is probable that the early Christians have the authors correct, making modern arguments from “reasoning” weak, whereas, modern arguments on which was written first is more likely to be able to correct a mistake by the early church.
Without reviewing the evidence, I conclude that it is very likely that the early church fathers have the author of Mark and Matthew right, but it is somewhat likely that modern arguments have the order of writing correct (Mark before Matthew). However, I am not certain that Mark was written before Matthew. I believe that a strong case can be made for the reverse. I have seen some pretty good arguments that Matthew was written first. However, for what it is worth, my opinion is that it is more likely Mark was written first.
You seem to be concerned that we may not be sure who wrote the gospels. Fair enough, but I am not particularly concerned about that. What is important is that the four gospels are inspired by God and are faithful records of the words and deeds of Christ. It is possible that the traditional author of Mark and Matthew are not correct. My faith is certainly not based on who is the author, and neither should yours. That is my input.