Q. If Luke was not an apostle, how do we explain why we view the Gospel of Luke and the book of Acts as inspired? And who is the author of the book of Hebrews? If the author of the book of Hebrews is unknown, how do we explain why we view the book of Hebrews as inspired?
How do we know the apostle Matthew was the author of the Gospel of Matthew? Also, was the Gospel of Matthew originally written in Greek? Or was the Gospel of Matthew originally written in Hebrew/Aramaic?
The premise behind these questions is that apostolic authorship is the criterion for NT canonicity. In other words, that only books demonstrably written by apostles are inspired and so should be included in the New Testament. But this is actually not the criterion for NT canonicity.
For one thing, as the questions show, it is not possible to apply this principle consistently to all of the NT books that Christians collectively accept as inspired. In addition to Luke, who was not an apostle (although he was a ministry partner and traveling companion of the apostle Paul), the NT authors include James and Jude, who were brothers of Jesus but not apostles. (Paul speaks of these as two different groups in his first letter to the Corinthians, “the other apostles and the Lord’s brothers.”) The NT authors also include Mark, who was not an apostle (although he was Peter’s co-worker and likely composed his gospel from Peter’s memoirs). And then, as noted, there is an anonymous book, Hebrews. If authorship is the criterion but we don’t know who the author of a book is, then such a book certainly does not qualify.
But as I said, authorship is not the criterion. Rather, the principle of canonicity is that the Holy Spirit bore witness to the church corporately, over a period of centuries, about which books were inspired. The books we currently have in the NT, along with some other books, were used widely in worship, teaching, and devotional reading in many different Christian communities throughout the Roman Empire. Over time, a consensus emerged among these communities about which books should be added to the Hebrew Bible to form the Christian canon.
The first complete list we have of the New Testament books as we know them today is in the Easter letter that Athanasius, bishop of Alexandria, wrote to the churches in A.D. 367. He wrote this list very intentionally, to distinguish canonical books from others with similar titles (some purporting to be apostolic, but actually not written by the supposed authors), to make sure that believers would not be “led astray.” Athanasius speaks of these books as “the divinely inspired Scripture, concerning which we have been fully persuaded, as they who from the beginning were eyewitnesses and ministers of the Word, delivered to the Fathers.” (So the criterion is not apostolic authorship, but the books nevertheless have to preserve eyewitness testimony of Jesus.) Athanasius describes these books as “handed down” and “accredited as divine.” He is speaking of the process by which the Christian communities received these books, test-drove them (so to speak), and recognized their inspiration. After listing the books, Athanasius concludes, “These are the fountains of salvation, that he who thirsts may be satisfied with the living words they contain. In these alone the teaching of godliness is proclaimed. Let no one add to these; let nothing be taken away from them.”
So there was a clear, thoughtful, thorough process that led to the recognition of the New Testament canon. But the principle behind that process was not apostolic authorship; it was the principle of the Holy Spirit’s witness to the church corporately.
(Regarding the authorship and composition of the Gospel of Matthew, I have studied it closely in Greek, and personally I can only conclude that it must have been written in Greek. Its use of the forms and terms of that language strikes me as original, the kind of thing we would not find in a translation from another language. The authors of most of the NT books give their own names, but that is not the case with this one, so we do not know for certain who wrote it. The apostle Matthew is considered the author by tradition.)
Good Question 
Christopher, I fully agree with the questions you asked. I have asked these questions for years and still do. I do not agree with the answers and conclusions you suggest. To study Greek was never of my interest, I studied Hebrew for few semesters in Salzburg, which led me to very different views and conclusions.
Thank you for your effort.
Best regards form Bavaria,
Christian
Thank you for your comment. If you have never studied Greek, I’m not sure how you are in a position to come to an opinion about whether the Gospel of Matthew reads like an original composition in Greek or a translation of a work into Greek from another language. Would you please explain? Thank you.
This begs the question of why do we trust Athanasius?
I cited Athanasius as someone who was summarizing the results of the church’s deliberations about the NT canon over three centuries. In other words, I cited him as a succinct expression of the consensus. Saying that the criterion of canonicity is the consensus that the Christian communities reached, directed by the Holy Spirit, of course represents a different presupposition than that of apostolic authorship. If one continues to hold to apostolic authorship as the criterion, then there is indeed a problem with the NT canon. I’m suggesting that the problem is with that presupposition, rather than with the books themselves.