There have been, and are, as you may know, persons who doubt whether the apostle Paul was the writer of the Epistle to the Hebrews. And there have been many more or less wild suggestions made, as to who was the author.
I only draw your attention to this for the purpose of warning you against the danger of guesswork in the things of scripture. The Bible is God's word; and all that we ought to know He has given us there, and what He has withheld is no concern to us. When therefore we begin to guess, we begin to go wrong.
But I feel I should like to put before you in a brief and simple way the principal facts that make it reasonably certain that the Epistle was written by the apostle Paul, and that the usual heading in our Bibles should stand as it does.
You know that Paul wrote letters to seven Gentile assemblies: — Romans, Corinthians, Galatians, Ephesians, Philippians, Colossians, Thessalonians; and in every one of those letters the first word is "Paul," and usually "Paul an apostle." But in the Epistle to the Hebrews there is a marked difference in the introductory phrases. Here we have as the opening words: "God, who at sundry times and in divers manners spake in time past unto the fathers by the prophets, hath in these last days spoken unto us by His Son."
Now, from this passage we may see two special characteristics of this Epistle: — (1) the ancient scriptures are used as possessing God's authority, and (2) the Son of God is God's spokesman, for the phrase, "spoken unto us by His Son," implies not only the words of the Lord Jesus, but His works, and indeed, all that He is. These features which run throughout the Epistle explain why Paul said nothing of himself as the writer, nor of his own apostolic authority. To the Gentiles he spoke with an authority which was his as an apostle whom God had set first in the church (1 Cor. 12:28), and had sent primarily to the Gentiles (Acts 9:15; Rom. 1:5; Rom. 11:13). But in this Epistle he appeals to the Jewish believers through the scriptures, which they all knew and loved and reverenced. There are more Old Testament quotations in the Hebrews than in any of the other Epistles. And the writer does not speak of what was made known to him personally by revelation, as he does when writing to the Gentiles, as for instance, in Galatians 1:12.
Beside, Christ is shown as the One Whom God had sent to make known and to do His will. In Old Testament times God sent prophets; but in New Testament times He sent His Son. In this Epistle Jesus Christ Himself is called the Apostle (Heb 3:1). Hence there was no need for Paul to introduce himself as an apostle.
Again, there was another reason:"Paul was the apostle whose mission was specially to the Gentiles and not to the Jews (Gal. 2:7-8).
I think, therefore, you will see that there would be what some might consider a want of propriety in the apostle of the uncircumcision bringing forward his authority to the believers among the circumcision (at any rate Paul would not give any ground for such objection), while it was perfectly legitimate, and likely to be more serviceable to him to show them the truth concerning Christ and His work out of the scriptures. It was, indeed, in this manner that he habitually spoke to the Jews in the synagogues (Acts 13:16-41; Acts 17:2; Acts 28:23).
It, therefore, may be well understood why Paul the apostle, in a wisdom wherein he was taught of God, refrained from attaching his name as weight and authority to an Epistle in which Christ and the scriptures were the authorities appealed to.
But, perhaps, you are asking how we know that the Epistle was written by Paul, if he did not give his name. The question is a fair one and very natural. And the strongest evidence we find in the scripture itself. We are told by another apostle that Paul did write to the Jews, and that this writing was well known in apostolic days. Peter, who was an apostle of the circumcision and who himself wrote two Epistles to the Jews who believed in the Lord Jesus, refers to what Paul also had written to them, in addition to his other Epistles. His words are: "Wherefore, beloved, seeing that ye look for such things [i.e. new heavens and a new earth], be diligent that ye may be found of Him in peace, without spot and blameless; and account that the longsuffering of our Lord is salvation; even as our beloved brother, Paul, also according to the wisdom given unto him hath written unto you [i.e. in the Epistle to the Hebrews]; as also in all his Epistles speaking in them of these things'' (2 Peter 3:14-16).
It is surely clear from this passage that Paul did write, in addition to his other Epistles, one to Israel's believing remnant. And it is really beautiful to observe from this reference the unity and harmony of the apostles in thus working together under the power of the inspiring Spirit for the edification of the church. Paul the apostle of the uncircumcision, writes to the saints of the circumcision; and so far from any feelings of jealousy or thoughts of interference, Peter, the apostle of the circumcision, commends the letter to them, and tells us who the writer is. Of a similar nature was the divine arrangement that Peter, the apostle of the Jews, should be the first to carry the word of grace to the Gentiles in the person of Cornelius (Acts 10).
There are other reasons which all tend to strengthen the Pauline authorship of the Epistle, but for those who are ready to accept the testimony of scripture this will suffice.