Objection 1. It would seem that Christ should have committed His doctrine to writing. For the purpose of writing is to hand down doctrine to posterity. Now Christ's doctrine was destined to endure for ever, according to Luke 21:33: "Heaven and earth shall pass away, but My words shall not pass away." Therefore it seems that Christ should have committed His doctrine to writing.
Objection 2. Further, the Old Law was a foreshadowing of Christ, according to Hebrews 10:1: "The Law has [Vulgate: 'having'] a shadow of the good things to come." Now the Old Law was put into writing by God, according to Exodus 24:12: "I will give thee" two "tables of stone and the law, and the commandments which I have written." Therefore it seems that Christ also should have put His doctrine into writing.
Objection 3. Further, to Christ, who came to enlighten them that sit in darkness (Luke 1:79), it belonged to remove occasions of error, and to open out the road to faith. Now He would have done this by putting His teaching into writing: for Augustine says (De Consensu Evang. i) that "some there are who wonder why our Lord wrote nothing, so that we have to believe what others have written about Him. Especially do those pagans ask this question who dare not blame or blaspheme Christ, and who ascribe to Him most excellent, but merely human, wisdom. These say that the disciples made out the Master to be more than He really was when they said that He was the Son of God and the Word of God, by whom all things were made." And farther on he adds: "It seems as though they were prepared to believe whatever He might have written of Himself, but not what others at their discretion published about Him." Therefore it seems that Christ should have Himself committed His doctrine to writing.
On the contrary, No books written by Him were to be found in the canon of Scripture.
I answer that, It was fitting that Christ should not commit His doctrine to writing. First, on account of His dignity: for the more excellent the teacher, the more excellent should be his manner of teaching. Consequently it was fitting that Christ, as the most excellent of teachers, should adopt that manner of teaching whereby His doctrine is imprinted on the hearts of His hearers; wherefore it is written (Matthew 7:29) that "He was teaching them as one having power." And so it was that among the Gentiles, Pythagoras and Socrates, who were teachers of great excellence, were unwilling to write anything. For writings are ordained, as to an end, unto the imprinting of doctrine in the hearts of the hearers.
Secondly, on account of the excellence of Christ's doctrine, which cannot be expressed in writing; according to John 21:25: "There are also many other things which Jesus did: which, if they were written everyone, the world itself, I think, would not be able to contain the books that should be written." Which Augustine explains by saying: "We are not to believe that in respect of space the world could not contain them... but that by the capacity of the readers they could not be comprehended." And if Christ had committed His doctrine to writing, men would have had no deeper thought of His doctrine than that which appears on the surface of the writing.
Thirdly, that His doctrine might reach all in an orderly manner: Himself teaching His disciples immediately, and they subsequently teaching others, by preaching and writing: whereas if He Himself had written, His doctrine would have reached all immediately.
Hence it is said of Wisdom (Proverbs 9:3) that "she hath sent her maids to invite to the tower." It is to be observed, however, that, as Augustine says (De Consensu Evang. i), some of the Gentiles thought that Christ wrote certain books treating of the magic art whereby He worked miracles: which art is condemned by the Christian learning. "And yet they who claim to have read those books of Christ do none of those things which they marvel at His doing according to those same books. Moreover, it is by a Divine judgment that they err so far as to assert that these books were, as it were, entitled as letters to Peter and Paul, for that they found them in several places depicted in company with Christ. No wonder that the inventors were deceived by the painters: for as long as Christ lived in the mortal flesh with His disciples, Paul was no disciple of His."
Reply to Objection 1. As Augustine says in the same book: "Christ is the head of all His disciples who are members of His body. Consequently, when they put into writing what He showed forth and said to them, by no means must we say that He wrote nothing: since His members put forth that which they knew under His dictation. For at His command they, being His hands, as it were, wrote whatever He wished us to read concerning His deeds and words."
Reply to Objection 2. Since the old Law was given under the form of sensible signs, therefore also was it fittingly written with sensible signs. But Christ's doctrine, which is "the law of the spirit of life" (Romans 8:2), had to be "written not with ink, but with the Spirit of the living God; not in tables of stone, but in the fleshly tables of the heart," as the Apostle says (2 Corinthians 3:3).
Reply to Objection 3. Those who were unwilling to believe what the apostles wrote of Christ would have refused to believe the writings of Christ, whom they deemed to work miracles by the magic art.
Source: Summa Theologica (New Advent)