The earliest explicit mention of infant “baptism” in the history of the church is from the African church father, Tertullian, who lived from about A.D. 160 to about 220. He was born in Carthage, studied in Rome for a legal career and was converted to Christianity in about 195. He was the first Christian theologian to write in Latin and exerted significant influence through his apologetic works.
The work, de baptismo (Concerning Baptism) was written, evidently between 200 and 206. In it Tertullian questions the wisdom of giving baptism to infants. He says,
According to everyone’s condition and disposition, and also his age, the delaying of baptism is more profitable, especially in the case of little children. For why is it necessary—if [baptism itself] is not necessary—that the sponsors should be thrust into danger? For they may either fail of their promise by death, or they may be mistaken by a child’s proving of wicked disposition…. They that understand the weight of baptism will rather dread the receiving of it, than the delaying of it. An entire faith is secure of salvation! (de baptismo, ch. xviii)
What we see here is that the first explicit witness to infant baptism does not assume that it is a given. In other words, at the turn of the third century it is not taken for granted, as it is 200 years later when St. Augustine addresses the matter. Tertullian speaks the way one would if the practice were in dispute, possibly as a more recent development.
When we look at the New Testament, the closest thing to infant baptism that we find is the reference to three “households” being baptized. In 1 Corinthians 1:16, Paul says, “Now I did baptize also the household of Stephanas; beyond that, I do not know whether I baptized any other.” In Acts 16:15, Luke reports concerning the new convert, Lydia, “When she and her household had been baptized, she urged us, saying, ‘If you have judged me to be faithful to the Lord, come into my house and stay.’” And in Acts 16:33, Luke tells us that after the earthquake in the jail of Philippi, the jailer “took [Paul and Silas] that very hour of the night and washed their wounds, and immediately he was baptized, he and all his [household].”
It is significant that in regard to the family of the Philippian jailer Luke reports in Acts 16:32, just before mentioning the baptism of the jailer’s household, “[Paul and Silas] spoke the word of the Lord to him together with all who were in his house.” This seems to be Luke’s way of saying that hearing and believing the word is a prerequisite to baptism. The whole household heard the word and the whole household was baptized. In any case, there is no mention of infants in any of these three instances of household baptisms, and it is an argument from silence to say that there must have been small children. It would be like saying here at Bethlehem that a reference to Ross Anderson’s household or Don Brown’s or Dennis Smith’s or David Michael’s or David Livingston’s or dozens of others must include infants, which they don’t.
Yet from these texts, Joachim Jeremias, who wrote one of the most influential books in defense of infant baptism, concluded, “It is characteristic that Luke could report the matter thus. For by so doing he gives expression to the fact that ‘the solidarity of the family in baptism and not the individual decision of the single member’ was the decisive consideration” (Infant Baptism in the First Four Centuries, 1960, p. 23, quoting Oscar Cullman, Baptism in the New Testament, 1950, p. 45). I would rather say that the entire drift of the New Testament, and many particular sayings, is in the opposite direction: it is precisely the individual in his relation to Christ that is decisive in the New Testament, rather than solidarity in the flesh. “It is not the children of the flesh who are children of God, but the children of the promise are regarded as descendants” (Romans 9:8).
Pastor John Piper http://www.desiringgod.org/articles/how-old-is-infant-baptism