In the context of discussing family backgrounds, our Gemara discusses how we can be certain that a child is a bekhor – the first-born child to his father – a status that would give him certain advantages with regard to inheritance, for example. Rav Nahman teaches that three people can be trusted to declare that one child is the bekhor – the midwife who delivers him, his mother and his father. The mid-wife is believed at the time of birth (if, for example, twins were born), the mother during the week after birth and the father at any point in time.
The Gemara supplies a source for the father’s believability with regard to declaring his son a bekhor. The passage in (21:17) that forbids the father from transferring the advantage of the bekhor to the first-born of his beloved wife (if there was an older child from a different wife), requires him to acknowledge the status of the oldest son. “Acknowledging” the oldest son implies that he is believed to say who is the oldest.
Rashi explains that the mother’s believability stems from the fact that she is in constant contact with the baby during the first week of his life. Once the baby is taken from her on the day of the brit milah – his circumcision – we can no longer be certain of her identification. The Meiri suggests that during the first week the mother makes certain to keep track of the first-born (e.g. if twins were born) in order to be sure which baby should be circumcised first, but after the brit takes place there is no longer any need to remember which child was born first.
According to the Talmud Yerushalmi, the midwife is believed based on the story in Sefer (38:28) where Tamar gives birth to twins and the midwife tied a string around the baby’s wrist saying “this one was born first.”