In Plato's Phaedo it said that we have memories from our past lives, and thus we have a longing for spirituality. I agree with you. However, it is hard to answer your second question. If we long for spirituality, then it could also be perceived that a belief in reincarnation would not be far behind. Let me re-phrase, believers in reincarnation argues that they want to purify their present lives (from the lessons of the past lives) so that ultimately they become spiritually purified and thus attain immortality (no longer needing the cycle of birth and rebirth!). Well, I am now confused! Anyway, here are excerpts from Wikipedia about Plato's Phaedo:
Plato's Phaedo is one of the great dialogues of his middle period, along with the Republic and the Symposium. The Phaedo is also Plato's fifth and last dialogue (the first four being Euthyphro, Apology, Crito and Meno) which details the final days of Socrates and contains the scene of his death. The dialogue is told from the perspective of one of Socrates' students, Phaedo of Elis. Having been present at Socrates' death bed, Phaedo relates the dialogue to Echecrates, a fellow philosopher.
In the dialogue set forth in Phaedo, Socrates discusses the nature of the afterlife. This dialogue is narrated to us, as well as to Echecrates, by Plato, through Phaedo. By engaging in dialectic with two of his friends, the Thebans Cebes and Simmias, Socrates explores various arguments for the soul's immortality in order to show that there is an afterlife in which the soul will dwell following death.
In total, Socrates presents four arguments for the immortality of the soul. The first three, though convincing, are insubstantial. These may be referred to as the cyclical, recollective, and affinity arguments. While valuable to the reader as examples of both invalid arguments and the difficulty of proving such a claim as the soul's immortality, none of the first three are sufficient either on their own or combined to satisfy either...