In the 1st millennium BC, they were the product of a mass eugenics program on the planet Sahndara. They were bred for contemplation, self-reliance, and longevity. The number of individuals generated by the program was pared down to thirty-eight. Before the planet's star went nova, this group resettled on Earth where they were exposed to the Greek civilization. When this civilization died, they moved on to Platonius where they founded their republic.
On Platonius, they continued to follow the teachings of Socrates and Plato, but soon they discovered that they had developed psychokinetic powers, attributed to the planet's kironide-rich food. By the time of the visit of the starship "Enterprize" the Platonians had inhabited their "Republic" for 2,500 years.
In the late 2260s, the republic's government was set up on the most powerful psionic, in this case Parmen being supreme. Considering that each Platonian's ability appeared to attuned to a different frequency, which made combining their powers impossible, Parmen's authority was unchallenged. (TOS: "Plato's Stepchildren")