Okay, well for that you have to know how the trait system works. I'll do this broadly at first and if you still need something more specific or explained better just ask.
First of all here is the code for the Greek Platonism trait line (Greek faction philosophy traits are different from the others actually).
Just focus on the lines I've bolded.
ExcludeCultures barbarian, roman, eastern, carthaginian
AntiTraits Iliterated, DarkPath, Ignorance, Anger, gStoic, gEpicurean, gCynic, gSkeptic, gPeripatetic, Pythagorean, gPythagorean
Effect Influence 1
Effect Law 1
Effect Influence 1
Effect Law 2
Effect Influence 2
Effect Law 3
Effect Loyalty 1
Excluded cultures does just as it sounds, factions of those cultures cannot get this trait line. A Barbarian can never get this trait no matter how long he stays in Athens at Plato's Academy.
NoGoingBackLevel is the point at which the character will not lose the trait. For this particular trait its 1, so once a character has started studying Platonism he will never lose the trait.
Antitraits are traits that are not compatible with this trait. If a trait has not crossed the "NoGoingBackLevel", one level of that trait will be subtracted if the character would earn an antitrait. This is how characters can lose traits (Losing "Social Drinker" if that character would have earned a "Sobriety" trait for example). If you do go over the NoGoingBackLevel, you can never acquire an antitrait. You shouldn't have gotten a character with Illiterate and Studies Platonism at the same time from this, but its possible he earned a point in a philosophy trait that got rid of his illiterate trait and enabled him to learn Studies Platonism.
Threshold defines the internal level of the trait for you to acquire a particular trait level. For this trait a threshold of 1 is need for you to acquire Studies Platonism, but to max out this trait tree you need a threshold of 6. In other words, your character has at least a gPlaton threshold of 2 (as he has Mastered Platonism) but not yet 6 (otherwise he'd be a Platon).
The code I posted for the Ludas Magnas is what determines how you actually earn and advance these traits. For the academies, basically keeping a character in a city for a turn gives that character a chance to learn any of a series of traits for example...
Affects LogisticalSkill 1 Chance 3
means any character waiting a turn in that settlement has a 3% chance to increase his threshold in the LogisticalSkill tree by 1. If that threshold gets high enough to earn him a better version of the trait, you'll see that in the next turn.