What is the platonic conception of Gatsby in The Great Gatsby?

Great question! 

In Chapter 6
of The Great Gatsby, our narrator, Nick
Caraway, says, "Jay Gatsby, of West Egg, Long Island, sprang from his Platonic
conception of himself." 

The term "Platonic" comes from the
Greek philosopher Plato who wrote mostly about the relationship between the ideal world
and actual world. Through his theory he established the idea of the "Platonic form"
which refers to the ideal (or perfect) form of something. So a "platonic form" (or
conception) equals an "ideal form." 

When Nick uses this
term to describe Gatsby, he is basically saying that Gatsby created his own identity by
thinking of the perfect, ideal form of himself, and committing to the ideal, despite the
realistic world around him.

It's helpful to look at the
rest of the quote in order to put this new understanding in
context: 

"He was a son of God—a phrase which, if it means
anything, means just that—and he must be about His Father’s business, the service of a
vast, vulgar, and meretricious beauty. So he invented just the sort of Jay Gatsby that a
seventeen year old boy would be likely to invent, and to this conception he was faithful
to the end."

Gatsby created himself into the person he
imagined himself to be when he was a kid.