Where Did Plato Go Wrong?
After eight chapters mostly dedicated to beating up Plato, in Chapter 9, Popper turn to his own theory of how to construct a better state: piecemeal social engineering - not revolution but evolution. The manifest advantage is the ability to discern effects without nearly so much disruption of society and human life. His model is based on the way science and engineering work and I agree completely.
To me, the advantages are manifest from the standpoint of the Twentieth or Twenty-First Century, but Popper then turns to why Plato went wrong. He attributes it to aestheticism and perfectionism, but I don't think this is quite right.
One of Plato's most important inventions was his Theory of Forms. He seems to have been led to it by puzzling over the question of common nouns. Proper nouns, like Plato, or the dog named Pluto, each have their specific referent, but what is the referent for "man" or "dog?" Perhaps overly influenced by geometry, he concluded that just as a circle constructed with a compass is only a necessarily crude model of the ideal circle, Pluto, or Bowser, or Fido are just models of the idea of dog. Such ideas, he concluded, were the creation of God, and Plato's Republic was an attempt to reify the ideal state.
Bertrand Russell, writing at about the same time as Popper, concludes that philosophy had not yet solved the problem of the common noun. I'm inclined to think that it is an artifact of the way our brains works: their is a generalized busy sorting and classifying things at the heart of our thought. To appeal to geometry lo these 25 centuries hence, think of them as principal component vectors of some data sets stored in the brain.
In any case, Plato could not see the advantage of piece-meal social engineering because the scientific method was still a couple of millenia away from being invented.