Platonism is hard to summarize concisely, but roughly amounts to an ontology (that is, the study of reality) which is grounded in the concept of the "form", that of an idealized and unchanging entity which is the pure, unadulterated essence of all objects in reality. Platonism then distinguishes right and wrong (deriving its ethics from an ontological basis) as being based in what is eternal, and what is subject to decay or corruption, as the eternal object most closely corresponds to its ontological form, and that which decays or is corrupt is constantly moving away from its form. Popper's argument is that Plato, in the Republic, says that forms of government which are directed and led by an educated minority of philosopher-kings, and in which there is no class stuggle because each caste, including that of slaves, knows it's place. Plato is shown to be arguing a form of reverse-evolution: that the justification for this position is the concept that all of society is a decayed version of a past, "golden" era. Popper refers to this position as "historicism", a yearning for the good old days when tyranny was just and slaves knew their place.
I have also completed George Soros' Open Society, and I am fascinated by the obvious parallel to Popper's analysis of the "Tyrant's successor problem". Soros' argument in Open Soiety is to expose the blind rhetoric of the cold war, whereby democracies such as the United States argued for years about the superiority of orderly democracy but then, after the fall of Communism, refuse to provide economic and military aid to the Balkan states and to Russia to help foster a nascent open society. Interestingly, he argues (to what appears to be a Clinton era government) that the role that the United States should take is one of "policeman to the world" (using that exact phrase). Interestingly, this is exactly the thing the Bush administration has been accused of being, and Soros had spent quite a fortune trying to prevent Bush from being re-elected last year.
The difference in treatment between teacher (Popper) and student (Soros) parallels in some ways the analysis Popper makes between teacher (Socrates) and student (Plato). For instance, Popper argues that Socrates criticizes democracy, but as an effort to improve it, and makes compelling arguments that Socrates believed in a form of egalitarianism/equalitarianism (for instance, proving that even a slave is capable of reason by teaching the Pythagorean theorem to a slave). Plato, on the other hand, rejects equal rights for people, and seeks to tear down democracy and restore the rule of the philosopher-king. Soros, while arguing often that his philosophy is at odds with his economic behavior, states that he is morally unobligated to invest consistent with his ethics as long as he obeys "the rules". Soros also makes clear the role of the "greater fool" in investing- that as long as you don't believe your own lies, you can make money on the upside while people believe you, and then as you prove yourself wrong, make money on the downside as others catch on. Popper seeks only philosophical ends, yet Soros seems to want the role of philosopher, and king (at least of markets). Popper argues that the Republic is a large argument why the people should make Plato himself the philosopher-king. Likewise, does it not seem that Soros' bullying to get western governments to "save the ruble" (which would have put money in his pocket) is just a philosophical dodge to make him ever-richer, and thus more powerful, in this society where economic might is the apparent source of political power?
Popper's most interesting but least developed point is one of the "paradox of democracy"- that given the choice to vote, people are free to vote in a tyrant. He argues that the system of democracy must be designed to prevent this from happening, but gives no clear reason as to the mechanism for this happening.
To my way of thinking, there are many tyrannies. First, a tyrant or totalitarian government with physical or political might can become sovereign and accrue power and wealth to themselves. Second, an oligarchy of tyrants can through economic might concentrate sovereignty among a upper-class minority. Third, a democracy can become a sovereign tyrant through majority rule- that a majority of citizens can enact rules which benefit the majority to the expense of the minority. The problem of tyranny in modern times has concentrated on the problem of the sovereign minority because this form has been the most common historical form of tyranny. However, as a swelling middle class and democratic institutions take hold, we must not only be vigilant against returning to the evil of the past, but also to be on guard for new forms and modes of the democratic tyranny.
Popper argues, somewhat naively, that a true democracy cannot become a tyranny, but seems to fall on his own sword in that he does not successfully define what a true democracy means. Soros argues against "market fundamentalism", the idea that free markets are an ethical end in themselves, and says that the state must regulate markets to prevent the rise of oligarchy or rule of the rich. Again, to this end, he states there must be a balance between state rule and state repression of markets, without clearly defining how this can be accomplished. Moreover, Soros is adept at continually proving himself wrong, aruing that this is a good sign since Popperian scientific method states that theories can never be proven correct, only incorrect.
To me, the hysteresis between continual tentative theories and their reversal/disproval in short order (which seems to be the end result of Popperian, Sorosian democracy) is no recipe for stable, successful, and ethical government. However, it would seem at some level this solution is very likely the worst, except for all the others. The poverty of philosophy is that is is inept at formulating solvable questions- what I wouldn't give for a new way of thinging about these things, such that answers were more achievable. It is ultimately this form of depression which leads to the root cause of the democratic paradox. People will always prefer a tyrant with apparent answers to everthing over the crushing indeterminism of true science.