What kind of questions did philosophers ask in Ancient Greece?

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Thank you for your question. I can't hope to answer it at all comprehensively. Instead I'll try to give a smattering of highlights, and some pointers as to where you might look to learn more. Among the questions that philosophers tried to answer at this time were: '

What is it to be virtuous, and can virtue be taught?

What is the soul, and does it survive the destruction of the body?

What is the best organization for a society to follow in order to be just?

What, at the most fundamental level, is the physical world made of? (Proffered answers included fire, water, and atoms.)

In what sense, if any, might the future be "real"?

There are lots of others. For a rewarding discussion that puts philosophical inquiry into a broader social and historical context, I would strongly suggest the classic _The Greeks_ by H.D. Kitto. Once you've gone through that, you can't go wrong by looking at some dialogues by Plato, in which the author purports to report various discussions his teacher Socrates had with his friends before being put to death by his fellow Athenians. See for instance Plato's _Five Dialogues_, translated by J. Cooper, and published by Hackett Publishing Co.

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