Roman men didn't begin studying philosophy until about 200 BC. At that time, the Romans were conquering Greece, and so a lot of Roman soldiers and generals spent a lot of time in Greece, and got a chance to talk to Greek philosophers.
The Romans found out that Greek philosophers like Socrates, Plato, and Aristotle had been doing a lot of thinking about philosophy just recently. Some Romans got interested, and by about 50 BC these Romans were even beginning to write philosophy themselves, though most of it was pretty much just translating Greek philosophy into Latin.
One of the first Roman men (Men wouldn't let women study philosophy) who wrote about philosophy was Lucretius. Lucretius followed Greek Epicurean philosophy. He left us a long poem, called On the Nature of Things, explaining Epicurean philosophy in Latin for men who couldn't read Greek.
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Cicero was another man who wrote about philosophy at just about the same time as Lucretius. Cicero was mostly a Skeptic philosopher. Like other Skeptics, Cicero thought that you should question any ideas or facts you heard about, and always ask "How do they know that?" or "How can they be sure?" or "What about this other thing?". Cicero tried to use philosophy to make men more logical thinkers, so that they would make better decisions about how to run the government. But Cicero also held some Stoic ideas, especially that men should try to be as good as possible.
About a hundred years later, in the time of the emperors Claudius and Nero, another philosopher called Seneca wrote another set of essays about Stoic philosophy. Seneca thought that men should not waste time on things that really didn't matter. Instead, they should use their time well, to help improve the world, and to improve their own minds by studying philosophy.
Soon after Claudius, many men and women began to look for a closer, more direct relationship to the gods or to God. Some people, like the Christian Gnostics, tried to use magic spells and secret knowledge to get closer to God. The Christian followers of Montanus thought you could get closer to God through prayer. Pagan Neo-Platonists used philosophical ideas that came from Plato's ideas about the perfect form to try to perfect themselves and get closer to God that way.
Later Christians developed their own philosophical ideas. St. Augustine and St. Ambrose both studied earlier philosophers and tried to create a Christian philosophy that would include both Christian ideas and Greek and Roman philosophy, including both Aristotle and Neo-Platonism.
The fall of the Roman Empire did not stop men (or a few women) from thinking about these ideas. In both the Islamic Empire and medieval Europe, men like al Tusi and Maimonides and Thomas Aquinas continued to try to make religion agree with philosophy, and to try to get closer to God through philosophy.
To find out more about Roman philosophy, check out these books from Amazon or from your local library: