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Why did the Romans attach so much importance to ancestor worship, patriarchy, and family honor?
The Roman attachment to fathers is essential to Roman identity. The Romans had a phrase, the mos maiorum, meaning "the customs of the ancestors" and our word, morality, comes from this. For the Romans, morality was doing what your ancestors would want you to do. The story of Rome's founder, Aeneas, encapsulates this beautifully. Aeneas escapes from Troy - he's a Trojan - and before he founds Rome, he escapes from Troy carrying his father, Anchises, on his back. And he also carries with him his household gods in the form of little statuettes of the gods he worships. For the Romans, you can see what a beautiful metaphor this is - Aeneas is pious and the word, pious, for the Romans means "devout towards his father." So, the real Roman is someone who is willing to carry the burdens of his ancestors forever.
For the Romans this resulted in a whole host of political practices and customs. In contrast to a lot of other ancient peoples who worshipped youth as the ultimate example of masculinity, the Romans worshipped age. The word for the Senate means "an assembly of old guys;" the word for "old" is senex. What the Romans wanted was a world that was governed by age and age was that which was worthy of reverence.
Any Roman in the republican period, any Roman of any social standing anyway, would have had in his house a shrine to his ancestors that would contain the funeral masks, the death masks of his ancestors, or little statuettes representing them. At a Roman funeral, as told by the Greek historian, Polybius, Romans would wear the death masks of their ancestors. Imagine this: you're a young man and you put on the mask of your dead father or your dead grandfather and what this was meant to show was that you are ready to embody that person and that your sense of honor and individuality was subsumed into this other being, your ancestor.
The fact that Romans gave the fathers of the household tremendous power - it was called patria potestas (power of the fathers) - meant that a father had the powers of life and death over his children, including his male children and even his grown children. It goes without saying that he also had these powers over his slaves.
Finally we can see the Roman attachment to fathers and ancestors in their naming practices. All well-born Romans had at least two names but the only name that mattered was the family name. Julius Caesar, for example, was called Julius. That is not his first name; his first name is Gaius, which is the Roman equivalent of Joe. The Romans only had around five first names and none of them were important. But the name, Julius, is his family name. It was the name of his father and his grandfather and his great grandfather before him. The name, Caesar, which is what we call him by, wasn't even a name at all. It was actually a nickname that meant "hairy," as in a hairy head. This was probably a joke because Caesar was bald. That nickname fascinatingly gets passed down to us as the word for "king" - Caesar, czar, in many languages - and for "emperor" because it is adopted by Caesar's own adopted son, who becomes Augustus Caesar and rules under that name, and he has to adopt that name because it's associated with his father.
Again, in all of these realms, in their funeral customs, in their legal system, in their government, in their religion, the Roman attachment to ancestors is visible in all aspects of their society.