Hellenism and Judaism seem politically opposed at first. Within Alexandria at least there is limited integration between the communities but at the same time there was a distinctive Jewish identity with the area. In the Holy Land itself those who claim to be defenders of Judaism wage war against the people they label as traitors against YHVH. In this sense they draw a political distinction between two communities. This distinction is further reinforced by the depiction of the Hellenists as the “fourth beast” which rules over Israel in contrast with an independent Israelite Kingdom.
However, at the same time Jewish theology and scripture is adapted, especially in Alexandria, is influenced by Greek. Scripture is translated and the idea of allegory is introduced. This shows that at the very least a cultural change was occurring as the Jews lived in a Greek world.
However, even as the identity of the Jewish people became a source of political antagonism what identity even means may have shifted. Non-Greeks were considered essentially Greek as is indicated by the way Alexander is referred to as being from “Kittim.” But these means that lineage doesn’t have much to do with what identity someone has. So someone’s commitments to particular cultural practices and perhaps to theological truths determine someone’s identity. This in turn would open up those not descended from Jacob directly to participate in the Jewish community contrary to what Ezra and Nehemiah would have wanted.
So the most important question to explain this relationship between Judaism and Hellenism is what the relationship is between cultural practices and lineage in determining membership in a community and identity. Someone interested in resolving who is and is not a Jew would have to put forward a theory which considers the relationship between these factors. She would also, perhaps, need to explain the role of political alignment and cultural influences in determining what someone is.