Different cultures have different florescences. Where are the other cultures when Egypt invented hieroglyphics, where were the other cultures, when the Greek were philosophizing, where were the other cultures, when the Mayas flourished. Where were the other cultures, when the Romans played with rethoric. What were the other cultures doing when the Chinese culture flourished.
Also these are not really competing, but more like morphing into another. During the Antique there was trade with India and China, subsahara parts of Africa. The Arabian mathematics was preserved in Greek literature. This was then rediscovered and translated into Latin, which kick-started European philosophy and sciences.
The same is true for religion. Christianity is the mix of Jewry and Greek philosophy. The Greek were heavy influenced from the Egypt culture. The Jews took their knowledge from the Arab peoples in their regions and also part from Egypt. This Monotheism came from earlier Polytheism and earlier natural religions.