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In Greek and Roman stories, snakes were often symbols of life after death, healing, immortality.
In the Hebrew Bible, in Numbers 21.4-9, YHWH tells Moses to create an image of a bronze serpent, and that anyone who looks at it will be healed.
The Christian author of the Gospel of John considered this healing serpent a symbol of Jesus, as he has Jesus declare: "Just as Moses lifted up the snake in the desert, so the Son of Man must be lifted up, that everyone who believes in him may have eternal life" (John 3.14-15, NIV).
This image should be compared to the rod of Asclepius, which is the symbol of the American Medical Association: a serpent entwined around a rod (not to be confused with Hermes' caduceus, which was erroneous adopted as a symbol by the U.S. Army Medical Corps in 1902). Asclepius was a Greek god of healing. He was born a man, son of a mortal woman and the God Apollo. He eventually began healing so many people, and even resurrecting the dead (like Hippolytus), that the balance of nature was overturned, since Hades was being depopulated and there was a possibility that Asclepius would make the human race immortal. Hades and Zeus agreed that something had to be done, so Zeus brought about his death through lightning. However, Asclepius did not remain dead. Zeus raised him to heaven as a God. He is represented by the constellation Ophiuchus ('serpent-handler' in Greek, Serpentarius in Latin). [This sounds similar to the Jesus story, right? Except it is much older.]
The Hebrews were not the authors of the concept either. The healing serpent associated with the Moses story owes its origin to pre-Israelite snake cults in Canaan. Bronze-age objects from religious rituals honoring snakes have been discovered at at least 4 sites -- Megiddo, Gezer, Hazor, and Shechem. See Karen Randolph Joines, "The Bronze Serpent in the Israelite Cult," Journal of Biblical Literature 87.3 (September 1968: 245-256). Also, Serpent symbolism in the Old Testament: A linguistic, archaeological, and literary study, by Karen Randolph Joines (Haddonfield House, 1974). One may learn, by reading between the lines at 2 Kings 18:4, that the Israelites had a history of snake worship, and it is probably directly related to Canaanite religion. 2 Kings 18.4: "He [King Hezekiah (r. 715/6 – 687 BCE)] removed the high places, smashed the sacred stones and cut down the Asherah poles. He broke into pieces the bronze snake Moses had made, for up to that time the Israelites had been burning incense to it. It was called Nehushtan" (NIV).
The serpent was also an important symbol in Egypt and was worn on the headdress of the pharaoh.
Snakes were important symbols in other myths around the world. Hinduism and Buddhism had them. Dragons were important in Chinese traditions. Aztec myths featured the "feathered serpent," the semi-divine human incarnation of the serpent sun who descended to earth as a great teacher.