Most educated people know that Christmas started out as a pagan festival, and was appropriated by the Christian church to better reach the masses. Chances are that the pagans the Christians stole it from had, in turn, stolen it from an earlier bunch of pagans, and so on, all the way back to a group of early humans huddling around a fire somewhere, seeing in the midwinter. Perhaps they exchanged some kinds of tokens, perhaps they imbibed fermented fluids our modern palates would find disgusting, perhaps they made propitiatory sacrifices to the gods of winter to encourage them to go away, though it's not unlikely that a burning log was involved.
So we had people marking midwinter and anthropomorphising the cosmic forces responsible for the season. Then more complicated religious systems came along and said, no, that's not the winter god, that's Zarathustra or Mithras or Sol Invictus. Then, around the fourth century, Christianity came along and decided that Jesus was born on the 25th of December. (Aside: according to some claims, the most likely date for the birth of Jesus would have been in August or September, assuming the thing about the shepherds being out in the fields was accurate.) Then along came secularism and the Enlightenment and Christianity receded somewhat to the background, though not quite disappearing; instead, becoming the default traditional-religious-meaning-of-Christmas which people complain nobody pays much attention to as they go gift-shopping.
So what we have today is a salmagundi of several different stories which don't quite fit together. We have, in particular, the Biblical story of the son of God being born in a manger in the Middle East, visited by wise men bearing gifts and so on. And beneath that we have a completely incongruous Arctic mythology of a fat man in a red suit who lives at the North Pole, rides flying reindeer and delivers presents. In some mythologies, he has armies of elves (an element of northern European mythology) helping him make and deliver the toys (presumably Apple and Nintendo have kindly signed some kinds of intellectual-property licensing agreements with them, allowing them to make iPhones and Wiis in their Polar chip fabs). In the Netherlands, he is accompanied by six to eight black men, whose job it is to thrash naughty children; in Switzerland and Austria, that task is performed by a demonic creature named Krampus. The man is known in English as Santa Claus or Father Christmas, though is generally identified as Saint Nicholas, a bishop from fourth-century Greece who is unlikely to have ever seen a reindeer. Similarities between Santa Claus and St. Nicholas of Myra are largely coincidental; some say that the bearded Arctic-dwelling man is derived from the Norse god Odin. Meanwhile, in Russia, he is known as Grandfather Frost, and in Finland, his place is taken by Joulupukki, the Yule Goat (which is actually a goatlike creature; the Finns are nothing if not metal)..
It would be complicated enough with just these two very different mythologies, awkwardly joined at the hip. But in the 20th century, as Christmas became an ever-greater secular and commercial milestone, even more elements were added. The general rule seems to be that anything goes, as long as it's vaguely wintery or snow-related. We got supernaturally animated snowmen (Frosty the Snowman, of the popular Christmas song, and Raymond Briggs' snowman), which have nothing to do with either Christianity or the old Nordic pagan mythologies. And more recently, other remotely polar elements have been appearing on Christmas cards, such as penguins. These, of course, live in the Southern Hemisphere, but if a fourth-century Greek bishop can travel the globe by flying reindeer, surely he can have a few penguins in his entourage. And I wouldn't be too surprised if, one of these years, someone threw in a polar bear or two for the more ecologically minded.
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