Beginnings are arbitrary, but important
Why does the year begin on the first of January?
Of course it’s convenient to have your months fit evenly into years, so that January 2016 is all in one piece. (But not absolutely required: the traditional year in England until 1752 started on the Feast of the Annunciation, the 25th of March, so part of that month fell in one year and part in another.) But a month is largely a bookkeeping device nowadays: originally fitting the phases of the Moon, they aren’t tied to it in our calendar any more. If they were, this year’s full Moon on Christmas would either happen every year, or never.
Why the first of January?
On the practical level, a year is a fuzzy length of time defined by the weather. One round of generally hot followed by generally cold is a year (with rainfall as a local cycle). But that’s not very useful, except maybe for hunter-gatherers. If you want to plant crops or get paid an annual salary, you want something more definite. Starting the year on, say, the minimum recorded temperature would be really hard to work: each region would start at a different time, years would be all different lengths, and you’d only know the year had started for sure a couple of months later. (And this year’s 70F in Alexandria during Christmas week would confuse things greatly.)
Astronomers, of course, have a precise answer: pick a solstice or an equinox, a very well-defined instant in the orbit of the Earth. In fact, the founders of the French republic did just that. They picked an equinox as the beginning of their calendar (and made all months thirty days long, all weeks ten days). Today is the 7ème Nivôse of the year CCXXIV according to that reckoning.
But the calendar on your wall doesn’t show that date. The failure of the French republican calendar underlines the fact that the beginning of the year is a social and psychological matter more than it is an astronomical one. There’s no point if having things worked out nicely if no one else uses your system. And January 1 does not correspond to any solstice or equinox.
Like many other things (for instance, the clock time of day), the start of the year is something that can be chosen arbitrarily beforehand, but must be absolutely fixed afterward. You can measure the revolutions of a wheel from any point, but you have to stick with the one you’ve chosen. So January 1, which possibly evolved from the winter solstice (something over a week before) through a year with a slightly inaccurate number of days, was fixed in place by Julias Caesar in a calendar reform and remains our starting-point. As long as everyone uses it, it doesn’t matter why it was chosen.
And the need for an exact starting-point goes deeper than science. People want to know fixed and exact answers—how many planets are there? When does the year start?—even when the questions move a scientist to sigh, and settle down for a long explanation. It is somehow important that, up to a definable instant—“I now pronounce you. . .”—two people are fiancés, and afterward a married couple.
And, of course, both a marriage and a New Year provide a good excuse for a party.