Why do we celebrate Jewish New Year 5780? why not 10037? or 32?
We first need to understand where our calendar comes from and how were the years counted in the old days...
The Bible, in the book of Exodus, tells us that "This month will be for you the first of the months of the Year". It does not refer to Tishri, but to Nissan, the month of Passover. In other words, the tale of Exodus is telling us that the Exodus itself is a turning point in History important enough to qualify as a beginning... in this case of the Jewish people. What we now know as Rosh HaShanah is mentioned in Leviticus as "Yom Truah" or the day of the blasting of the Shofar. It is intended to mark the beginning of Fall. Why was that important in Biblical times?... because we were a fundamentally agrarian people and the passing of the seasons were to be mark based on their relation with the agricultural cycle. Beginning of the Fall was the beginning of the economic year in the ancient Middle East. This is why contracts and economic agreements were dated considering Tishri the first month of the year, when in fact is the seventh according to the Biblical count. It was also the reference when refering to the Jubilee Year. There is also the fact that for political purposes, Nissan remained the New Year... when a king was in power, the count of their reign was governed by Nissan - not Tirshri.
While there are other "New Years" in Jewish tradition, they tend to be for very specific purposes, not a generic calendar statement. The continuous use of Tishri as a reference for economic purposes became even more prevalent during the Baylonian exile, during which the Jews in the Diaspora had to refer to the Baylonian calendar for dating contracts - and the baylonians, of course, use Tishri as a reference.
But the replacement of Nissan by Tishri as the Jewish New Year is not the only thing we took from the Babylonians. The very names of the month of the year are of Babylonian origin. In the Bible the count of the month is only ordinal: First, Second, etc... just like the days of the week. The names of the months were adapted from Babylon. An interesting case is Cheshvan. The name itself is Babylonian, and is traditionally refered also as "Marcheshvan". The "Mar" in the beginning is a short reference to "Marduk", the main God in the Baylonian panteon. Marduk's holiday took the whole month, but his birthday has been calculated to have been around what we call today December 25. Babylon had a strong influence in Middle Eastern Cultures beyond its fall as an Empire.
But what about the count? In old days, counting of the year began anew with any trascendental event. The Bible talks about the "First Year of King David" for example, and renews the count with each king's ascencion to power. Later on, Jews refer to the "So and So year of the baylonian exile" for example. A significant turning point came with the destruction of the Second Temple. From that point on, Jews referred to the "Year so and so from the destruction of the Temple and Jerusalem". By that count, we are now in the year 1950!.
It was Maimonides who standardized the rules for counting the years and the months and codified them in the XII Century. He also started referring the year as "The Year of the Creation of the World"... which is the count reference we use today. The number 5780 signifies in our tradition 5780 years since the creation of the world.
A Calendar is a cultural element that reflects the cycle of the annual renewal time in a specific society, and how we look at it defines in a way how we see ourselves. The life, death and believed resurrection of Jesus of Nazareth defines for Christians their core identity, so it makes sense that they use the year of Jesus' birth as the beginning of their count (leaving aside errors and questionable documents). For Muslims, Mohammed's move from Mecca to Yatrib Al Medinah marks the truning point in the evolution of Islam, so they count their years starting from that central event. For us Jews, as we count today, the Creation of the World is our reference - and we see ourselves as Human Beings first (A Mentsch, Ben Adam, etc) and we are Jews as a secondary definition.
Some trivia about the Creation fo the world according to our tradition: The Year 1556 of the Creation marked the Flood, and the Year 1556 in the Christian Calendar marked the death of Charles V Habsburg (Charles I of Spain).
The Year 1948 of the Creation of the world markes the birth of Abraham, and of course we all know the meaning of 1948 in the common calendar. Just trivia... don't read any maning into it.
SHANAH TOVAH TIKATEIVU V'TECHATEIMU
May you be inscribed and confirmed for a Good Year!