הַחֹדֶשׁ הַזֶּה לָכֶם רֹאשׁ חֳדָשִׁים רִאשׁוֹן הוּא לָכֶם לְחָדְשֵׁי הַשָּׁנָה
This month shall be to you the beginning of months; it shall be the first to you of the months of the year. (Exodus 12:2)
This is the first commandment in the Torah given to the entire nation of Israel. (Three of the 613 mitzvot appear in Genesis, but they're all given to individuals: "Be fruitful and multiply" to the first humans, circumcision to Abraham, and the prohibition of the gid hanasheh just appears as a descriptive statement.) Therefore, the very first question that Rashi asks, at the very beginning of the Torah, is "Why didn't the Torah just start with 'hachodesh hazeh lachem'?". And ok, he answers the question. But the point is that "hachodesh hazeh lachem" is considered important enough that it could have been the beginning of the Torah.
According to rabbinic tradition, "hachodesh hazeh lachem" represents the commandment to sanctify each new month, done originally through witnesses observing the new moon, and now through mathematical algorithms. Therefore, all of our calendar geekery is in the service of this mitzvah.
The word "zeh" ("this") is often understood by the rabbis to mean that someone is pointing to something. Thus, the Rambam (in Hilchot Kiddush Hachodesh 1:1, quoting the Mechilta) says that the Holy One showed Moses the image of the moon in a prophetic vision and said "See it like this and sanctify it". However, Hilchot Kiddush Hachodesh 5:1 goes on to say:
Everything that we have said about determining Rosh Chodesh by sighting [the moon] and intercalating the year based on time or need, is only done by a Sanhedrin in the land of Israel, or a court ordained in the land of Israel whom the Sanhedrin has given authority. For this was said to Moses and Aaron: “This month shall be to you the beginning of months”, and via oral transmission one person learned from another from our teacher Moses that this is its interpretation: This testimony is delivered to you, and anyone who stands in your place after you. But in a time when there is no Sanhedrin in the land of Israel, new months are determined and years are intercalated only by the calculation by which we calculate today.
Nevertheless, we're the ones who are authorized to set up the calendar algorithm. The U.S. Constitution says "Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors...", commanding the state legislatures to set up a system for choosing presidential electors, whose results are binding on the federal government (until the Supreme Court trampled this, but that's another story). Likewise, the Torah says "hachodesh hazeh lachem", commanding Israel to set up a system for sanctifying the new month, whose results are binding even on God. That's right, you read that correctly. Rosh Hashanah 8b interprets Psalm 81:4-5 ("Sound the shofar on the new moon ... for it is [1st] a law for Israel, [2nd] a ruling of the God of Jacob") to mean that the court above does not enter into judgment [on Rosh Hashanah] until the court below has sanctified the new month.
According to the algorithm we have set up, Nisan (the 1st month) begins this Thursday.