According to halakha, when the beit din cannot sanctify months based on testimony, Rosh Ḥodesh is sanctified according to the calculation of the Hebrew calendar. In 359 CE (4119 from creation), almost three hundred years after the destruction of the Second Temple, Hillel the Second, the nasi of the beit din, realized that it was no longer possible to sustain the beit din that sanctified the months. He and his colleagues instituted a fixed calendar and, based on it, sanctified all the months until the restoration of the beit din (MT, Laws of Sanctification of the New Moon 8:2; Peninei Halakha: Zemanim 1:3 n. 3).
Seemingly, at that point there was no longer a need to keep two days of Rosh Ha-shana. After all, everyone knew, based on the calendar, when the first of Tishrei would be. However, just as the Sages ordained that those living in the diaspora should continue to follow their custom and keep each holiday for two days, so too, they ordained that residents of Eretz Yisrael should follow their custom and keep Rosh Ha-shana for two days. One might suggest that the fact that it was often necessary to keep two days even when the month was sanctified based on witnesses implies that this is the proper way to observe it. As usual, halakha reflects spiritual reality. Zohar (III 231a) explains that because of the severity of the judgment, the Sages saw fit to add a day, so the judgment contain compassion and thus be mitigated. If Rosh Ha-shana were only one day, the world might be destroyed due to the severity of the judgment.
Furthermore, the reason all holidays must be celebrated for two days in the diaspora is because the people there are living far away from the manifestation of holiness. In Eretz Yisrael, however, where holiness is more accessible, the holiness of the holidays can be absorbed in only one day. This can be compared to a flashlight: When illuminating a nearby location, the light is strong and focused on a small area. In contrast, when it is used to illuminate a distant location, the light is weaker and more diffuse (Derekh Mitzvotekha 114:1). Because Rosh Ha-shana is a day of hiddenness and concealment, two days are needed to absorb its light, even in Eretz Yisrael.
Perhaps we can suggest that the prayers of the first day are primarily directed toward collective matters – that we may merit that God’s kingship manifests over His people, Israel, and over Zion, the dwelling place of His glory, so that every living being will say, “The Lord, God of Israel, is King, and His kingship has dominion over all.” Through this, the whole world will attain blessing and peace. In contrast, in the prayers of the second day (which was established by the Sages), we ask for all these lofty ideals to be realized in our individual lives as well. We ask to be partners in revealing God’s glory in the world in our day-to-day lives, and thus to merit divine blessing.
. Zohar (Pinḥas 231b) explains that the first day is one of uncompromising judgment (dina kashya), while the second is one of milder judgment (dina rafya). The underlying rationale is clear: The first day is of Torah origin while the second is rabbinic. Sha’ar Ha-kavanot (Derushei Rosh Ha-shana #2) states that the first day is for rectification of the inner ze’er anpin and the second for rectification of the outer ze’er anpin. Ramḥal writes that the first day is a corrective for Leah and for the keter aspect of Raḥel, while the second is a corrective for the ḥokhma aspect of Raḥel (Kitzur Ha-kavanot, Rosh Ha-shana, p. 110). Shem Mi-Shmuel (Rosh Ha-shana 5673, s.v. “ita be-kitvei”) suggests that the first day is one of uncompromising judgment about divine matters, while the second day is one of milder judgment about mundane matters. See the Harḥavot.