1) "KIDUSH HA'CHODESH" WITHOUT WITNESSES
OPINION: The Mishnah here describes how Beis Din interrogated the witnesses who said that they saw the new moon. The Mishnah later (24a) relates that Raban Gamliel showed the witnesses models of the moon in different shapes and facing different directions in order to clarify exactly how the witnesses saw the new moon. These Mishnayos imply that the declaration of the new month was determined exclusively by authenticating the testimony of the witnesses.
RABEINU BACHYE (Shemos 12:2) cites a novel opinion in the name of RABEINU CHANANEL that Kidush ha'Chodesh is entirely independent of the testimony of witnesses. Rather, the first day of the new month is determined by Beis Din solely in accordance with the calculations of the Sod ha'Ibur given to Moshe Rabeinu at Har Sinai. Witnesses do not affect the determination of the new month at all. This opinion is also cited in the name of RAV SA'ADYAH GA'ON (see TORAH SHELEIMAH 13:293, and OTZAR HA'GE'ONIM to Beitzah, p. 3).
Rabeinu Chananel cites a number of proofs for this view. First, for the forty years of the Jews' sojourn in the Midbar they were covered and protected by the Ananei ha'Kavod, the Clouds of Glory, and thus it was impossible for anyone to see the new moon. Accordingly, the new month must have been established without testimony about the new moon.
Second, the verse (Shmuel I 20:5) relates that David ha'Melech told Yehonasan that on the next day there would be a Rosh Chodesh meal, and on the following day there would be a meal for the second day of Rosh Chodesh. How did he know that the next day would be Rosh Chodesh if witnesses had not yet come to testify that they saw the new moon? Moreover, if the next day would be Rosh Chodesh, then why would the day after that day be a second day of Rosh Chodesh? It must be that the new month was established based on calculations. (Both days were considered Rosh Chodesh since the calculations showed that the new month started towards the end of the thirtieth day. See Rambam, Hilchos Kidush ha'Chodesh 8:4.)
Third, the Mishnah (25a) states that Raban Gamliel sanctified the new month even when the witnesses' testimony did not conform with the physical reality. (See Insights to Rosh Hashanah 25:2
Fourth, the Mishnah (25b) states that when all of the Jewish people see the new moon but there is not enough time to sanctify the new month before nightfall, Beis Din sanctifies the following day instead. If the new month depends on the testimony of witnesses, why does Beis Din not declare on the next day that the previous day was Rosh Chodesh (retroactively)? It must be that the establishment of the new month is unrelated to the sighting of the new moon, but rather it depends solely on the calculations.
According to Rabeinu Chananel, why do the Mishnayos in Rosh Hashanah discuss witnesses in the first place if Kidush ha'Chodesh is independent of their testimony and is based solely on calculations?
He answers that witnesses are necessary because the Tzedukim did not accept the calculations of Beis Din. They did not accept the veracity of the calculations of the Sod ha'Ibur and they claimed that the new month is determined solely by when the new moon actually appears. They claimed that it was not possible for Beis Din to know -- based on mathematical calculations alone -- exactly when the new moon would appear every month. In order to disprove the view of the Tzedukim, the Chachamim enacted that Beis Din should sanctify the new month based on the testimony of witnesses which would show that their mathematical calculations are correct. Hence, it is a Mitzvah d'Rabanan to sanctify the new month based on the testimony of witnesses. This is why Raban Gamliel kept models of the moon in his chambers. Every month on the day before the new moon appeared, he would show the Tzedukim ("Hedyotos") where the new moon would be seen the next day. When witnesses arrived in court the next day and testified that they saw the new moon, their testimony substantiated the prediction of Beis Din with regard to where and when the new moon would be seen. All present proclaimed in unison, "Mekudash, Mekudash!" -- "the calculation of Beis Din was correct!"
The Chachamim eventually triumphed over the Tzedukim, but nevertheless Beis Din retained the practice of accepting testimony from witnesses about the new moon in order to commemorate how the Chachamim overcame the Tzedukim.
According to the opinion that the establishment of the new month depends solely on calculations and not on witnesses, why are there two days of Yom Tov outside of Eretz Yisrael (or two days of Rosh Hashanah even in Eretz Yisrael)? Since the exact date of Rosh Chodesh is based on calculations, there should be no grounds for doubt and it should not be necessary to observe a second day of Yom Tov. The Ge'onim (cited in OTZAR HA'GE'ONIM to Beitzah, p. 3) answer that the obligation to observe two days of Yom Tov outside of Eretz Yisrael is a Halachah l'Moshe mi'Sinai.
The RAMBAM (in Perush ha'Mishnayos) rejects this opinion because it contradicts many explicit statements in the Gemara. It is obvious that Beis Din sanctified the new month based on witnesses. Only when there were no witnesses did they sanctify the new month based on calculations. The Rambam suggests that perhaps Rabeinu Sa'adyah Ga'on expressed this opinion only because of the Karaites in his time who pestered him, saying that Kidush ha'Chodesh based on calculations is unreliable. In response, Rav Sa'adyah Ga'on said that not only is Kidush ha'Chodesh based on calculations completely accurate, but Kidush ha'Chodesh based on witnesses is not even mid'Oraisa.