What is the significance of having two heads?

Ha'Boneh: This hints to one who has conflicting thoughts of Heavenly matters and worldly matters. On which does he wear Tefilin - which should he subjugate to Hash-m? Rebbi was angry - surely both most be l'Shem Hash-m! A case occurred in which a baby truly had two heads; both are to Hash-m, until they are redeemed.

Why did Rebbi say that Flimo must exile himself, or be excommunicated?

Rashi: You mock me!

Tosfos: We do not find like this [nowadays]. However, a Midrash says that Ashmedai (the king of the demons) brought out of the ground a man with two heads. He married, and had sons. Some had two heads, like their father, and some had one head, like their mother. They came in front of Shlomo for Din - the former claimed double shares of their father's inheritance [due to their extra heads].

Shitah Mekubetzes, ha'Boneh: Shlomo commanded to put hot water on one of the two heads. Also the other head screamed; this shows that they are one. Had the other not felt it, this would show that they are two brothers, and receive two shares.

Daf Al ha'Daf citing Or Zaru'a (424): Often the Gemara asks about cases that never occur, e.g. one who has two heads, one who makes an animal or barrel of liquid a Golel for a grave (Eruvin 15a-b, Ohalos 15:9)... (NOTE: Rashi says that Golel is the cover of a coffin. Tosfos (Shabbos 122b) disagrees; one of his objections is that people do not make a living animal a coffin cover! - PF) Ohr ha'Yashar asks, the Kesef Mishneh (Hilchos Talmud Torah, Sof Perek 6) learns from here that we excommunicate for asking something impossible. Rebbi himself codified Mishnayos about impossible matters. Why did he decree [Galus or] Niduy on Flimo? Some say that a two-headed baby would not live to 13, to wear Tefilin. [The questioner in] Shevus Yakov (1:4) testified that he saw two one-year olds joined at the back of their heads; Shevus Yakov said that when they mature, both must wear Tefilin [if they were Jewish].

Kusnos Ohr (Bamidbar, cited in Gilyon ha'Shas): Flimo held like Teshuvas ha'Rashba (1:98), that a tradition teaches that if lack of a certain limb [or organ] makes a Tereifah, Yeser (an extra such limb) is Tereifah, but the Klal 'a Tereifah cannot live more than a year' does not apply to it. Rebbi held that it is like all Tereifos, and cannot live more than a year. If so, Flimo contradicts Chachamim's tradition! Hash-m caused that the case of Pidyon ha'Ben arose, and the Beraisa taught that he pays 10 Sela'im. We do not redeem a Tereifah! Rather, this shows that a Treifah of Yeser is unlike other Tereifos.

Daf Al ha'Daf: Teshuvas Chasam Sofer (YD 294, citing R. Shmuel Krakavar) brings that in the count of Bnei Yisrael in Parshas Bamidbar, it says "l'Gilgelusam", but not in the count of Levi, lest we say that a baby with two heads would count as two. This applied only to Levi, who were counted from one month old. Bnei Yisrael were counted from 20 years; such a baby is Tereifah, and could not live past one year. Chasam Sofer said that perhaps "Omer la'Gulgoles" teaches that two portions of manna would fall for such a baby. (NOTE: Hash-m could have enabled babies to subsist on manna. However, He withheld from manna tastes that are bad for nursing women. This implies that women nursed in the Midbar! Rather, extra manna fell for his mother (a nursing woman eats more - Kesuvos 64b). We know that [extra] manna fell for babies. If a woman remarried within three months of being divorced [or widowed], and gave birth nine months after divorce, it is a Safek who is the father. The manna settled the Safek; the father received extra manna (Yoma 75a). - PF) The Rashba [and Isur v'Heter 49:3 and Sha'arei Dura 84:3] say that a [Tereifah] Yeser can live more than a year; it is not included in the Klal. If so, why is l'Gilgelusam omitted only for Levi? Hash-m did not say anything l'Vatalah; surely someone in the Midbar had two heads, but not in Levi. After the question about the baby with two heads, Rebbi ceased his anger. This is a proof for the Rashba! Merafsin Igra asked, since such a baby counted as two, it should have said l'Gilgelusam also for Levi! He answered that even so, he would count only as one in the count of Leviyim. The episode with Shlomo shows that they are one. Also, Flimo asked only about on which head to put Tefilin; he knew that he need not wear Tefilin on both.

NOTE: There is no support from Flimo. Perhaps he was unsure if he must wear Tefilin on both. He cleverly asked which head, for if both require Tefilin, Rebbi would say so. Had he asked whether he must wear on both heads, and Rebbi said no, he would not know which head must wear! Merafsin Igra implies that even if only one Yisrael had two heads, it properly said l'Gilgelusam also for other Shevatim, for they were counted together. It was omitted for Levi, for they were counted separately. (PF)

Tzon Kodoshim: The Zohar (Bereishis p.180) says that in the place to which Kayin was exiled, people gave birth to babies with two heads. Exile yourself to there and ask your question. Here, your question is like a mockery!

Why must he pay 10 Sela'im to redeem a Bechor with two heads? If twins were born at once, he pays only five (Bechoros 48a)!

Rashi: There, Iy Efshar Letzamtzem (surely, one was first; the other is not a Bechor). Here, it is one baby; Efshar Letzamtzem, and both of them are Peter Rechem.

Shi'urei R. Yechiel Michal Feinstein: Birth of one head is like birth of the entire body. Therefore, it is as if they were born exactly together.

Chidushei Basra: We could have explained that the majority of the bodies came out before the heads. Once the majority left, it is as if the entire baby was born.

Daf Al ha'Daf citing Chacham Tzvi (97): Why must he be redeemed at all? Anything extra, it is as if it was removed. If the head was sealed, there is no Tum'as Leidah (Nidah 24), and the next child is a Bechor (Rambam Hilchos Bikurim 11:14), and all the more so if it is as if he has no head! Rather, it is not truly as if it was removed. This is merely a Klal to teach when extra limbs or organs make a Tereifah.

Daf Al ha'Daf: Minchas Chinuch (392) asked why the Rambam did not bring that one must redeem a Bechor with two heads for 10 Sela'im. I answer that the Beraisa that obligates is like the opinion that a Tereifah can live. We hold that a Tereifah cannot live, so there is no Mitzvah to redeem him.