Gemara

67b (Mishnah): If a man became unable to speak, and others asked him if we should write a Get for his wife and he nodded his head, we test him three times. If he answers properly no and yes, we write and give the Get.

70b - Question: Perhaps a sickness is causing him to nod his head like people nod to indicate 'no', or to indicate 'yes'!

Answer (Rav Yosef bar Minyomi): We test him in a way that requires him to alternate answers of 'yes' and 'no'.

Question: Perhaps a sickness causes him to nod alternatively in these two ways!

Answer: We do not ask in the order of a simple alternation.

(Tana d'Vei R. Yishmael): We ask him questions relevant to the season.

Question: If we ask if he would like to be covered with a coat or sheets, failing this test does not show that his mind is disturbed. Perhaps he was struck by a chill, or felt hot!

Answer: Rather, we ask if we should pick for him fruits that are out of season.

71a (Rav Kahana): If a Cheresh (deaf person) can communicate by writing, we write and give a Get to his wife if he requests this.

Question (Rav Yosef): Why must Rav Kahana teach this? We know this from our Mishnah!

Answer (R. Zeira): The Mishnah discusses a mute. He is not like a (total) Cheresh:

(Beraisa): One who can hear but not speak is a mute. He is like a Pike'ach (sound-minded person) in every respect.

72a (Beraisa): If a scribe wrote a Get and witnesses signed it, it is Batel unless the scribe and witnesses heard the husband's voice command them.

This teaches unlike Rav Kahana (71a, who said that if a Cheresh can communicate by writing, we write and give a Get to his wife).

Rishonim

The Rif and Rosh (6:2) bring the discussion on the Mishnah.

Rosh (ibid.): R. Tam says that even if a Pike'ach wrote that we should write and give a Get to his wife, we do not do so. The Mishnah allows writing a Get for one who nodded his head. This is because nodding is better than writing. Alternatively, it is a leniency for one who became unable to speak, to enable his wife to remarry.

Rambam (Hilchos Gerushin 2:16): If a man became mute but his mind is sound, and others asked him if we should write a Get for his wife and he nodded his head, we test him three times alternating. If he answers properly, we write and give the Get. Similarly, If he wrote 'write and give a Get to my wife', if his mind is settled we do so. One who became mute is unlike a Cheresh.

Kesef Mishneh: A Beraisa teaches that a Get is valid only if the scribe and witnesses heard the husband's voice command them. ''The husband's voice' teaches unlike Rav Kahana, who said that if a Cheresh can communicate by writing, we write and give a Get to his wife. The Gemara (71a) connotes that if he can hear or speak, he can authorize a Get through writing. It seems that the Rambam agrees that we rely on gestures and writing of anyone except for a Cheresh, who is like a lunatic.

Kesef Mishneh: The Ran that we must test even one who writes. If we will say that the Rambam agrees, Rav Yosef asked what was Rav Kahana's Chidush because he was sure that Rav Kahana agrees with the Mishnah, which requires testing. R. Zeira answered that Rav Kahana teaches that we write even for a Cheresh. The Mishnah discusses a mute to teach about nodding, which does not apply to a Cheresh. Alternatively, Rav Kahana teaches that writing is better than nodding. We write for a total Cheresh only if he writes, not if he gestures.

Poskim

Shulchan Aruch (EH 120:5): Even if one wrote to a Sofer instructing him to write a Get and wrote to witnesses that they should sign, they should not obey unless they hear from his mouth. This is whether he is Pike'ach, or became mute, or cannot hear, whether or not he can speak. Some permit to write and sign if he became mute and commands the Sofer and witnesses through writing, and his mind is settled.

Beis Yosef (DH v'Lich'orah): It would seem that if the Rambam relies on writing of one who became mute, all the more so he relies on writing of a Pike'ach. However, the Tosefta (2:10) is Posel a Get unless the scribe and witnesses heard the husband's voice command them. R. Yerucham cites R. Tam, who says that this disqualifies the written command of a Pike'ach for Gitin. It helps only for one who became mute. Presumably, it does not help for a Pike'ach because he can command by speaking; likewise, gestures do not help. It appears that the Rambam disagrees, and holds that also a Pike'ach can command through writing or gestures. However, the Rashba, Rosh, Ran, Mordechai and other Poskim apply the Tosefta to a Pike'ach husband, and all the more so if he cannot speak or hear.

Question: The Mishnah allows writing a Get for one who nodded!

Answer #1 (Beis Yosef, ibid.): The Rashba learns from the Yerushalmi that nodding is like speaking.

Answer #2 (Beis Yosef, ibid.): The Rosh says that it is a leniency for one who became mute, to enable his wife to remarry. According to this, also writing helps. Even though the Beraisa (72a) unlike Rav Kahana says that the Get is Batel, i.e. mid'Oraisa, the Rosh says that we are lenient for one who became mute. This is because the Tosefta says that the Get is Pasul, i.e. mid'Rabanan. The Gemara brought the Beraisa instead, for it disqualifies mid'Oraisa (regarding a total Cheresh) a Get that Rav Kahana is Machshir. The Tur rules like Answer #1, for the Yerushalmi supports it. Also, the Yerushalmi brings a Beraisa like the Tosefta, and concludes 'it is not a Get'. This suggests that 'Pasul' in the Tosefta means Batel mid'Oraisa.