One of the most practical toldot of Dash of choleiv (milking) – at least according to most rishonim. Defining the exact parameters of the prohibition of milking is not simple. We begin with a machloket tannaim as to whether or not this is a Torah prohibition at all. According to the majority position of the chachamim milking is only a rabbinic prohibition. R. Eliezer disagrees. (Shabbat 95a)
החולב והמחבץ והמגבן כגרוגרת, המכבד, והמרבץ, והרודה חלות דבש
.שגג בשבת – חייב חטאת, הזיד ביום טוב – לוקה ארבעים, דברי רבי אליעזר
.וחכמים אומרים: אחד זה ואחד זה אינו אלא משום שבות
It is an attempt to understand this berayta and to synthesize it with the limitation of dash to things that grown from the ground that generates the complexity of the sugya of choleiv.
After a complicated analysis we come to following positions:
Rashi, Tosafot and Rambam all hold that choleiv is a Torah prohibition. Rashi and Rambam as a tolda of dash, Rabbeinu Tam as a Toldah of memachek.
Ramban, Rashba, Ritva, Ran, Nimukei Yosef all hold that it is only a rabbinic prohibition. See here for the mekorot as well as material dealing with gidulei karka.
The mechaber appears to side with the Rambam that it is a Torah prohibtion but the Magen Avraham paskens like the Ramban. There are a few achronim (see the Divrei Malkiel here) that want to at least use this shita as a leniency and the Emden assumes that choleiv is d’rabanan (see his teshuva here).
The Tzitz Eliezer in a brilliant essay outlines all the various shitot. You can find his piece here with my emphases added (link). In this masterful teshuva / article he lays out all of the issues at length and with great clarity.
These issues are extremely important for religious kibbutzim and nursing mothers. It is important to note an important language distinction that Hazal make. Choleiv refers to the removal of milk from a cow and yonek refers to the removal of milk from a woman. It is taken for granted that the same set of halakhot that apply to one will apply to the other. I recognize the problematics of that fact.
The question that is often asked by nursing mothers who are expressing is how to do this on Shabbat. This is not a simple issue as it may be a d’orayta issue. If we take the mechaber as agreeing with the Rambam then there is no way to save the milk that is expressed. Only by accepting the minority read of the Emden, Magen Avraham and Ramban can we permit a women to save the milk that is expressed on Shabbat.
This is only in a case where the baby no longer needs that milk. That is to say that she is expressing to relieve her own discomfort but the baby has no real need for the milk. If the baby needed the milk then it would be possible to even permit the d’orayta for the sake of the baby.
Here are the four shiurim that I gave on this topics: