Our Gemara on amud aleph relates the machlokes between the house of Shamai and the house of Hillel regarding what is the age of a child whom the parents are obligated to bring up for the festival pilgrimage.

Who has the status of a minor with regard to this halakha? Any child who is unable to ride on his father’s shoulders and ascend from Jerusalem to the Temple Mount; this is the statement of Beis Shammai. And Beis Hillel say: Any child who is unable to hold his father’s hand and ascend on foot from Jerusalem to the Temple Mount, as it is stated: “Three times [regalim]” (Exodus 23:14). Since the term for feet is raglayim, Beis Hillel infer from here that the obligation to ascend involves the use of one’s legs.

Torah Temimah (Shemos 23:12) explains the reasoning for The house of Hillel. Tge Mitzvah of chinuch it’s only active in the way that if the child were an adult, he would be obligated. Since even an adult who is unable to walk is exempt from the obligation of pilgrimage as we saw in the Mishna, A minor can only be obligated once he is able to walk on his own two feet.

What shall we say to explain the house of Shamai’s position? In general we find that the house of Shammai takes a more extreme approach toward the obligation of chinuch, seeming to make the requirement more about habituation and exposure without even yet understanding. For example, we find in Mishna Succah (2:8) that the elder Shamai made a hole in the roof to create a small Succah for his newborn grandson and daughter-in-law. Obviously, an infant cannot possibly have any intelligence to yet learn about or understand the mitzvah, yet according to some commentaries indeed Shammai consider this a part of the mitzvah of chinuch (see for example Tiferes Yisrael Yachin 37.)

Malbim Deuteronomy (19:1) offers an ingenious understanding of the tendency of the rulings of the House of Shammai to be stricter while the rulings of the House of Hillel to be more lenient, what specific teleological ends it serves, as well as a fascinating allegorical interpretation of the famous Oven of Akhnai controversy in Bava Metzia (59b) and several other aggados which we will discuss more of later. Malbim suggests that The House of Shammai represents a spiritual ideal. Namely, that the intellect can be elevated through engagement in spiritual matters and abstention from physical lusts and desires to an extent that it can attain an Edenic state of Adam prior to eating from the Tree of Knowledge. If this is attained, a higher level of immortality of the soul is achieved insofar as the intellectual soul can divest from the body at will, allowing for transcendent states even prior to death, and higher levels of attainment in the World to Come. This accounts for why the House of Shammai’s rulings are often more stringent because the foci are on abstinence and spiritually pure states. The House of Hillel endorses a more temperate approach; serving G-d in this world by engaging in physical acts while also in devotion to G-d. 

So in regard to chinuch of a young child, Bais Shammai goes for the ideal and pure state.  Even the soul of the infant can appreciate the mitzvah. (In terms of the pilgrimage, the child must be safe, so he needs to at least sustain being carried.) But the House of Hillel requires that child be old enough to perform and understand the mitzvah on some level

Thus as one example, the dispute between the House of Shammai and the House of Hillel in Mishna Sukkah (2:7) can be understood on a halakhic level as well as metaphysical: 

One whose head and the greater part of his body were within the sukkah and his table within the house: Beis Shammai say: it is invalid and Beis Hillel say it valid. 

Here we find typically the House of Hillel choosing the more lenient position. Malbim adds a mystical layer of meaning to this Mishna. The House of Shammai allows for no compromise; one must completely disengage from the comforts of this world (the home) and dwell on the spiritual plane (the Sukkah). However, the House of Hillel advocates moderation. One may have his head and the greater part of his body in the Sukkah (spiritual sphere) but still stay partially engaged in the physical world (the lower part of his body resides in the home).

This idea is also reflected in a teaching of the Arizal that in the World to Come, the halakha will be in accordance with the House of Shammai (Sefas Emmes, Korach 5647). Meaning, in a post-Messianic world, people can live on this ideal plane of disengagement from physicality, in accordance with the principles of the House of Shammai.

We can add to the Malbim’s mystical take on Beis Shammai and Beis Hillel in regard to another of their disputes found in Mishna Keylim (18:1)

A wooden chest that is large enough to contain forty se’a is not susceptible to contracting ritual impurity, since it is no longer considered a vessel. In determining its capacity, Beit Shammai say that it is measured on the inside, and Beit Hillel say that it is measured on the outside so that the volume of the walls of the chest itself is included in the measurement. 

Once again, Beis Shammai is more strict, requiring forty se’a of volume to be reached only on inside actual capacity before the vessel can be exempted from impurity. While Beis Hillel allows one to use the total areas including the outside and thickness of walls to reach the volume of forty se’a. Metaphysically, Beis Hillel allows the outside (earthly experiences) environment to be included in the measurements to achieve purity. While Beis Shammai requires uncompromising internality (only the inside volume) to achieve purity. 

In addition, one can apply the Malbim’s ingenious principle to another dispute between Shammai and Hillel found in regard to Shaatnez and Tzizit. According to BeisHillel, one may wear a linen garment, even though the fringes that contain the blue thread are made out of wool, which is ordinarily a forbidden mixture (Deuteronomy 22:11) of wool and linen. However, according to Beis Shammai, this mixture is forbidden and thus a linen garment is exempted from the obligation of Tzizit (Menachot 40a). 

Let us consider the sources of both of these materials. Wool comes from sheep and linen comes from flax plant fibers. In other words, a shepherd manufactures wool, while a farmer manufactures linen. What is the Biblical understanding of these two professions, farming and shepherding? Notably our forefathers, Abraham, Isaac and Jacob, as well as Moses and David were all in the shepherding profession. And, it is most telling, that the two rivalrous sons of Adam and Eve, Cain and Abel are pitted against each other in these opposing professions. “Abel became a keeper of sheep, and Cain became a tiller of the soil.” Genesis (4:2). The commentaries attribute significance to each brothers’ chosen occupation. Rabbenu Bechaye (Genesis 46:32) reflects on the nomadic qualities of the shepherd. It involves a life of separation from cities and metropolitan life, where often that life is sinful and materialistic. The farmer, on the other hand, is tied down to his possessions and is in a fixed location, usually part of a city/state structure. We moderns are used to thinking of farmers as the opposite of city folk, but in ancient times, agriculture was the core of a city, as opposed to nomadic hunter-gatherers.  

As the Malbim has explained, Shammai advocated for an ideal withdrawal from the pleasures and pursuits of this world, while Hillel advocated moderation. Therefore, we may also understand the argument about the permissibility or prohibition to wear Shaatnez and tie Tzizit to a linen cloak as stemming from this same truth. Shammai affirms the suitability of the ascetic nomadic life represented in the shepherding and wool garments. Only such garments are considered the cloak of a Jew, and only those are obligated in Tzizit. Hillel accepts that in this world one cannot fully withdraw. One can be a part of society, and belong to an agrarian city/state, as symbolically exemplified by the linen garment. This cloak, or metaphorically this persona, can also be worn by a Jew and can also fulfill the mitzvah of Tzizit, along with and not in contradiction, to any other spiritual pursuit.  Also note, Tanya, Iggeret HaKodeh (13) offers a similar understanding of Beis Shammai and Beis Hillel, though he does not develop it explicitly on a practical level.

Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation

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