Some acharonim (see Mishnat Ya’avetz, O.C. 74) suggest that the dispute between Tosafot and the Rambam is over the relationship between “mehadrin min ha-mehadrin” of Chanukah and the general concept of “hiddur mitzvah” (beautifying mitzvot) applicable to other mitzvot (Shabbat 133b). The latter concept appears to have a requirement that the enhancement of the mitzvah be visible to the world (see Yoma 70a). Accordingly, some rishonim believed that the parshiyot of tefilin, which are always covered, have no requirement of hiddur (see Mordechai, Hil. Tefilin; Tosafot, Menachot 32; Darkhei Moshe, O.C. 32:1.). Thus, Tosafot, concerned with visibility, is assuming “mehadrin min ha-mehadrin” to be an example of the general rule of hiddur, where such a requirement exists. The Rambam, though, would believe “mehadrin min ha-mehadrin” to be an independent rule, not associated with hiddur mitzvah, that is an integral part of the mitzvah of Chanukah candles and does not require visibility.
Evidence to this approach is taken from the fact that both the Rambam (Chanukah 4:12) and the Shulchan Arukh (O.C. 671:1) rule that one is required to take extreme measures, such as begging, to ensure that they are capable of acquiring Chanukah candles. However, later commentaries (respectively, the Ohr Sameach, and the Chemed Moshe cited by the Biur Halakhah) perceive different nuances in the rulings: the Shulchan Arukh requires the extreme measures only for one candle, as that is the core mitzvah; the Rambam requires them even for the additional candles, apparently feeling the hiddur to share a status with the base mitzvah.
It would seem, though, that a slightly different perspective on the Rambam and Tosafot can be offered. In truth, the concern for visibility would appear to be shared by all rishonim; no one is advocating lighting in a manner that will not be observed. The issue that is debated is whether the pattern of the candles must be discernible.
It might thus be suggested that both the Rambam and Tosafot agree that the hiddur spoken about is a special one, distinct from the general concept of hiddur mitzvah. This is evident from the fact that the normal standard of hiddur is an enhancement of one third (Bava Kama 9b), while in this case the cost of mehadrin min ha-mehadrin is many times that of the base mitzvah. Thus, they would agree that the “hiddur” involved here is unique and particular to the Chanukah candles (and thus could justify the extreme measures discussed above). Their argument, though, may be on the nature of the hiddur: is it functionally comparable to the hiddur applied to other mitzvot, while differing in significance, or is it even in function fundamentally different?
From this perspective, it would seem that the Rambam is the one following a more standard model, while Tosafot is pursuing one that is wholly unique. The Rambam is unconcerned with creating a discernable pattern. In his view, the hiddur is primarily creating a larger flame. This is of a kind with other mitzvot, where, for example, a larger etrog would be considered more of a hiddur. Tosafot, in insisting on a recognizable message within the hiddur, is expressing a completely different type of hiddur, one that is not merely “more…” but is different (following the Talmud’s lead; the very notion of both a “mehadrin” and a “mehadrin min ha-mehadrin” is already unusual).
The idea that Chanukah, of all mitzvot, should contain a unique element of hiddur, may stem from the fact that many theories place hiddur at the center of the miracle of the oil: some explain thusly the need for the miracle, despite the fact that “tumah hutrah b’tzibbur” (impurity is waived for communal needs); or the need for more oil when oil could have been conserved by using thinner wicks and foregoing hiddur (see Beit HaLevi al haTorah); or that the long journey was necessary in order to obtain specific oil that met the needs of hiddur (see Gevurot Yitzchak, Chanukah and Purim, 13). Further, as noted by the Hod Tzvi (6), for most mitzvot, the hiddur is an ancillary component to the process; in the case of Chanukah, the mitzvah is the publicizing (pirsumei nisa), and in that the greater the flame, the greater the publicity, the mehadrin of Chanukah directly enhances the fulfillment of the mitzvah itself.
The Ri’s focus of an ability to discern the pattern affects other issues as well. R. Shlomo Kluger (Chokhmat Shlomo 671 and Resp. HaElef Likha Shlomo, 380) rules that there is no harm in adding to the relevant number of candles (based on a ruling of the Rama, O.C. 263, regarding Shabbat Candles.) The Resp. Mishneh Sakhir (II, 199), among others, disagrees, referencing the significance of the specific number of candles involved (see also Resp. Shevet HaKehati, I, 208). Similarly, the Resp. L’Horot Natan (II, 51) explains the inapplication of “sfeka d’yoma” to Chanukah as due to the importance of accuracy in the count (based on D’var Avraham I, 34, regarding Sefirat HaOmer). Further, there is some discussion as to whether one who has not been lighting “mehadrin min ha-mehadrin” may begin doing so in the middle of Chanukah (again, a comparison to Sefirat HaOmer; see Shiltei Giborim, Shabbat II, 5; Resp. Riv’vot Ephraim I, 444). Further, poskim debate the question of one who does not have enough candles for the mehadrin min hamehadrin of that night: should they light just one candle (the base obligation) or as many as they have (see Beit HaLevi al HaTorah; Mishnah Berurah, 675:5; Avi Ezri; Resp. Kinyan Torah B’Halakhah, VI, 50; Resp. K’tav Sofer, O.C. 133.)
The assumption that the hiddur of Chanukah is unique and central to the mitzvah affects other questions as well, including: the position of the Shulchan Arukh (676:5) that the lighting should start from the newest candle (see Biur HaGra; Resp. Chatam Sofer, O.C. 187; Resp. B’er Sarim, III, 72:8; Mishnat Ya’avetz); reciting the berakhah of “ha-roeh” on the additional candles (see Resp. L’Horot Natan, II, 53 and Resp. Kinyan Torah B’Halakhah, III, 98); training children in lighting with the additional candles (see Mishnah Berurah, 675:14, and Birkat Shimon, glosses to Shulchan Arukh); the question of reciting the berakhah when only lighting additional candles (see Resp. R. Akiva Eiger, II, 13; Pri Chadash, and Elyah Rabbah, 672; Birkhat Mordechai, II, 40; Resp. Binyan Shlomo, II, 59); someone giving up his hiddur so another can light the base mitzvah (see Da’at Torah, 671:2; Resp. Torah L’Shmah, 162; Migd’not Eliyahu, 200:4); and weighing the value of the additional candles against that of performing mitzvot expeditiously (zrizin makdimin l’mitzvot) (see Resp. Minchat Asher [Goldenberger], III, 73; Resp. Nachalat Binyamin, 133).
It should also be noted that the assumption that the Rama rules according to the Rambam requires further analysis, as there is an important distinction: the Rambam advocates the head of the household lighting candles per person and per number of nights, while the Rama refers to each member of the household lighting on their own, per number of nights (see Chiddushei Maran Riz HaLevi; Galyei Masechta, 6; Resp. Binyan Shlomo, II, 58; Gevurot Yitzchak, 10; Sha’arei Torah [Yahav], 12; Moadim U’Zmanim, I, 132; Avi Ezri).