UPDATED****** (I changed some wording around in order to make the second part more easily understandable)
My brother-in-law, Rabbi Dovid Gottlieb (not the PhD in mathematics), was the Rabbi of congregation Shomrei Emunah in Baltimore Maryland. He wrote a book with the title of Ateres Yaakov that deals with various topics in Jewish Law. I decided to translate and comment on the section that has to do with Chanukah. It is separated into 14 sections.
Section 1: The relationship between the laws of Mehadrin (beautifying the Commandment) and Mehardin min Hamihadrin (even more beautifying of the Commandment).
It states in the Gemara in Shabbos (21b) "The Rabbis teach us that the commandment of Chanukah is that every house requires a candle. A beautification of the commandment is that every person in a house requires a candle. An even further beautification is either; as Beis Shammai says, the first day eight candles are lit and every subsequent day one candle is removed; or it is, as Beis Hillel says, the first day one candle is lit and every day an extra candle is added." We follow Beis Hillel (Shulchan Orech, Orech Chaim, 671:2).
There is a relationship between the beautification of the first level of beautification and the beautification of the second level of beautification. Tosfos says (Ibid. heading Mehadrin Min hamihadrin) the law of adding an extra candle every day(Beis Hillel's opinion) is in place of (instead of) a candle for each and every person in the house. To fulfill the commandment of Mihadrin min Hamihadrin (An even further beautification of the commandment) only the man of the house has to light, but he has to light in a manner that he adds an extra candle every day (until there are eight candles).
Tosfos explains that it must be so (that the Mihadrin min Hamihadrin commandment is performed with the parameters of only one person lights and adds an extra candle every day) because if it wasn't so then there would be no way to recognize what is the current day. For, if every person requires a candle to be lit for them then the observer who sees the candles will think that the number of candles represents the number of men in the house and the observer will not recognize that the candles represent the number of days [of Chanukah].
However, Rambam (Chanukah 4:1) argues on Tosfos and says that the law of adding an extra candle each day is included in the laws of every person requiring a candle be lit for them.(Everyone has an extra candle added each day, not just one person for the whole house) See the Shulchan Orech (Orech Chaim 671:2) where he holds like Tosfos and see the Rama (Ibid) where he argues on the Shulchan Orech and holds like Rambam. Also take a look at the Taz (Ibid) where he points out that this is an exception where the Shulchan Orech follows Tosfos and the Rama follows Rambam.
The position of the Rambam is a little bit of a question. It appears to be that Tosfos is correct that if every person has candles lit for them and the candles represent the number of nights of Chanukah there is no recognition [by an observer] that a person is lighting an extra candle for each day. The beautification of the Mitzva (commandment) requires further investigation because really there is a Tannaic dispute of whether the beautification of the mitzva needs to be recognized. However, it teaches in Tractate Yoma (70a) that the beautification of the mitzva (commandment) is "In order for the masses to see its appearance. (See Rashi there)"
In the GRA's commentary (671 heading viyeish omrim) he explains that the argument (Between Tosfos and Rambam) is dependent on two possible reasons that can be applied to the opinion of Beis Hillel that is found in the Gemara. The first line of reasoning is that [the reason we light an extra candle each night is] to count the amount of days of Chanukah that have gone by. The second line of reasoning is that one always goes upward in holiness and not downward. The GRA explains that the question of [whether or not the candles count] needs to be recognized is dependent on this argument. For, it is only if we say that the main reason [for lighting extra candles] is to count the days of Chanukah that we would need the amount of candles to be recognizable. However, if the main reason is because we always go upward in holiness and not downward, then no recognition of the amount of candles is needed.
It appears from the continuation of the Gemara, where it talks about a happening of two elders, that it explains the main reason (we increase the amount of candles every day) is because we always go upward in holiness and not downward. Therefore, it seems like we can answer according to Rambam's opinion because according to his opinion it doesn't matter if the number of candles are recognized as the amount of days for there is no problem with [them not being recognized]. If this is the case we can imply from the words of the GRA that, essentially, the Rambam agrees that there are reasons one should be careful that the beautification of the Mitzva (commandment) be recognized, but there are also reasons why the beautification being recognized is unnecessary.
See Part 2 here.
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