There are three types of benefit derived from an apple: taste, sight, and smell. This is symbolic of our appeal for sustenance in the three aspects of children, health, and livelihood for the entire year (Ben Ish Chai, first year, Parashas Nitzavim).
It is written in Zohar (Parashas Shmini 4a) that after one drinks wine, one eats an apple to prevent harm from the wine. Wine represents gevurah (severity), and the apple calms the severity. Therefore on Rosh Hashana we eat apples to pacify the harsh judgement. (Ben Ish Chai, ibid.)
The Zohar (above) states that the source of all fruit is the Divine aspect of malchus ("royalty," considered feminine), except for the apple, which is sourced in the aspect of tiferes ("splendor," considered masculine). The Arizal says that the summer months represent malchus and the winter months are tiferes. Therefore at the start of the “masculine” months we eat an apple which comes from a “masculine” source. (Ben Ish Chai, ibid.)
According to Rabbeynu Yona (brought inquoted by Rosh on Brachos Perek 6 Siman 35:35) that, the nature of honey is to change everything that comes into it –- even something impure –- into honey. (Therefore, one can consume honey even though the legs of the bees may be mixed in.) The holiness of Rosh Hashana should convert all bad to good. (Shem Mishmuel Mo'adimShem Mishmuel, Mo'adim, Rosh Hashana 5674.) (Some add that the sin of Adam Harishon and the Tree of Knowledge was with an apple, and we therefore dip an apple into the honey to rectify the original sin.2)
The Gematriagematria (numerical value) of "תפוח" (apple) is equivalent to that of "פרו ורבו" ("Be fruitful and multiply."), and "דבש" (honey) is equal to "אשה" (woman), symbolizing that Rosh Hashana is an auspicious time for barren women to be remembered (Imrei Noam Chelek BeisImrei Noam volume 2, Likutim Besof Hasefer Oisend note 9).
Gershon Gold added: Taamei HaMinhagimTaamei HaMinhagim 706 says it is done for Kabalistic reasons. In the notes, he mentions in the name of the Imrei NoamImrei Noam that the Gematriagematria of the word "Tapuach""tapuach" (the Hebrew word for apple) is the same as the Gematriagematria of "S'e Akeida""S'e Akeida" -- so we eat the apple to recall the Akeida (Binding of Isaac).
- Original question: mi.yodeya.com/q/20334
- Midrash Rabba Breishis 15, 7 brings an opinion that the fruit in the garden was a תפוח"תפוח." However, Tosfos on Shabbos 88a writes that תפוח"תפוח" in Tanach means an אתרוג (citron).
- http://www.etymonline.com/index.php?term=apple