Who knows thirty-two?
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32 degrees is the (approximate) latitude of Jerusalem. (Rambam, Hil. Kiddush Hachodesh 11:17)
This information is necessary in order to calculate the moon's visibility and altitude on a given night. The Sanhedrin needed to do this in order to cross-check witnesses' reports of having seen the moon, which would then be used to set the dates of Rosh Chodesh and of the holidays.
32 are the dotted letters in a Sefer Torah:
The significance of each of these is explained in Sifri, Behaaloscha sec. 69. Rashi cites most of these explanations in his commentary on the respective verses.
Every book of Tanach contains some word with gematria 32 -- except for Song of Songs. Go figure, such a love story, and yet no "heart."
(While Jewish tradition counts Ezra/Nechemiah as one book, each of its halves also contains a 32-word. Each sub-book of Twelve Prophets contains a 32-word, except for Obadiah.)
Yes, it's the gematria of "lev"="heart", and the Talmud says that the Avnet atoned for sinful thoughts of the heart. Similarly, I've also heard (don't recall in whose name) that our daily thanks to God for "girding Israel with strength" refers to self-control.
The Malbim explains the role of the Avnet as taking our most animalistic emotions -- lust (מכנסיים) and rage (כותונת) -- and keeping them subdued (sublimated?) deep down in the soul, so they don't bubble up. (I'm not doing justice to Malbim's prose here.)
32 is the greatest number of civil dates that can be part of a single Jewish month.
If a Summer Rosh Chodesh for a 30-day month happens to fall, somewhere in the extreme North or South, on a day when nightfall is before midnight, and if by the following Rosh Chodesh, nightfall has moved to after midnight, then your Jewish month has contact with 32 civil days! Note that this works whether you use sunset or your favorite value for Tzeit (though you don't have to go as far north for the latter).
32 cubits is the length and width of the base of the Altar in the Second and (soon to be built) Third Batei Mikdash. (Rambam, Hil. Beis Habechirah 2:5)
32 cubits is also the length (measured along the floor, not the slope) of the Altar's ramp. (ibid. 2:13)
32 strings in the tzitzis. (Although actually 16, there are 32 hanging from the tassles.)
Although Maseches Megillah only has 31 dapim, it has 32 mishnayos.
32 is the greatest number of extra calendar days between a regular year and a leap year that follows it.
The shortest year (chaser) is 353 days. The longest year (me'bar maleh) is 385 days. Thus, a 32 day difference.
The 32 Paths of Wisdom concept is derived from the 32 times that the name "Elohim" is mentioned in Genesis, Chapter One.