They are just strings. How are strings supposed to change what I do?
The Torah gives the reason for tzitzit in one line: "You will see it, and remember all the mitzvot of Hashem, and do them, and you will not go exploring after your heart and after your eyes." That is a big claim for a few strings.
The Gemara reads the verse as a chain. Seeing leads to remembering. Remembering leads to doing. Rashi, the classic Torah commentator, explains what tzitzit is up against: the heart and the eyes are the body's spies. The eye sees, the heart wants, and the body does it. The Torah's answer is to give the eye something else to see.
Sefer HaChinukh (a book from Spain in the 1200s that explains each mitzvah) compares it to wearing your master's seal fixed to the clothes you always have on. Your eyes and your mind keep landing on it all day. Nothing in the world, he says, works better as a reminder.
Does it work? The Gemara tells one story, and it is not a polite one. A man who was careful about tzitzit heard of a prostitute overseas who charged four hundred gold coins. He sent the money and made the trip. At the last moment his four tzitzit slapped him in the face. He slid to the floor and sat there. She demanded to know what was wrong with her. Nothing, he said. But there is a mitzvah called tzitzit, and just now the four of them looked to him like four witnesses.
The story does not say tzitzit stopped him from wanting to go. He paid, he traveled, he walked in. It says that when they hit him in the face, he remembered. The Gemara ends: she gave away most of what she owned, came to Rabbi Chiya's study hall, converted, and married the man from the story.
The Shulchan Aruch (the main code of Jewish law) says that when you put tzitzit on, you should have in mind that God commanded this so that we remember all His mitzvot and do them. A reminder only works if you know what it is reminding you of.
Bamidbar 15:39,Menachot 43b,Rashi on Bamidbar 15:39,Sefer HaChinukh 386,Menachot 44a,Shulchan Aruch, Orach Chaim 8:8Does the 613 math actually work?
The second paragraph mentions "the 613 mitzvot that depend on it." The best-known explanation is a calculation. Rashi, the classic Torah commentator, brings it, and it also appears in the Midrash: the letters of the word tzitzit add up to 600. Add the 8 strings and 5 knots on a corner and you get 613, the number of mitzvot in the Torah.
The Ramban (Nachmanides, a leading Torah commentator in Spain in the 1200s) checked the numbers and wrote: "I did not understand this." In the Torah the word is spelled without its second letter yud, so it adds up to only 590. And by his reading of the Gemara, the Torah itself requires fewer strings and fewer knots than that.
There is an answer to the spelling problem. The Tur (a code of Jewish law from Spain in the 1300s) quotes it from a midrash: the word appears three times in the passage, and one of those times it has the letter lamed in front, which equals 30. Split the 30 three ways and each word gets its missing 10.
Siftei Chakhamim (a commentary on Rashi from the 1600s) takes up the strings and knots. He says outright that the Torah did not have this count in mind. The rule of exactly five knots, no more and no less, was set afterward, he writes, so that the total would come to 613 and seeing the strings and knots would do a better job of reminding you of all the mitzvot. So the math was built in on purpose, by a later rule and not by the verse.
The Ramban himself went another way. For him the reminder is in the blue thread. He points to the Gemara's line that this blue looks like the sea, the sea looks like the sky, and the sky looks like God's throne.
Either way, the idea does not depend on the arithmetic. The Gemara says tzitzit is "weighed against all the mitzvot" without any numbers. It learns that straight from the verse: see it, and remember all the mitzvot of Hashem.
Rashi on Bamidbar 15:39,Midrash Tanchuma, Korach 12,Ramban on Bamidbar 15:32,Tur, Orach Chaim 24,Siftei Chakhamim on Bamidbar 15:39,Menachot 43bWhy do only boys wear tzitzit?
It was not obvious, and the Sages argued about it. One teaching in the Gemara lists everyone who is obligated in tzitzit, and the list includes women. Rabbi Shimon disagrees and exempts them.
His reason comes from the words "you will see it." Clothing for the night, when you cannot see, is exempt. That makes tzitzit a mitzvah tied to a time of day, and the general rule is that women are exempt from positive mitzvot (commands to do something) that depend on time.
The other view was not only theory. The Gemara reports that Rav Yehuda attached tzitzit, blue thread included, to his wife's garment.
The law follows Rabbi Shimon. The Shulchan Aruch (the main code of Jewish law) rules that women are exempt. Can a woman choose to wear them anyway? The Rambam (Maimonides, the great codifier of the 1100s) writes that women who want to wrap themselves in tzitzit do so without a blessing.
The Rema (Rabbi Moshe Isserles of Krakow, whose notes on the Shulchan Aruch record Ashkenazi practice) agrees that in principle a woman may wear tzitzit, and even say the blessing. But he writes that it looks like yuhara, a show of extra piety, and so women should not. His reason: tzitzit is not a duty on the person. Even a man does not have to buy a four-cornered garment in order to become obligated.
That became the accepted practice. The Aruch HaShulchan (a code of Jewish law from around 1900) writes: this is the custom, and it should not be changed. If that leaves you with questions, the question itself is old. The Gemara records both views.
Menachot 43a,Shulchan Aruch and Rema, Orach Chaim 17:2,Rambam, Laws of Tzitzit 3:9,Aruch HaShulchan, Orach Chaim 17:3