Without courtesy there is no karate.
When students and teachers bow to each other it is to show courtesy and respect. A bow from a student to a Sensei shows courtesy thanking him for the time he takes to train the student; a bow from a Sensei to a student shows courtesy thanking her for the lesson he will learn in the teaching; a bow between students shows courtesy thanking what they learn from one another in class.
Kata, a structured way to teach combat technique based on specific principals of self-defense, teaches respect and courtesy first, karate as a last resort.
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