Great Regulars: Here, then, is part of what it takes

to be a great Torah scholar: the ability to perform feats of memory and logic, to reason strictly from premise to conclusion. Two things strike me about this sacralization of the intellect. The first is that, for the rabbis, this kind of thinking is not just impressive; it is itself the supreme expression of piety, since Torah study is the highest Jewish obligation. We please God most not by feeling or even praying, but by thinking. The second is how very unusual this value system is, historically speaking.

from Adam Kirsch: Tablet: Rabbinic Mind Games

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The obligation to tear your clothes applies only in the case of the death of a relative; to tear your clothes for that reason would indeed violate Shabbat, because it would be carrying out a positive obligation. What the Mishnah has in mind is tearing your clothes over a non-relative, which is not obligatory and is therefore strictly destructive. It is a seemingly inverted logic: On Shabbat, you are not allowed to do what you are supposed to do, and you are allowed to do what you are not supposed to do.

But the Gemara is not done turning over this question.

from Adam Kirsch: Tablet: Things Broken and Repaired

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