Just a side note that pertains to the original post, though not necessarily where the discussion has gone: in his autobiography, "Karate-Do, My Way of Life," Funakoshi describes his method of dealing with illness, which was to "perform kata until sweat runs from the body." (Not an exact quote; I don't have my copy handy.

When I feel like I'm getting a cold, or have an allergy attack, I've had some success with sweating it out this way, though I can't say that kata would work any better than another sustained physical activity. When I've got a stomach thing, I don't even bother to try.

Someone else mentioned improvement of posture and balance; the curriculum I study contains mostly kata that are pretty symmetrical, which I think is beneficial for balance and for equally developing both sides of the body. I have noticed that many people spar in only one direction (I don't; I'm ambidextrous, meaning that I'm equally incompetent in both directions), so I can see where being forced to use both sides in a symmetrical kata could be beneficial as well.
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