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I agree about thinking "in system", but your "as fast as possible" comment is interesting, because I think this actually highlights the kind of problem that mathematical thinking is good at fixing.

Why "as fast as possible"? Usually when programmers say this it's because they think "the faster, the better!". But obviously there is some speed beyond which there's no discernible improvement. At that point, pursuing the as-fast-as-possible mandate at the expense of other concerns is the wrong thing to do.

Therefore, there's an assumption built into this statement that the system will never be that fast, and therefore "as fast as possible" is a good target. Needless to say this isn't a safe assumption.

The thing is that we're built to be really bad at knowing when we're making assumptions. Thinking mathematically is to some extent a way of trying to overcome that limitation.

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