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How about solving differential equations numerically. Such programs are used all over the place in mathematical modeling, with applications ranging from economics to electronics.

And you can't just "write a computer program that tries different parameters". You have to prove that your numerical method solves a certain class of equations first, otherwise your rocket will fly sideways, if at all.

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The intelligent way to do that nowadays is to use automatic differentiation followed by optimization "plugins". And that has little to with proving theorems unless you consider all computer programs as proofs and all computer programmers as mathematicians.

I'm not sure what automatic differentiation has to do with solving diff. equations but...

I assume that all those automatic methods you mention are passed down from above in some sort of holy scriptures that we're supposed to blindly believe and use? Or maybe some "pencil pushing" mathematician came up with them first and _proved_ that they actually work?

If you're solving DE's by hand you were probably just in a class taught by members of the Mathematician-Teaching Complex (allusion to Military-Industrial Complex).

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