[wanderingmind]

Its like how people use tools, like gears, knowing when to use what type of gears without needing to understand the microscopic material structure that provides its strength, or the mathematical definition of it's kinematics.

[pnin]

I think if you use gears, it would be well to know some Newtonian mechanics. The material structure might not be relevant, but in mathematics you also don't need to go down to the foundational level for every problem. You don't need to go deep into set theory to understand the proofs in calculus (though it is true that Cantor's investigations into set theory started from the question of convergence of certain infinite series, which is a calculus problem).