My claim is the exact opposite. I claim you don't need to understand the mechanics ( just blindly abide by the rules of the group or abelian group or finite simple group or whatever), which is why the approach is better. If you show a monkey red means stop and green means go and reinforce these rules by rewarding with a banana, eventually the monkey will stop when he sees the red. Not because he understands the mechanics of traffic management. Simply because he is abiding by the rules. Similarly, large portions of math can be approached by either the definitional route ( ie. rules ie. define propositions & theorems that logically follow if those props held ) or via trying to understand actual mechanics by mapping everything to real world phenomena ( x = distance, dx/dt = velocity, d/dt(dx/dt) = acceleration etc. ) which are problematic because the mapping breaks down due to the nature of physical reality ( like friction etc. )
How would one explain say Hilbert's 7th problem via the actual mechanics ?
If a is algebraic and b is irrational show a^b is transcendental.
What does that even mean when you map them to the real world ? Instead, the solution is to build upon theorems that logically follow from the axioms you start out with. Problem: http://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem Solution: http://terrytao.wordpress.com/2011/08/21/hilberts-seventh-pr...