That depends on what you define as a "deeper understanding" doesn't it?
The question at hand is not that, though, it's far simpler. It is simply "Does a person who understands the math in addition to the other material have a deeper understanding than one who has not been at all exposed to the math?"
With that, I do not see how it is possible to say that non-exposure is the same.
By and large I would agree, however I think that there's range of "satisfactory" depth of understanding for our discussion. Non-Exposure to the math is also essentially meaningless if someone isn't conversant in math. Pictoral, and written descriptions might provide a deeper understanding to that
specific person rather than simply glossing through the heiroglphys of the algebra. I say that as someone who's fairly decent at math. I think its easy to lose sight of the forest for the trees in this regards. For example, a lot of people could tell you the core principles behind the theory of relativity, they could tell you about warping space, Lorenz contraction, etc. but give them a bunch of equations, and ask them to point out the significance of gamma in these equations, or how they relate to the overall structure of the universe, and most will be out of their league.
The problem is that to
truly understand the mathematics, you have to have more than simply an algebraic understanding of why something does something. You have to know a little calculus to know "why" there's a 1/2 or "why" there's a 2 in front of it, then to truly understand lift you have to know how to work in the Complex Plane and do the Calculus of Complex Variables so that you can calculate how the Jukowski airfoil sections apply forces in different directions. Then you have to be able to apply that to surfaces which aren't necessarily perfect airfoils, then you might have to use Navier-Stokes equations to try to get a grasp on the turbulence in the flow. See, at some point the information
about the physical aspect of flying gets lost in the noise of all the other BS that you're having to sort through to get it. Its not a truly deep understanding if you know "this is the equation that governs lift, this is how it works," if you don't know where it comes from. By the time you get to a truly complete understanding of the mathematics of it, you're so worried about the minutiae you stop thinking the big picture. Depending on the student, someone who's less versed in the mathematics of the subject might actually have a worse understanding of how flying works if they're mired in formulaic jargon. Knowing that lift increases with the square of velocity doesn't tell me a damn thing about "why" it does so. So really, all I've really done is rote memorized some equations at the low end of the mathematics, then
maybe, maybe correllated them to the act of flying. I don't really see that as "understanding."
