It would be wonderful to have the privilege and dedication to learn all of this.
Do you think it would be possible to construct a high level treatment that would impart a rough idea of to the layman? One that omitted all the business about finding solutions and stuck to merely tracing the structures?
I have seen that lower-level concepts like the fundamental theorem of calculus and the Fourier transform can be easily explained in a matter of minutes with the help of diagrams. It is my hunch, but I lack proof, that the same could be done for all of mathematics. Of course I have been told a few times that it would be impossible.
I think that the book "Q.E.D" by Richard Feynman does about as good a job as is possible at explaining quantum electrodynamics to the layman (I first read it when I was 17, and found it very understandable, with the possible exception of the final chapter).
As to whether you could do this for all mathematics - I'm not sure. It's quite easy to 'visualise' the FTC or the fourier transform, and they have immediate applications to things that non-mathematicians care about. I'm not quite sure how one would go about explaining e.g. representation theory of lie algebras, since all of the motivating examples would only be of interest to mathematicians.
It's a bit like the wall I hit when I tried to study category theory. It's perfectly possible for someone with very little math background to learn the basics, but until you've seen a lot of mathematics you won't understand what the point of it all is.
About representation theory of Lie algebras: physicists actually care about that quite a bit, as the the theory of spin is intimately tied up with the subject. Your point stands, though. I don't know of any major applications outside quantum mechanics, and other parts of mathematics have even fewer applications. (I have always found integration theory kind of tedious for exactly that reason---some results turn out to be handy, but it's more like you're laying the groundwork for background material for stuff that'll be useful to physicists.) Also, the way in which physicists care about the math is very different from the way mathematicians do: we want the moral reasoning---we want to have some intuition for why the result is true---but (to grossly stereotype) we don't really care about the detailed proof.
(I must add that I heartily second your recommendation of /Q.E.D./)
I dont know your level, for someone like me who studied mathematics years ago but not applied it for years, I found "In pursuit of unknown, 17 equations that changed world" by Ian Stewart very helpful. While it is not a detailed math book, it gave me enough pointers to refresh and prepared me to dive deeper through other material.
Do you think it would be possible to construct a high level treatment that would impart a rough idea of to the layman? One that omitted all the business about finding solutions and stuck to merely tracing the structures?
I have seen that lower-level concepts like the fundamental theorem of calculus and the Fourier transform can be easily explained in a matter of minutes with the help of diagrams. It is my hunch, but I lack proof, that the same could be done for all of mathematics. Of course I have been told a few times that it would be impossible.