Serving your country while in academia, or before going into academia
I think you should really consider what you are willing to do in the military.  Are you going to enlist or apply to OCS?  What is your desired MOS?  If you want to join the military while continuing in academics, this is very field specific.  For example, the Navy/Army/Air Force employs a number of medical physicists and will pay for their masters degrees.  You need to specify exactly what your goal would be while in the military.

"I am not, of course interested in industry, at all. I would sooner do math on the streets than go into any other job where I cannot continue what I am doing 24/7."

If you join the military, you will almost absolutely break your 'string' of math achievements.

"the wars that are fought are antithetical to my conception of American values"

If you are enlisting and you branch infantry, what will you do?  If you become an armor officer, what will you do?

All academics aspire to 'serve' humanity in their own way.  Research pushes mankind forward, not nations.