I think it would be nice to have a second calculator, different
from my TI-85 and any other calculator that does pragmatic
calculations and numerical analysis type stuff. This new
device would be useful in theory classes, for doing proofs and
geometric constructions based on fundamental axioms
and
theorems from modern algebra and geometry.

I believe you're looking for a "symbolic math package"; examples include Mathematica, Maple, and MathCAD. They come as desktop computer software, since (unlike adding machines) they benefit from nontrivial computing power and require a high-bandwidth display.

They typically support numeric approximations as well as exact symbolic manipulation. Theorem provers and verifiers are separate and generally more experimental tools.

You could certainly run such a thing on a subcompact laptop, tablet PC, or perhaps even a high-powered PDA.

Perhaps I should have mentioned that I have used math software. My idea isn't for software. It's for a "calculator", a handheld device for students to use. About 150$US. Maybe solar powered. =^)

If you've read Issac Asimov's book, Foundation, there is what we would now call a tablet device were Hari Seldon does a derivation, a psychohistorical calculation to show a young mathematician that the galactic empire was inevitably going to collapse under its own weight.

After reading Foundation, I've always wondered if someone would ever develop such a calculator. When I heard about Mathematica, I thought this would be it, but not so much. I love Mathematica, but the idea is different.

The concept is you enter the equation, just as with a calculator you enter the numbers, and you tell the calculator which algebra and calculus steps to take, just as with an ordinary calculator you press buttons for arithmatic operations.

So you do the thinking, the real brain work, but the calculator does the steps without making a mistakes.

Steps would be like: cross multiply, factor, solve for, whatever.