a) You don’t do this full time.
b) By “bottoms up” you just mean “with firm grasp on fundamentals”, not logic/set/category/type theory approach.
c) You are skilled with programming/software in general.
In a way, you’re ahead of math peers in that you don’t need to do a lot of problems by hand, and can develop intuition much faster through many software tools available. Even charting simple tables goes a long way.
Another thing you have going for yourself is - you can basically skip high school math and jump
right in for the good stuff.
I’d recommend getting great and cheap russian recap of mathematics up to 60s  and a modern coverage of the field in relatively light essay form .
Just skimming these will broaden your mathematical horizons to the point where you’re going to start recognizing more and more real-life math problems in your daily life which will, in return, incite you to dig further into aspects and resources of what is absolutely huge and beautiful landscape of mathematics.
I'm very glad to see more books being modeled after it, and I hope this trend will continue with things beyond math. My only hope is that those new books will match the quality of the original.
 - http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...
Its certainly too advanced for a 6 year old (or even a 16 year old, TBH) but just having it around is really great, I think. I remember when I was younger, I would look up stuff in more advanced books even if I couldn't understand them right away. The feeling I had was always: "Someday, I will be able to understand this..." which made me learn more physics and math.
"How to Solve it" is especially great if you do/will teach her in the future.
It's a good way to skim a lot of different mathematical topics for further exploration.
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