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.
Obligatory 'zoomout' recommendation: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V..., which I learned about from HN (http://hackernewsbooks.com/book/mathematics-its-content-meth...). Wish I had read/pondered this before grad math classes.
Just from a search, there are some great results:
1. Mathematics for Computer Science - HN Submissions: https://news.ycombinator.com/item?id=9311752, https://news.ycombinator.com/item?id=3694448
2. How to Read Mathematics - HN Submissions: https://news.ycombinator.com/item?id=4030812, https://news.ycombinator.com/item?id=1576969
Here are some other excellent mathematics books:
1. Mathematics: Its Content, Methods and Meaning
Containing the thoughts and direction of numerous mathematicians including Kolmogorov, this is a great survey of the field of mathematics. It touches upon Analysis, Analytic Geometry, Probability, Linear Algebra, Topology, and more. [1.]
2. Concrete Mathematics: A Foundation for Computer Science
Containing the thoughts and direction of mathematician and computer scientists such as Donald Knuth, this is a great reference for computer science related mathematical concepts focusing on continuous and discrete concepts. [2.]
One of the masterpieces that has gone the opposite direction is:
Mathematics: Its Content, Methods and Meaning (three volumes bound as one) by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent'ev (18 authors total)
This book is really good companion for autodidacts. It's basically overview of mathematics.
Covers something like three years of an undergraduate degree in mathematics. Lots of words - but that text is used to develop an understanding of the concepts and images. Considered a masterpiece. An enjoyable read.
For a more "math for general culture" I'd recommend this one: http://www.amazon.ca/Mathematics-1001-Absolutely-Everything-... which covers a lot of fundamental topics in an intuitive manner.
I have both books on the shelf, but not finished reading through all of them so I can't give my full endorsement, but from what I've seen so far, they're good stuff.
Math books rarely move from the Soviet Union to west, but this did and for really good reason. Just look at the list of writers included. So far I have not seen any math books that come even close to this. Reading this book together with the The Princeton Companion to Mathematics was real treat.
The Russian school put out some pretty massive volumes, like
but I found those to be a little too caught up in the proposition/proof cycle to be useful as a guide to the uninitiated.
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