[throwaway55533]

+1, pg is using a pretty typical argument you see in analysis/topology.

If you want to get to real analysis/topology the typical sequence is

1. Logic and Set theory (recommendation: How to Prove It, Velleman)

2. Linear Algebra (don't have a good recommendation)

3a. Real analysis (recommendation: PMA, Rudin)

3b. Topology (recommendation: Topology, Munkres)

I'm not sure I'd recommend learning math. It's an extremely expensive skill -- though pretty valuable in the software industry. People who go learn math are generally just drawn to it; you can't stop them even if you wanted to.

But be aware, (1) you'll have no one to talk about math with. And (2) you'll be joining a club of all the outcasts in society, including the Unabomber.