Regardless, I've experienced this effect a lot when writing design docs. Iteration and objective criticism on a tangible thing (a doc) is an extremely effective way to see the problem from all sides.
The correct statement from a logical point of view is: if A implies B, then not B implies not A.
In this case, even if writing down your ideas makes them more precise, there might be other methods that make your ideas more precise. Again this is just the logical point of view, out of context.
The statement in TFA is not that though. Instead, it is "if A implies B, then not A implies not C."
A: writing about thoughts
B: thoughts become more complete
C: thoughts are most complete
f: a function indicating how complete the thoughts are.
A: writing about thoughts.
EDIT: shortened sentence
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.
But the quote above says "If..." and then makes a statement that isn't true and then having a conclusion based on that false premise. I can tell you it isn't true because I can recall countless times in the last few months alone where writing down my ideas has resulted in a muddier thought; lost ideas while writing them down; confusing me and missing some parts; it does not "always make them more precise and more complete". So the rest of the statement is just silly.
Sure, sometimes writing down ideas helps clear things up. Most times even. But always?! Definitely not.
* Although it's not really right to use this kind of language here (premise, conclusion, deduction). It's a casual statement, so I suppose people can somewhat reasonably argue about it, but the assertion is tautological ('if something is incomplete, it isn't fully formed').
Similarly, if walking North always brings you closer to the North Pole, then you can never reach the North Pole without walking North, or at all. But look out for oceans.