"That's what rationality is: having explicit and conscious standards of validity, and applying them in a systematic way. It doesn't matter if we possess an inner conviction that something is true - if we can't demonstrate that it can be generated from basic principles according to well-defined rules, it's not valid."

What you're talking about here is "system 1" vs "system 2" (http://www.overcomingbias.com/2006/11/why_truth_and.html)

You need to make a distinction between "we can't generate this from math because we're stupid" and "we can't generate this from math because the math gives a different answer". If your car isn't running well, it's good to look under the hood, but you have to learn the difference between "I don't know how this thing works" and "It's broken because the ignition coil is unhooked"

While "system 2" may give better results in general than "system 1", it is an oversimplification to decide we only need system two. Ignoring system one seems to be one of, if not the most common way for a wannabe rationalist to shoot himself in the foot.

If you ask someone that isn't a wannabe rationalist why they aren't, a very common response is "you need some emotions" or "being rational doesn't always give the right answer". This seems to come partly from seeing this error, but oversimplifying and erring on the other side.