It was precisely with Popper's scheme in mind (and, to a lesser extent, Hume's simpler "How do you know?") that I wrote my previous comment, although, as you said, very few scientists work with said scheme in mind, given its obvious impracticality and lack of precisely defined methods.
I didn't even know Quine's holism existed, thank you for providing me with the link.

As for the rather liberal use of the words 'empiric' and 'scientific' in my previous post, it was because I wanted to stress the importance of the 'applied' part of this article's title.
When it comes to applicability, the old engineering adage says it all:
"In engineering, you're not done when there's nothing more to add, but when there's nothing more to take away".
When developing a philosophy aimed at giving some sort of justification for the applicability of abstract mathematical principles to concrete, physical reality, shouldn't the prospective scientist/philosopher/logician guide his reasoning by that same adage?
I can't think of a more gross violation of the most basic principles that guide applied mathematics than literally inventing a whole world to justify itself.

I apologize if this all sounds rather vague, but I lack the preparation or background to give a more rigorous and detailed defense of my case, assuming I have one at all.

P.S. I'm currently trying to study this subject on my own using Mendelson's "Introduction to mathematical theory", Pierce's "Introduction to information theory" and Devlin's "Information and logic".
Are there any other good books I could use to study this, preferably mathematical in format?