You wrote:
IMHO, first order logic is quite handy for computer science but fails in natural science and thus would not be of a necessity portable to another universe or domain – which would be subject to other physical laws.
I'd call this a bit of an over-reach. Predicate logic doesn't really "fail" in natural science. It just isn't called into the game very often. Occasionally, one sees a truth table imployed where we've made a system with so many conditionals it's hard to think about without turning our brains to putty, but for the most part, most daily reasoning, including most technical and scientific reasoning, takes place without the need of formal logic, because what most people think about most of the time isn't complex enough to require formal tools to avoid errors of logical conflict. We still obey the rules of logic (which apply pretty well to large, gross objects in our local environment) as laid down by Aristotle and Boole, we just don't deign to notice. In our own universe, we have examples of useful systems of logical thought (notibly in the sub-nuclear realm of quantum mechanics) which obey a different set of fundamental rules.
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You wrote:
Unfortunately, this philosophy was proven unfit by Gödel's incompleteness theorem
I had to think about this for a bit--it seems to me that the formalist school hasn't suffered any worse than the rest of the big league logical sports teams. What Godel demonstrated certainly takes the wind out of Grand Project of formalizing all of mathematics. But this limitation is about equally sobering, in my opinion, for any players in the game. If I were betting on this event, I'd put my money against Platonists in this regard--in that we have now I would suppose, less of a clear vision as to what sort of ghostly reality math and/or logic represents. Just a hunch, of course.
At any rate "unfit" seems a bit strong. Hilbert's agenda of producting logical systems divorced from any domain of discourse (clearly a formalist agenda) proved, in the end to be highly fruitful, and is far from running out of steam as we speak. Establishing isomorphisms between disparate domains of discourse under any given set of logical rules has been a discipline that started as way to escape the type conflict dilemma, and the undecidability dilemma (which it failed to do) but in the end, has turned out to be quite helpful in sub-nuke and topology studies, amongst others.
Predicate logic doesn't really "fail" in natural science. It just isn't called into the game very often.
I probably should have used better phrasing. I know there are narrowly focused models for natural and human systems.
The statement that formalism was proven unfit by Gödel's incompleteness theorem was not mine, it comes from the linked article. However, since I'm of the Platonist school, I do agree with the observation.
IMHO, Platonism should be the least effected by Gödel because, under Platonism, the math already exists, waiting discovery. To me, that inoculates mathematical discovery from nay-sayings of all stripes.
I guess mathematical "ghosts" are ok with me (LOL!)