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.
For lurkers: there are various schools of mathematics and tortoise and I are at odds because I fall in the Platonist school.
Formalism is the view that mathematical statements are not about anything, but are rather to be regarded as meaningless marks. The formalists are interested in the rules that govern how these marks are manipulated. Mathematics, in other words is the manipulation of symbols.
Unfortunately, this philosophy was proven unfit by Gödel's incompleteness theorem [see below] ...
A break off of Formalists school is the Logicists school, championed by Bertrand Russell and Gottlob Frege, who sought to show that are knowledge of mathematical truth was as certain as our knowledge of logical truth. They attempted to define mathematics in the language of logic. Their efforts resulted in some important ideas, such as the relationship between number theory and set theory, but ultimately this enterprise was found faulty as well due to paradoxes such as Russell's Paradox, an important principle of set theory which could not be based on logic.
Inventionism
Inventionism, also sometimes called Constructivism, holds that true mathematical statements are true because we say they are. Mathematicians do not discover mathematics, as the Platonists claim, they invent new mathematics.
Intuitionism
The easiest way to define intuitionism is that it is the corollary of logicism. The Logicists want to define mathematics in the language of logic. The intuitionists want to define logic in the language of mathematics.
Platonism
The view as pointed out earlier is this: Mathematics exists. It transcends the human creative process, and is out there to be discovered. Pi as the ratio of the circumference of a circle to its diameter is just as true and real here on Earth as it is on the other side of the galaxy.
The above definitions are from What is Mathematics?
Incompleteness Theorum (Gödel)
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.
...
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.
I'm not so sure that it is a difference in schools of thought as it is a difference in mathematical backgrounds. I tend to look at everything in mathematics through the eye of Kolmogorov information theory and related fields of computational theory. When people posit things, I immediately frame everything in the context of those fields (which fortunately have very broad application and fairly penetrating theoretical value).
I'm pretty pragmatic about mathematics, probably because my real background is engineering and science, though I'm far better known for my applied mathematics work. My "school" lives somewhere between Formalism and Platonism. Incidentally, I don't really see how Godel's IT is a serious problem for Formalism, at least no more of a problem than it is for anyone else. There are many important theorems in other areas of mathematics that are analogous to GIT (including some extremely useful variants with respect to computation theory found in information theory). The work of Chaitin, Fisher, and others really puts a nastier limitation on our knowledge than Godel does in my opinion. Godel merely asserted that there was a limit, but others have shown exactly what the nature of those limits are and to the extent that we are regularly bumping up against those limits. Bertrand Russell's "Principia Mathematica" has been known to be a fool's errand for some time, at least in its original intent, and I don't think many people are working on a mathematical Theory Of Everything.
In the event you are trying to reduce cognitive experience to formal constructs – since we are blessed on this forum to have an expert in Artificial Intelligence - I am pinging him for his comments.
Heh. Ironically, I am quite probably the most qualified expert on AI theory on this board, though I don't spend too much time on it here. None of the rest of the guys that are recognized experts in the field are Freepers that I am aware of, and I am at least acquainted with most of them. And if any of them read my posts on AI, they'd be able to name me pretty quickly from familiarity with my theoretical work. :-) Interestingly enough, I know for a fact that there are a number of famous scientists and physicists who have been Freepers for a long time. A lot of really well-known and interesting individuals from the academic community hang out here incognito, including individuals we even occasionally talk about in threads -- heaven forbid it gets out that they are regulars on FreeRepublic!
Back to the topic, you can reduce "intelligence" in all meaningful forms to the same formal constructs (and some related proofs have been published in the last couple years regarding this), but not in the sense that most people imagine when they make the assertion that "you can't reduce cognitive experience to formal constructs". A lot of the really cool work is recent, and to a great extent, unpublished. The formal constructs that are emerging are extremely elegant, but not something you can explain to people in an elevator pitch. Explaining it to people who are very competent theoretically still takes several hours for me; people take considerable time to wrap their heads around the math despite its relative simplicity. They just aren't used to thinking about some things in the directions it takes you. Its good stuff, though, and just starting to produce really interesting results in practical application.
Thank you for the discussion, tortoise!
No, thank you! :-)
Seems pretty true to me, reason is limited and those who claim one can arrive at the ultimate truth through it are wrong.