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! :-)
I regret I haven't read enough of your posts to realize that you are an expert on Artificial Intelligence. The field is fascinating to me, though I am somewhat skeptical for the same reasons Penrose mentioned in Emperor's New Mind. LOL!
I do believe a Mathematical Theory of Everything is possible - along the lines outlined by Wolfram's A New Kind of Science. Some Freepers laugh at me for being hopeful that a primordial algorithm will be discovered.
I truly believe that is as far as we can go in understanding origins in the natural realm and that it is possible --- if only theoretical physicists can conquer the "there be dragons there" qualms that lead to head-scratching theories such as multiple universes from multiple quantum fluctuations.
Current work on mathematical physics is fascinating to me. I've been watching this consortium for years now: Space-Time-Matter Consortium.
Thank you oh so very much for letting us know that some leaders in science are lurking here! What a thrill! I do hope they will offer some links now and again - we Freepers are sponges for science!
Again, thank you oh so very much for the great discussion!