Günther Greindl wrote: > I just reject the notion of some understanding "beyond the machine" > which is usually invoked, but I see that this is not what you mean.
Right. I (in my more reductionist moments) reject that, too. I'm not claiming that there is anything _other_ than formal systems. I am claiming that somehow ... I don't know how ... humans can put on and take off different formal systems as if they were hats or shirts. When we hop out of a formal system (into a different formal system) and begin counting the sentences in the first one, we are doing something that we can't (yet) mechanically automate. But we are _still_ engaging in math. (Witness "model theory".) > Yes, very interesting, this hopping is just what would be required! But > I think in computer science just this is being done! I mean, an > operating system is doing nothing but switching contexts, right? > > But only in a haphazard way (trying to optimize processing throughput) - > if you could devise a mechanism for formal-system jumping in a directed > way depending on environmental requirements, I think you would have > solved the problem of Artificial General Intelligence. It seems to be > difficult *grin* Right. We _do_ this type of hopping around all the time in math and computer science ... in fact, we do it all the time even psychologically. (I think it's what allows people to hold strongly to obviously contradictory or incommensurate convictions.) The trick is that we don't have a full-blown _measure_ (or better yet, metric) of the set of formal systems and a grammar for hopping in and out of them. Again, I have to qualify that with "as far as I know." There could easily be whole branches of math I've never run across and perhaps they do exactly what I'm saying hasn't been done.... for all I know. Now, I don't know if I'd go so far as to claim that obtaining such rigor for a description of "formal systems space" would solve the problem of artificial (general) intelligence. But, it would help us build robots smart enough to do the general work that can be done by most humans. Designing a robot that could adopt and abandon formal systems when its interactions with the environment start to go wonky beyond some threshold would definitely go a long way to achieving general intelligence. Of course, I don't believe that's possible in the _abstract_, though. As I've said before, I think the ability to don and doff formal systems goes hand-in-hand with our embeddedness ... the fact that our CNS is concretely embedded in the environment via continual, real-time, sensory motor feedback loops. Without such embedding, I don't think we can construct robots that don and doff formal systems at "will." >> No matter how high or low in the hierarchy you may go, you will still be >> using math, but you will not be locked within any given formal system. >> Hence, math is somehow more than (or outside of) formal systems. > > Here I disagree - you are reifying the word "math" - but the collection > of all formal systems is not a thing which is good to speak about I think. > > "the dao that is named is not the eternal dao" ;-)) [grin] "Good"?? Why wouldn't it be good to speak about that collection? Do you think it's ill-defined? I admit that it's vague and that I'm just yappin' without making any significant contribution... but I don't think it's necessarily "bad" to talk about such a mathematical object... even if it's only hypothetical. I take heart from things like universal algebras that we can make progress. But, then again, I'm an optimist. I'm hoping that you mentioned the loopy nature of taoism purposefully. There's a lot of deep analogy between assuming a fixed formal system (reductionist -- guilty of the fallacy of the perfect solution) versus allowing math to contain some ill-conceived "glue" between formal systems and Western versus Eastern thought. But, again, the path to enlightenment usually involves being swatted upside the head when you ask or attempt to answer a loopy question. That swat in the head that you get from, say, your Zen master (now only $39.95 from your local Wal-Mart! ;-) is a reminder that the answer lies in the sensory-motor coupling between inferential and causal entailment and NOT isolated on either side. > We have not formalized this hopping about, but it surely is > formalizable. It may not be formalizable. ... And don't call me Shirley. ;-) > We humans think we can hop about as we like because we > live in tightly constrained environments (our universe, more > specifically Earth, heavily industrialized/civilized/conformed to > primate living requirments) - I would guess our cognitive systems would > crash if sufficiently alien environments were provided (probably > dropping you on Pluto in a spacesuit would be enough to make most people > go insane). So I think this "unmechanistic" jumping is an "inside view" > cognitive illusion. I'm not so sure. I think there is plenty of inter-individual variability between humans to argue that _some_ people would go insane if they were dropped on Pluto but others would not. Whether the ones who go insane do so because they believe in a universal formal system or because they flip out in some runaway hopping process is another more refined question. Ultimately, given my previous opinion, if they go insane it won't be because humans _need_ some fixated formal system. It'll be because living in a space suit with no other living beings with which to interact would be an extreme abstraction from the necessary sensory-motor embedding we all use. > So, we have the hypothesis that in the end it all boils down to formal > systems (=mechanism; which is nicely defined via computability); > or that it somehow goes beyond the formal - but what should this be? I > wonder... The minimal conjecture would be that math consists of a set of formal systems plus methods for describing relations between formal systems. But such a thing can't be proven because, really, math is just a _language_ humans invented and use. > Of course, computability (I equate it with mechanism here) has one thing > speaking for it: the Church-Turing Thesis. This is a deep principle, > requiring much thought. I would like to end on this philosophical note. > Thanks for your remarks Glen, very stimulating! Yep! That's a good place to end... in spite of all my rambling above. Ultimately, when big weighty principles like that enter the discussion, it's usually an indicator that we've traveled very far down the rabbit hole and any further descent will be useless (except for mumbo-jumbo mind expansion ;-). Regardless of whether or not the universe is ultimately mechanical or not doesn't remove the (apparent) fact that lossy compression (as in a human's understanding of the world) is useful for prediction. So, in the end, even if everything's ultimately mechanical, there's still a place for things like "magical thinking", wisdom, and intuition. Hence, it's down right silly and wasteful for die-hard reductionists to spend so much time brow-beating the holists. And thanks for your patience and vision. Normally, people just call me a silly person and walk away. [grin] -- glen e. p. ropella, 971-219-3846, http://tempusdictum.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
