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Gunnar_Zarncke comments on Debunking Fallacies in the Theory of AI Motivation - LessWrong
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I confess that it is would take me some time to establish whether weak constraint systems of the sort I have in mind can be mapped onto normed linear spaces. I suspect not: this is more the business of partial orderings than it is topological spaces.
To clarify what I was meaning in the above: if concept A is defined by a set A* of weak constraints that are defined over the set of concepts, and another concept B has a similar B, where B and A* have substantial overlap, one can introduce new concepts that sit above the differences and act as translation concepts, with the result that eventually you can find a single concept Z that allows A and B to be seen as special cases of Z.
All of this is made less tractable because the weak constraints (1) do not have to be pairwise (although most of them probably will be), and (2) can belong to different classes, with different properties associated with them (so, the constraints themselves are not just links, they can have structure). It is for these reasons that I doubt whether this could easily be made to map onto theorems from topology.
Thanks for your answer. I trust your knowledge. I just want to read up on the math behind that.
Actually it turns out that my knowledge was a little rusty on one point, because apparently the topic of orderings and lattice theory are considered a sub branch of general topology.
Small point, but I wanted to correct myself.
Hm. Does that mean that my reference is the right one? I'm explicitly asking because I still can't reliably map your terminolgy ('concept', 'translation') to topological terms.
Oh no, that wasn't where I was going. I was just making a small correction to something I said about orderings vs. topology. Not important.
The larger problem stands: concepts are active entities (for which read: they have structure, and they are adaptive, and their properties depend on mechanisms inside, with which they interact with other concepts). Some people use the word 'concept' to denote something very much simpler than that (a point in concept space, with perhaps a definable measure of distance to other concepts). If my usage were close to the latter, you might get some traction from using topology. But that really isn't remotely true, so I do not think there is any way to make use of topology here.
I think I recognize what you mean from something I wrote in 2007 about the vaguesness of concepts:
http://web.archive.org/web/20120121185331/http://grault.net/adjunct/index.cgi?VaguesDependingOnVagues (note the wayback-link; the original site no longer exists).
But your reply still doesn't answer my question: You claim that the concepts are stable and that a "o gotcha" result can be proven - and I assume mathematically proven. And for that I'd really like to see a reference to the relevant math as I want to integrate that into my own understanding of concepts that are 'composed' from vague features.
Yes to your link. And Hofstadter, of course, riffs on this idea continuously.
(It is fun, btw, to try to invent games in which 'concepts' are defined by more and more exotic requirements, then watch the mind as it gets used to the requirements and starts supplying you with instances).
When I was saying mathematically proven, this is something I am still working on, but cannot get there yet (otherwise I would have published it already) because it involves being more specific about the relevant classes of concept mechanism. When the proof comes it will be a statistical-mechanics-style proof, however.
OK. Now I understand what kind of proof you mean. Thank you for you answer and your passion. Also thanks for the feedback on my old post.