Thanks a lot for the thought provoking response.

Admittedly I am not a mathematician, and didn’t give any detail of geometrical properties and operations in this post, but I am talking about geometry here. The post is really about applying ideas from cognitive science to computer science/AI. In particular ideas about conceptual properties being convex regions (perhaps generated from concept prototypes as voronoi points) within integral quality dimensions. The post was getting long, I ran out of steam so just dropped in a link to Gärdenfors’ fine book, where these ideas are lifted from.

“point[ing] at the various dimensions and show what their units are and how they interrelate” is what many people are trying to do in music psychology (including myself), and I hope results there can be applied usefully in the field of computational creativity.

Gärdenfors sets geometrical conceptual space as a level of representation separate from symbolic representation. I’m struggling to understand what you mean by “describing the data itself”. What’s the data? As I understand it Gärdenfors says the percept/concept exists as geometry and the symbolic level (i.e., a word) is just a name for it.

This sounds interesting and good because I actually tried something like this a while ago. I was trying to evaluate poker opponents by putting their moves and playing patterns in a multi-dimensional space. Then by taking the distance between points in the space a “goodness” of a certain move could be evaluated. This experiment did seem promising, but didn’t work out that well after I tried it in practise. Perhaps the “shape” was missing and taking a distance between points was too straightforward :)

IMHO engineering-oriented programmers really benefit from this sort of writings floating around internet. Reading through, and understanding, an awful lot of really complex AI research papers and books is not usually an option unless you are really into AI research. A lot of people just “need to get things done” and have a practical engineer’s view into the issue. Unless it’s certain that some complex writing really helps in “getting things done”, they won’t bother reading it carefully. Heck, I know that even for a researcher it’s a load of work to skim through piles of papers because most of them are actually useless crap :)

Though talking about “geometry” might be technically wrong here, I think it’s fine because it gets the message delivered. And that’s what counts if you step out of the researcher community. This is a blog after all, not a presentation or a research paper :)

With geometric algebra and regular geometry being relevant to a fair number of AI subfields, it’s maddeningly annoying to see yet another already established term being overloaded, instead of a more appropriate word or description being used.

Geometry is beautiful, and has very little to do with multidimensional representation of data as it’s used in AI (naturally, any dimensioned data space allows for geometry, but unless you can point at the various dimensions and show what their units are and how they interrelate, it’s not geometry… it’s abusing the word).

That said, technically this just another symbolic representation, since “symbolic” just means we are describing the data itself. An apple can be symbolically represented by the world ‘apple’, but it can also be symbolically represented by a property/value set -that’s still symbolic representation.

Using property/value sets, or just property sets, is more functional for data processing in settings where natural language is hard or even meaningless, but the symbolic representation used only makes explicit what natural language assumes known. In effect, it’s not beautiful because it suddenly all makes sense, it *is* that sense, like a dictionary lets us make sense of complicated words. It’s the explicit description of any otherwise obfuscating description language.

Sadly, it can also lead to massive headaches, like when the idea is taken further, to frame descriptions for natural language (which take the idea much further) or higher order logic in general for reasoning in multidimensional data spaces.

AI’s pretty well defined by the fact that every problem allows for a plethora of approaches to solving it, but the major divide between operating on data (using symbolic representation) and operation on mappings (using neural nets) is still the same. Symbolic representation just falls apart it more approaches than going the neural net way.

(Plus, in symbolic approaches you can at least say what the algorithms mean, and thus explain why it works beyond showing mathematical validatity. Using neural nets, that benefit is lost; you can show the mathematics behind the mapping is valid, but at the cost of not being able to say what the mapping itself represents)

We are, after all, working with analogies and descriptions here and those tend to break down when they don’t precisely line up with our needs. If I write a “drummer” class it may work very well but just because it’s a analogy for a drummer doesn’t mean I can feed it a beer if it doesn’t sound spontaneous during a recording session.

I think that when we create abstractions that precisely cover our discussion domain using these various methods then the methods are already equivalent (mathematically speaking, of course, one might be much easier to work with than another). I’m fairly certain of this, I strongly suspect it’s even obvious ;¬)

I also think that we can combine them and that we actually do so all the time, yet in a very informal and imprecise way (then we get bugs) as they will all be slightly inaccurate descriptions. To again point out something obvious (but non-trivial!); modern formal logic logic grew out of language philosophy.

What is it that you (Alex) are after here? Are you interested in how we think, how we creatively deal with these formal constructs in our code and looking to increase your effectiveness in thinking?