informal provability

[gusl]

My current mindfuck is Hannes Leitgeb. He makes me want to abandon the "semantic web" for the "schema-ic web".

I enjoy it because he seems like a AI-logician who is thinking about mathematics as a cognitive phenomenon... this is the perspective he uses to shed light on the apparent paradoxes that arise when we simultaneously consider its Formalistic and Platonistic characters¹ He focuses on the following issue: if formal and informal provability are extensionally equivalent, then what do the incompleteness theorems imply about provability? "Are There True but Informally Unprovable Statements?" a.k.a. "Are there absolutely undecidable statements?" My take is: well, if you see one, you surely won't recognize it as such! How would you know it to be true? (I assume axioms are not included)

To get there, he builds a theory of cognitive representations -- how mathematicians might represent graphs, integers, etc.

I'm halfway done, might post highlights later.

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^{1 - At least that's what I was into, when I was into these ideas.}

Comments

[easwaran.livejournal.com]

Hannes Leitgeb does so much interesting stuff - I haven't heard this stuff though, but it's related to a paper idea I've been trying to develop.

[jcreed.livejournal.com]

I think the phrase would be "schematic web"? At least that's the default adjective formation of "schema". Unless you want to create a definitely-new word to avoid existing connotations of "schematic"...

[gustavolacerda.livejournal.com]

<< Unless you want to create a definitely-new word to avoid existing connotations of "schematic"... >>

exactly!

[roseandsigil.livejournal.com]

This sounds a lot like what I am thinking about these days. I will try to remember to take a look.

[http://users.livejournal.com/_greg/]

I picked my graduate school and advisor because I was hot to study cognitive schema theory where it was being most vigorously pursued, at the "West Pole", UCSD under David Rumelhart. I was very disappointed when I got there to discover that they were winding down schema theory in favor of unstructured connectionism. This was in 1980. Suddenly representation was a dirty word; everything was just going to "emerge". Very little has emerged; after nearly 30 years neural networks on our fastest systems cannot compete with schema-based systems running on machines with only a few MIPS. Schema theory was hot with Bartlett in the 1880s, then hot with Piaget in the 1940s, then hot with AI researchers and cognitive scientists in the 1970s who discovered that a direct application of logic is computationally intractable. The semantic web is just a repeat of the enthusiasm for first order logic and semantic networks of the 1960s and early 1970s. Yet connectionism has been even less fruitful. I'm hoping that sense will prevail and we'll see a return to progress.

_Greg

[gustavolacerda.livejournal.com]

I'm very fond of schemata!

Leitgeb (probably following others) uses it as a weakening of what I would call "formal concept"... maybe it's also a more interactive idea than formal concepts: you understand mathematical concepts when you see them in action... just like you can understand a machine by seeing its motion, rather than its implementation. This is for the same reason why it is possible to think of an image but not be able to paint it: we don't immediately know how to ground our high-level percepts into low-level stimuli.

Painters spend years learning to dominate their own illusions.

[gustavolacerda.livejournal.com]

did you actually study/do research with Rumelhart?

[http://users.livejournal.com/_greg/]

Yes - I was around him a lot when I was studying AI before I returned for graduate school and then later I was his Ph.D. Student. Before I returned for grad school I built a successful semantic network based AI system for tutoring and learning, but I was clear it would not scale well. I started studying schemata, frames and similar systems. Frames seemed to rigid to me. I courted Rumelhart and was accepted as his student in a Ph.D. program in Cognitive Science in the early 1980s at UCSD. Unfortunately for me he left for Stanford for a sabbatical year putting me under the wing of Jay McClelland - a nice guy but not at all into schema theory or representation of any kind. Rumelhard accepted a position at Stanford and I was kind of stranded. Just about everybody but me came down with neural network fever. For those and other reasons I left the program.

There are very few people who seem to have taken seriously the notion that intelligence is a graceful coordinated use of multiple forms of knowledge embedded in different representation systems. Raj Reddy's Blackboard architecture and some of the Schema and Frame systems are the rare exceptions. It is particularly important to unite declarative and procedural views, i.e. to give strategic procedural support to a declarative system, e.g., automatization creating procedures which can be reviewed and updated as needed.

I'm pleased that you are aware of schemata and of this kind of work. Schemata were much more than weak concepts, but they could include a very flexible fuzzy model with partial commitment to more formal models and procedures. They are the most advanced knowledge representation system I am aware of, and just waiting to be exploited in new forms.

_Greg