One Chomsky Essay/Lecture Down, Five To Go

Tonight I finally finished the first section of the first part of Chomsky's Rules and Representations. I've started and stopped it at various points over the past month, picking up and finishing other things along the way. While I could get into my general sense of wonder at the content (an introduction into the ideas behind general innate mental structures of language) or distaste for the vehicle (the language of pseudoscience/philosophy academia, in all it's convoluted, obfuscating glory), I think I'm more inclined at this point to jump into a train of thought that sprang up in the last few pages that sent me on a few tangents. So to get into, excerpt #1:

-"When we turn to the mind and its products, the situation is not qualitatively different from what we find in the case of the body. Here too we find structures of considerable intricacy, developing quite uniformly, far transcending the limited environmental factors that trigger and partially shape their growth. Language is a case in point, though not the only one. Think for example of the capacity to deal with the number system, common to humans apart from pathology and as far as we know unique to humans"

So the first bit. Chomsky seeks to make a case that we are not a completely blank slate. There are kernels of hard code that determine how particular portions of the mind essentially work. These kernels may need experience to fine tune and mold the developing structures into their final shape, but there exists some central bit that dictates what and how these mental capacities develop. The initial bit of this was interesting to me, but the number bit set the stage for the next bit that got my current tangent going:

-"The very essence of the number system is the concept of adding one, indefinitely. The concept of infinity is not just 'more' than seven, just as human language, with its discrete infinity of meaningful expressions, is not just 'more' than some finite system of symbols that can be laboriously imposed on other organisms (nor, by the same token, just 'less' than an essentially continuous system of communication, like the dance of bees). The capacity to deal with the number system or with abstract properties of space is surely unlearned in its essentials."

So from this I got two things stuck in my head. One, the number system as the concept of adding one indefinitely. Two, both the concept of quantifying (the number system) and abstract properties of space as innate mental structures (at least in terms of some basic structures that can become more complex). And one more excerpt, this time from Steven Pinker's The Language Instinct, which has a chapter called "How Language Works" that goes through some of Chomsky's technical work on language in a more approachable manner:

-"A grammar is an example of a 'discrete combinatorial system.' A finite number of discrete elements (in this case, words) are sampled, combined, and permuted to create larger structures (in this case, sentences) with properties that are quite distinct from those of their elements.
...
In a discrete combinatorial system like language, there can be an unlimited number of completely distinct combinations with an infinite range of properties."

The chapter goes onto explain some of the broader bits and pieces of Chomskian thinking on structures of language. At the core is the notion of recursion. Basically our lexicon of words fits into various structures (noun phrases, verb phrases) that can recursively nest creating an infinite number of possible output.

So to tie it back, I saw the bit on the number system as adding one indefinitely as an instance of recursion at work (I still haven't gotten around to reading Principia Mathematica, but I wonder if this is where Russell and Whitehead started). The idea of recursion was initially (or at least in terms of viewing it as a useful tool) introduced to me in computer science classes in college as a means of manipulating mathematical formulas such as Fibonacci numbers and such. But I've also seen it poke it's head in biology classes, and obviously more recently in Pinker's explanations of Chomskian theory on language structure. Which brings us to the questions that come to mind.

Is it possible that the structure surrounding "abstract properties of space" are also built on recursive definitions? And if so, would an underlying ability to handle recursive properties be a base mental structure in itself upon which other structures grow upon? Or would it be a property rather then a structure (which happens to be shared by at least two - if not three - innate structures)?

When we look at nature, math, language, etc... we find recursion (or means of representing things recursively). There is a pervasiveness here. Do we see it because our minds are designed in such a way that this relationship can easily be created (a pattern overlaid on chaos), or because this is an actual fundamental principle that explains the structure of the universe. If the former, then it stands to tell us something about how our minds actually work. I would assume, based on the popularity of LISP with AI researchers, that there is a nice body of research that would at least make a case for this. If the latter, then it's quite possible that you could make a case for looking for recursive relationships in any and every field of study, from hard science to social science (recursive relationships as underlying rules of sociological phenomena?).

I wouldn't be surprised if recursion as a core concept didn't have a great degree of viability. Recursion is an elegant way of building complexity and infinity out of a finite number of simple parts.

And a final link. I tend to overuse Wikipedia for such things, but it typically provides a good overview with citations, plus a few jumping off points. This is their topic on recursion:

http://en.wikipedia.org/wiki/Recursion