Doyle I would guess you mean that certain kinds of transformations are possible to describe with mathematical methods and the cerebellum seems to be the body knowledge source which would fall under the category of such transformations. Such mathematical rules in the brain as innate or hard wired would be big news I'm sure. There are of course inherent structures to the brain, but the neural networks don't seem to have much built in with respect to hardwired mathematical rules. That is why Chomsky has so many gnats swarming around his theories, because he proposes innate structures where the neural networks don't seem to have them. Instincts for certain kinds of activities seem there, but nothing I've heard of like E(8). This seems to be a product of selection due to human dependence upon language. In other words mathematics is external where it belongs.
Doyle Some kinds of perception such as in vision in the optic nerve track seem to a common "ruled" structure to humans. The structure which is inherented seems to mediate seeing color, and other generalized qualities of vision, and I could accept an inheritence of the these structures. Mathematics in the brain needs to demonstrated in the material of the brain for me to accept such a claim. regards,
Doyle Saylor
----------------
This gets waaay complicated. Let me say, first, I'll give you the argument, but I am not sure I believe it--so call it provisional. The primary problem boils down to a philosophical and scientific issue. Is the world structured and that's what we learn or are we structured and thats why the world seems to make sense?
Here is my quick take. The answer is both in reciprocity as and through evolution. So the question is how can the world contain anything like a Euclidean geometry?
Go back to a primitive topological manifold of practically no structure at all (bi-continuous arbitrary point set). Now, ask how can you
embed a group structure? This requires an orientation, an orientable manifold, i.e. more structure. So, where is this structure supposed to come from?
Here's hint. What never changes and never dies? What always points in the same direction? Gravity. We live in a gravitational field and it always points down. Viola! The spatial orientation is the gravity vector which for every suitably large piece of matter acts as a pole or axis.
Now one of the representations of a Euclidean space are the reflections and rotations of some symmetric star (intersecting lines) about an axis (gravity) and there are a minimum of eight of these symmetries--isomorphic to the Euler Angles or O(3) [you can further show a homomorphism between O(3) and SU(2). After that practically the rest of kinematics follows]. So the structure is in the world by virtue of the presence of mass and gravity. This structural feature of our space relieves biology from the obligation of explaining a lot, but more particularly the various morphologies of organisms, including our bi-lateral symmetry (oriented as reflections about the gravity vector).
You don't have to imagine we are born with 2 + 2 in our head. Think about this. Every organism has to be able to orient itself in space and time. How to they do this? Well they don't really, except to react to the pre-existing physical parameters of their environment. However this amounts to the most primitive of perceptions, no? Well, say yes for now. So, you built up from there. Remember we have a complete gyroscope literally between our ears--anatomically this is a region that includes the semi-circular canals, the optic chasma to the lateral geniculate bodies, insertion points of the cranial nerves, the union of the spinal cord with the brain stem and thalamus. In other words the primary orientation/motor/perception junction--the big intersection.
If you drop a plumb line through the body, and intersect that line with vision and hearing you end up exactly in this region and something that looks like a star drawn about a axis. So, the brain is literally build and organized about this region, which is of course known as the 'most primitive' part. That is we share this general arrangement with just about everything that has a knob at the end of its spinal cord.
Now there is a certain amount of evidence that this region is pre-wired in fetal development through global electrical impulses which trace out electrical potentials or patterns that are used by developing neurons to make their necessary connection--particularly in this region (see, Marion Diamond et al. developmental neurophysiology). This arrangement then provides the basic hardwired format about which motor control, sensory perception, and basic regulation of the rest of the body are constructed. After birth, there's nothing to look foreword to but positive re-enforcement--i.e. out in the world of EMS, mass, and gravity.
Now, Piaget picks up a little of this trail when he describes the stages of learning that begin with basic kinesethic knowledge--this is here, that is there, we turn the red ball this way and that, etc. But all of that learning depends on this other more rudimentary structure that comes with the package.
So, this form of structure doesn't require anything from language, since all that comes later and arrives/derives from out of this more basic architecture.
I don't know anything about Chomski's linguistics, except that it follows a structuralist line derived from Saussure and depends on a Whorf-Sapir hypothesis--language constructs reality (?); that the commonality of thought between peoples derives from some innate patterning mechanism of the thought to be found in genetics.
I don't know what I think about Chomski, since I haven't read his technical work. But, of course I have an opinion. Any commonality between people and languages has to derive from the shared biological origin of our experience (birth, yelling, eating, fucking, and death) and the simple fact we all live in the same physical world of gravity, light, temperature, water, etc. In a sense there is enough organization, rules, and structure to go around, without accounting for it all in genetics.
What I am most interested in is dis-placing or eliminating structure, rules, organization, and other extremely difficult to account for appearant aspects of the biological world. I want to displace all this paraphenalia out into the world and the environment. There is far too much concentration on trying to discover hidden design in biological processes, particularly in genetics. I am aware this motive is almost completely counter to what I just wrote in the above--but not entirely. See, the anatomy of the body and brain follow some sense of design, and the question is where did that come from? Well saying genetics, doesn't quite answer the question, so that is why I want to discover the roots of that patterning mechanism in the world. That relieves genetics of any pre-figuration and allows living things to just follow the rules of the world through a history of evolutionary paths. The basic idea is if you can account for something in the physical world, then you have simplified biology.
Chuck Grimes