Hmm... I have not read either of those books... I have tried reading Shadows of the Mind and I found it far too difficult to follow with a healthy amount of scepticism simply through lack of mathematical understanding (despite having a maths degree!). Elisha, I am interested in your thoughts on this regarding his other books...
However, in a similar vein of thought; I used to agree with the idea that the universe "adheres" to mathematical principles. This was until I took a course in mathematical logic in which the lecturer mathematically proved that 1+1=2. It was a long time ago and I am very rusty on this but it all came down to boolean logic (implies, if and only if, not etc.) A central principle of mathematics is that it grows through the application of mathematical proofs. If a new statement can be shown to fit in with proven mathematical principles, it is considered to be true. However, this meant it was necessary to go to the source of maths (numbers) and proove that 1 is one and that two is thus 1+1. From this basis, it can then be claimed that all mathematics has been "proven" to be "true".
Putting it another way; a lot of mathematical principles/proofs were well established before it was proven that 1=1 and 1+1=2 etc. These latter concepts were "proven" through the retrospective application of mathematical principles to mathematics itself as a science. Once this was done, mathematicians could prove that maths "is real" (that is, it is not a process of creation through human inquiry but is actually a real phenomenon awaiting discovery through human exploration). I hope this makes sense.
Unfortunately, there was a major flaw in the proof that "maths is real". As I said, it was prooven that one is 1 and that two can therefore be considered 1+1; however, as I said again, this proof relied on the use of boolean statements like "if and only if". Importantly, these statements were not "proven" to work in the way accepted, it was simply assumed as a tautological fact that statements like "implies" or "if and only if" work in an intuitive way. That is, one cannot prove that "if and only if" works in such a way that T<=>T=T, F<=>T=F, T<=>F=F and F<=>F=T. Mathematicians just accept that the operator <=> works in this way and that it is self-evident that it should do.
However, as I see it, this means that these concepts are purely abstract human creations. Given that maths as a discipline rests entirely upon the assumption that these logical statements operate in the way we expect them to, maths is essentially dependent upon conscious observation and it is thus a model of the universe NOT the universe itself.
Thus, the idea of mathematics elucidating a the Ideal Form seems a bit rediculous: if maths can encapsulate the Ideal Form one can say that maths=the Ideal Form. BUT maths would not exist without *our* assumptions about logical statements. Thus, no consciousness => maths=0 = non-existent => the Ideal Form is non-existent - assuming that => works the way I think it does! 
Bah! So much more sensible to simply exist that maths models reality (only reasonably well - see statistical mathematics) and that we will never have a formula for how the entire (I mean absolutely EVERYTHING) universe works but that's OK because that means the search for understanding is never-ending fun and excitement  .
Phew, my head hurts.
J. |
