We are only feeding a reply from Alan at this point, but here's my view in regard to your question:
1. Dick's paper is not peer reviewed, therefore it isn't even within our jurisdiction to accept or deny his mathematical conclusions. Most scientists who look at it are so turned off by the metaphysics that they immediately stop reading it, so we don't even have any real indication whether the math is correct or not.
2. Most of physics is derivable from symmetries of nature, so it isn't any great feat on Dick's behalf to obtain known physics equations. The fact that there are probably more physics equations in the future that will be obtained in this manner suggests that Dick has no knowledge of what those future symmetries are, and which is why we don't see those future equations in his paper (a good thing I might add since can you imagine how cocky he would be if he predicted future physics equations?).
3. The jury is still out on what the mathematization of nature actually means. Does it mean that nature is mathematical as Alex suggests, or does it suggest that we are limited somehow in how we as a species can best describe nature, or are there other possibilities? It is perhaps possible that Dick's assumptions are somewhat, mostly, or entirely correct, and that he has stumbled upon something grand. However, the assumptions are still faulty in that he treats them as absolutely necessary and eliminating all doubt for being ill-founded assumptions. This is my main point against Dick. We don't know if nature is ultimately 'logical' or ultimately 'mathematical'. To assume that it is becomes only a possibility (might be/might not be). It might be that his assumptions are correct, but we certainly can't assume that they are.
4. There are also many coincidences in physics. It is not too uncommon to derive an equation if you know what that equation is up front, and then you can play with the variables to get it right. I imagine that the then young Dick had some reason for developing this 'model of reality' back in the 1960's, and he probably had many renditions to his paper. If we could time travel back to that point, we would probably see his paper basket full of failed attempts, near Schrodinger equations that weren't quite right, and many obvious math errors. Maybe we are seeing one success out of a hundred failures. He then could work backward to the assumptions that he would need to make in order to get the equations he derived, and then he could declare his 'model of reality' of success. He then spends the next 40 years telling physicists all of over the world how much smarter than Einstein he is. He leaves physics entirely (ba humbug) and preaches his gospel according to Dick. End of story.
Warm regards, Harv