Mathematical Physics as Topological Numbers, Types and Quanta by Lawrence B Crowell

I think you made reference to how mathematics by itself is not a minimal guide for physics. Many mathematical areas are vastly complex systems, which might not serve as an effective foundation to reality.

Maybe a part of the problem is that we do not have a "unified theory of mathematics," assuming such a thing is possible. I think the foundation of the universe involves zeta functions, 8-fold periodicity related to Bott periodicity, homotopy indices, Langland number theoretic correspondences and so forth. This may involve some unification of some subjects in mathematics. I am not sure if this is comprehensive though. Physics in one sense involves working on a similar type of problem on deeper levels, while mathematics often involves pursuing the study of entirely different sorts of structures. These new structures can come into play with physical problems, but the method of thought is often very different between how a mathematician works out the consistency of some type of structure, and how a physicist frames a type of theory or solves phenomenology.

I would agree that Tegmark's MUH does not appear to satisfy certain minimal conditions we would prefer. In that I would tend to agree with you. I am not sure if this is exactly a proof though. For all we know our requirement for simplicity could be a bias that is wrong. Maybe the universe is vastly complex at is foundation. There are areas of mathematics that have physical implications which involve a huge level of complexity. The universe might in fact have some extremal Kolmogoroff complexity condition, which means the foundations are not only bewilderingly complex, but unknowable. I am not saying I think this is the case, but on the other hand I do not know that this is not the case.

Cheers LC

The Rodin book looks to be a very long read. It could have some interesting insights into things. There are connections through groupoids to category theory and Grothendieck type of theory and cohomology.

It is hard to know what the totality of mathematics is. It could be infinite in extent, which of course makes it difficult to know how this applies directly to physics. That would be difficult with respect to the Tegmark MUH conjecture. The one thing that is apparent is how many areas of mathematics are mapped into each other according to functors and categories.

Cheers LC

With respect to Rodin's manuscript, Voevodsky is a major developer of the HOTT.

I think in one sense this will probably have to be simplified or made more applicable for it to be widely used in physics.

LC

Dear Alma,

Thanks for the positive assessment of my essay. Fortunately a number of people seem to share your opinion. It has been near the top since the beginning, and I have been in #1 and 2 spot for nearly a week.

I see there being a sort of two fold system. Standard mathematics might be thought of as the "soul," or a "ghost," and mathematics that is restrained by concerns of Kolmogoroff complexity, types and so forth as the "body." It may not be possible to express all numbers between 10^{10^{10}^{10}}} and 10^{10^{10}^{10^{10}}}}, but this just means the body is not able to construct or contain the information space necessary to do so, but this still leaves room for the "soul." Mathematicians are then free to "pick their poison," where a pure mathematician may prefer to stay with the standard approaches to math, while a more practical minded analyst might prefer to stick with the "body."

I don't particularly get into the argument over whether the soul of mathematics exists or not. This involves things such as infinities, infinitesimal or even finite numbers that can't ever be computed. I am agnostic on the idea of there being a Platonic realm of ideals. The idea seems in one sense compelling, but it also seems to lead to some mystical notions that are not entirely comforting.

Cheers LC

There are some questions concerning foundations of mathematics. I am not a great expert on this, but it does seem that as mathematical physics develops that it will embrace concepts that are not as tied to many aspects of point set topology with infinitesimals and the rest.

I don't say there is something wrong with the foundations, and it appears that we are increasingly in a time where there are several such foundations. These things seem in some ways to be model dependent, with different proof methods and the rest.

LC

Dear Laurence,

Thank you for taking interest in my essay. The idea is that the quantum vacuum as a set of qubits, say (0, 1) set to a|0> = 0, has a phase structure based on how qubits are transformed into each other. We normally think of the vacuum as invariant under a certain symmetry group, but underneath that it could just be a vast self-referential loop, where there are "accidents" that occur where the vacuum has a symmetrical structure. This means zones exist where there are dynamical structures, where symmetries are aspects of division algebras.

These self-referential qubits, or loops of them, form a strange basis for the universe, or multi-verse, that can't be derived or computed. We can't then know what is not computable. It is similar to Chaitin's halting probability; we can know there are incomputable symbol strings in a set of them of length N, but we can't compute with certainty which are not computable (Turing's halting problem), we can't compute the number of them that are computable or not computable, or the probability for any of them to be incomputable or nonhalting. We are then faced with a bit of a conundrum; this would be a theory that tells us that this state of affairs exists, but we can't compute much of anything with it.

If physics and cosmology reaches this state of knowledge it might be the end of these foundations. The end of scientific foundations might occur this way, though I suspect we have quite a ways to go before progress in physical foundations stops at this point.

Cheers LC