Or physics problems unanswerable due to a math paradox?

From Nature:

In 1931, Austrian-born mathematician Kurt Gödel shook the academic world when he announced that some statements are ‘undecidable’, meaning that it is impossible to prove them either true or false. Three researchers have now found that the same principle makes it impossible to calculate an important property of a material — the gaps between the lowest energy levels of its electrons — from an idealized model of its atoms.

The result also raises the possibility that a related problem in particle physics — which has a US$1-million prize attached to it — could be similarly unsolvable, says Toby Cubitt, a quantum-information theorist at University College London and one of the authors of the study.

The finding, published on 9 December in Nature, and in a longer, 140-page version on the arXiv preprint server2, is “genuinely shocking, and probably a big surprise for almost everybody working on condensed-matter theory”, says Christian Gogolin, a quantum information theorist at the Institute of Photonic Sciences in Barcelona, Spain. More.

Here’s the abstract:

We show that the spectral gap problem is undecidable. Specifically, we construct families of translationally-invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lower-bounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.

This implies that it is logically impossible to say in general whether a quantum many-body model is gapped or gapless. Our results imply that for any consistent, recursive axiomatisation of mathematics, there exist specific Hamiltonians for which the presence or absence of a spectral gap is independent of the axioms.

These results have a number of important implications for condensed matter and many-body quantum theory.

(Public access)– Toby Cubitt, David Perez-Garcia, Michael M. Wolf

Early overheard comments say that the issue turns on undecidability and could have big implications for naturalism.

*Note:* Posting light until later this evening, due to O’Leary for News’ alternate day job.

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I think it’s a

physicsproblem that is unanswerable due to amathematical“paradox”.Peter Rowlands and Bernard Diaz have found a way to circumvent Godel’s theorem.

It’s based on deriving maths from rewriting. This means that 1 is a rewrite (“copy”) of 0. So the 1 is derived from 0, rather than the 1 is obtained by counting. The 0 and 1 therefore have a boolean interchangeable relationship. This way all the mathematical operators and numbers are derived.

I think pure mathematics is the obvious theory of everything. There will never be any theory of physics which is not mathematical.

http://arxiv.org/ftp/cs/papers/0209/0209026.pdf

“Mathematics can be shown to be constructible using this mechanism, with an order which is more coherent than one produced by starting with integers. By rejecting the ‘loaded information’ that the integers represent, and basing our mathematics on an immediate zero totality, we believe that we are able to produce a mathematical structure which has the potential of avoiding the incompleteness indicated by Gödel’s theorem. (Conventional approaches, based on the primacy of the number system, havenecessarily led to the discovery that a more primitive structure cannot be recovered than the one initially assumed.) From this mathematical structure, we have been ableto develop an insight into how physics works, and using this to suggest a process that leads naturally to a formulation for quantum computation”

I like physorg’s write up of the paper:

In other words, this is bad news for reductive materialists who would like to describe everything in the universe, as well as the universe itself (i.e. inflation), in a bottom up fashion.

Around the 13:20 minute mark of the following video Pastor Joe Boot comments on the self-defeating nature of the atheistic/materialistic worldview in regards to providing an overarching ‘design plan’

A few more notes:

Georg Cantor’s part in incompleteness is briefly discussed here in this excerpt from the preceding video

Kurt Godel’s part in bringing the incompleteness theorem to fruition can be picked up here in this excerpt:

A bit more solid connection between Cantor and Godel’s work is illuminated here:

An overview of how Godel’s incompleteness applies to computer’s is briefly discussed in the following except of the video:

As to the implications of his incompleteness theorem, Godel stated this:

Notes as to how Godel’s work relates to ID

Verse and Music:

mohammadnursyamsu @ 2 –

They don’t provide any evidence (like a proof) for this. They don’t even get started on this, and in mathematics just stating a claim isn’t enough.