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Math problems unanswerable due to physics paradox?

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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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4 Replies to “Math problems unanswerable due to physics paradox?

  1. 1
    daveS says:

    I think it’s a physics problem that is unanswerable due to a mathematical “paradox”.

  2. 2

    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”

  3. 3
    bornagain says:

    I like physorg’s write up of the paper:

    Quantum physics problem proved unsolvable: Godel and Turing enter quantum physics – December 9, 2015
    Excerpt: A mathematical problem underlying fundamental questions in particle and quantum physics is provably unsolvable,,,
    It is the first major problem in physics for which such a fundamental limitation could be proven. The findings are important because they show that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.,,,
    A small spectral gap – the energy needed to transfer an electron from a low-energy state to an excited state – is the central property of semiconductors. In a similar way, the spectral gap plays an important role for many other materials.,,,
    Using sophisticated mathematics, the authors proved that, even with a complete microscopic description of a quantum material, determining whether it has a spectral gap is, in fact, an undecidable question.,,,
    “We knew about the possibility of problems that are undecidable in principle since the works of Turing and Gödel in the 1930s,” added Co-author Professor Michael Wolf from Technical University of Munich. “So far, however, this only concerned the very abstract corners of theoretical computer science and mathematical logic. No one had seriously contemplated this as a possibility right in the heart of theoretical physics before. But our results change this picture. From a more philosophical perspective, they also challenge the reductionists’ point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description.”
    per physorg

    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’

    “If you have no God, then you have no design plan for the universe. You have no prexisting structure to the universe.,, As the ancient Greeks held, like Democritus and others, the universe is flux. It’s just matter in motion. Now on that basis all you are confronted with is innumerable brute facts that are unrelated pieces of data. They have no meaningful connection to each other because there is no overall structure. There’s no design plan. It’s like my kids do ‘join the dots’ puzzles. It’s just dots, but when you join the dots there is a structure, and a picture emerges. Well, the atheists is without that (final picture). There is no preestablished pattern (to connect the facts given atheism).”
    Pastor Joe Boot – Defending the Christian Faith – video
    http://www.youtube.com/watch?v=wqE5_ZOAnKo

    A few more notes:

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

    Georg Cantor – The Mathematics Of Infinity – video
    http://www.disclose.tv/action/....._Infinity/

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

    Kurt Gödel – Incompleteness Theorem – video
    http://www.metacafe.com/w/8462821

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

    Naming and Diagonalization, from Cantor to Godel to Kleene – 2006
    Excerpt: The first part of the paper is a historical reconstruction of the way Godel probably derived his proof from Cantor’s diagonalization, through the semantic version of Richard. The incompleteness proof-including the fixed point construction-result from a natural line of thought, thereby dispelling the appearance of a “magic trick”. The analysis goes on to show how Kleene’s recursion theorem is obtained along the same lines.
    http://www.citeulike.org/group.....le/1001747

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

    Alan Turing & Kurt Gödel – Incompleteness Theorem and Human Intuition – video
    http://www.metacafe.com/watch/8516356/

    Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems (1931), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system. This topic was further developed in the 1930s by Alonzo Church and Alan Turing, who on the one hand gave two independent but equivalent definitions of computability, and on the other gave concrete examples for undecidable questions.
    – wiki

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

    “Either mathematics is too big for the human mind, or the human mind is more than a machine.”
    Kurt Gödel As quoted in Topoi : The Categorial Analysis of Logic (1979) by Robert Goldblatt, p. 13

    Notes as to how Godel’s work relates to ID

    “In an elegant mathematical proof, introduced to the world by the great mathematician and computer scientist John von Neumann in September 1930, Gödel demonstrated that mathematics was intrinsically incomplete. Gödel was reportedly concerned that he might have inadvertently proved the existence of God, a faux pas in his Viennese and Princeton circle. It was one of the famously paranoid Gödel’s more reasonable fears.”
    George Gilder, in Knowledge and Power : The Information Theory of Capitalism and How it is Revolutionizing our World (2013), Ch. 10: Romer’s Recipes and Their Limits

    Conservation of information, evolution, etc – Sept. 30, 2014
    Excerpt: Kurt Gödel’s logical objection to Darwinian evolution:
    “The formation in geological time of the human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field is as unlikely as the separation of the atmosphere into its components. The complexity of the living things has to be present within the material [from which they are derived] or in the laws [governing their formation].”
    Gödel – As quoted in H. Wang. “On `computabilism’ and physicalism: Some Problems.” in Nature’s Imagination, J. Cornwall, Ed, pp.161-189, Oxford University Press (1995).
    Gödel’s argument is that if evolution is unfolding from an initial state by mathematical laws of physics, it cannot generate any information not inherent from the start – and in his view, neither the primaeval environment nor the laws are information-rich enough.,,,
    More recently this led him (Dembski) to postulate a Law of Conservation of Information, or actually to consolidate the idea, first put forward by Nobel-prizewinner Peter Medawar in the 1980s. Medawar had shown, as others before him, that in mathematical and computational operations, no new information can be created, but new findings are always implicit in the original starting points – laws and axioms.
    http://potiphar.jongarvey.co.u.....ution-etc/

    Evolutionary Computing: The Invisible Hand of Intelligence – June 17, 2015
    Excerpt: William Dembski and Robert Marks have shown that no evolutionary algorithm is superior to blind search — unless information is added from an intelligent cause, which means it is not, in the Darwinian sense, an evolutionary algorithm after all. This mathematically proven law, based on the accepted No Free Lunch Theorems, seems to be lost on the champions of evolutionary computing. Researchers keep confusing an evolutionary algorithm (a form of artificial selection) with “natural evolution.” ,,,
    Marks and Dembski account for the invisible hand required in evolutionary computing. The Lab’s website states, “The principal theme of the lab’s research is teasing apart the respective roles of internally generated and externally applied information in the performance of evolutionary systems.” So yes, systems can evolve, but when they appear to solve a problem (such as generating complex specified information or reaching a sufficiently narrow predefined target), intelligence can be shown to be active. Any internally generated information is conserved or degraded by the law of Conservation of Information.,,,
    What Marks and Dembski prove is as scientifically valid and relevant as Gödel’s Incompleteness Theorem in mathematics. You can’t prove a system of mathematics from within the system, and you can’t derive an information-rich pattern from within the pattern.,,,
    http://www.evolutionnews.org/2.....96931.html

    What Does “Life’s Conservation Law” Actually Say? – Winston Ewert – December 3, 2015
    Excerpt: All information must eventually derive from a source external to the universe,
    http://www.evolutionnews.org/2.....01331.html

    Verse and Music:

    John1:1
    “In the beginning was the Word, and the Word was with God, and the Word was God.”

    of note: ‘the Word’ in John1:1 is translated from ‘Logos’ in Greek. Logos is also the root word from which we derive our modern word logic

    Joy Williams – 2000 Decembers ago
    https://www.youtube.com/watch?v=4W8K3OhxVSw

  4. 4
    Bob O'H says:

    mohammadnursyamsu @ 2 –

    Peter Rowlands and Bernard Diaz have found a way to circumvent Godel’s theorem.

    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.

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