Robert Marks on the math paradox challenging physics

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Dr. Robert MarksYesterday we noted new findings that a math paradox might make physics problems unanswerable be unanswerable (and thus maybe turn the physics problems into paradoxes too).

Robert Marks II, computer science prof at Baylor U and editor-in-chief of Bio-Complexity , offers some thoughts:

Anything algorithmic can be done by a computer. Give me a recipe for doing something, and I can whip it up in the kitchen. There are things which are not algorithmic the most celebrated of which is Turing’s halting problem: there exists no algorithm able tell whether or not a computer program runs forever or halts. (The halting algorithm must work for any and all computer programs.)

But a computer program will halt or won’t halt. But since there is no algorithm to figure this out, the halting problem is undecidable. We don’t know before running the program whether or not it will halt. It could run trillions of years and then halt long after we’re dead. If it doesn’t halt, we may never know (unless we know the so-called busy beaver numbers which is the same as knowing Chaitin’s number which is unknowable. But I digress.)

I’ve always thought this is a strong statement about determinism. A computer program that doesn’t use randomness is deterministic. There it is. All 12,465 lines of code. Yet we don’t know whether or not this deterministic program will halt or not. Rice’s Theorem extends undecidability to any “non-trivial” property of the computer program. We can’t even write an algorithm to tell us whether or not the number 3 will be printed by a computer program! So undecidability is much greater than only the problem of halting.

Chaitin wrote a book called The Unknowable that addressed this view of undecidability. (His other books are better.) Chaitin’s number is a single number between 0 and 1 that, if known, would allow solution of ALL the unsolved problems in math that can be resolved with a single counterexample (assuming we had gobs of time and computer memory.) Its construction uses the halting problem. So the C++ on your computer has a deterministic Chaitin’s number. But it is unknowable. Bummer.

Lastly, Roger Penrose in Shadows of the Mind and The Emperor’s New Mind makes the case the human mind, through creativity and the creation of information, does nonalgorithmic things (and is therefore not merely a computer).

I am starting to believe creation of information requires a nonalgorithmic process, hence intelligent design.

Creation of information as a nonalgorithmic process? Most writers you’d ever want to read would understand that. See also: How DOES creativity happen? Have we found the answer via neuroscience?

File:A small cup of coffee.JPG The creation of new information is not incomprehensible. But it often cannot be understood in the way many insist on trying to understand it. On the other hand, their attempts do often generate suitable subjects of fun.

See also: Math problems unanswerable due to physics paradox? Or versa vice? Early overheard comments say that the issue turns on undecidability and could have big implications for naturalism.

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5 Replies to “Robert Marks on the math paradox challenging physics

  1. 1
    wd400 says:

    Both this post and the discussion of the physics result seem to skipp this point

    (The halting algorithm must work for any and all computer programs.)

    There are plenty of programs that can be shown to halt or not. The halting problem only tells us that it’s not possible to generate a general solution. Likewise (as I understand it) the physics results tells there is no general solution that will allow us to predict arbitrarily small spectral gaps. That doesn’t mean specific solutions can’t be found.

  2. 2
    Gordon Davisson says:

    Furthermore, the undecidability result only applies to an infinite lattice. From the Nature article:

    For a finite chunk of 2D lattice, however, the computation always ends in a finite time, leading to a definite answer. At first sight, therefore, the result would seem to have little relation to the real world. Real materials are always finite, and their properties can be measured experimentally or simulated by computer.

    Unsurprisingly, this is similar to the underlying halting problem: the question of whether a given program halts within N steps is computable. The halting problem only becomes undecidable if you remove the limit and ask whether it halts given infinite time.

    Essentially, you can’t always answer infinite questions in a finite computation. Looked at this way, it’s not really surprising, and it’s hard to see why it would have any particular metaphysical implications.

  3. 3
    bornagain says:

    Although I do not know about the details of the halting problem in particular, I do know that Godel’s incompleteness theorem does have some fairly profound ‘metaphysical implications’ overall.

    As to the implications of his incompleteness theorem:

    “Note that despite the incontestability of Euclid’s postulates in mathematics, (ref. on cite), Gödel’s incompleteness theorem (1931), proves that there are limits to what can be ascertained by mathematics. Kurt Gödel (ref. on cite), halted the achievement of a unifying all-encompassing theory of everything in his theorem that: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove”. Thus, based on the position that an equation cannot prove itself, the constructs are based on assumptions some of which will be unprovable.”
    Cf., Stephen Hawking & Leonard Miodinow, The Grand Design (2010) @ 15-6

    Excerpt: we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel’s critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.

    “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

    “In materialism all elements behave the same. It is mysterious to think of them as spread out and automatically united. For something to be a whole, it has to have an additional object, say, a soul or a mind. “Matter” refers to one way of perceiving things, and elementary particles are a lower form of mind. Mind is separate from matter.”
    Kurt Gödel – Hao Wang’s supplemental biography of Gödel, A Logical Journey, MIT Press, 1996. [9.4.12]

    “If the world is rationally constructed and has meaning, then there must be such a thing [as an afterlife].”
    Kurt Gödel – Hao Wang, “A Logical Journey: From Gödel to Philosophy”, 1996, pp. 104–105.

    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.

    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.,,,

    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,

    Verse and Music:

    “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

  4. 4
    aqeels says:

    Bornagain @3

    Turnings halting problem is a basic reformulation of Godel’s incompleteness theorems. Essentially, any algorithm is nothing more than a formal mathematical system so the ideas are the same. Roger Penrose also gives his own version in Shadows Of The Mind.

  5. 5


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