A proof against determinism?

Could this actually hold?

Please disprove the following:

  1. For a system to be deterministic, its underlying rules must be consistent.
  2. For a system to be deterministic, its underlying rules must be complete.
  3. No system of rules can be both complete and consistent per Gödel’s Incompleteness Theorems.
  4. Thus no system can be deterministic.

FYI: This proof dawned on me while researching Gödel’s Incompleteness Theorems – but I realize that points 1 & 2 above may perhaps be invalidated by presenting cases for deterministic systems with incomplete or inconsistent rule sets. Can such cases exist?

Update (2011-01-30): After some good comments, I offer this:

  1. Thesis: The universe contains all there is and all there ever will be, it is a complete and closed system and causally deterministic (Laplace’s demon)
  2. According to Gödel’s Incompleteness Theorems, such a system would have to contain paradoxes (inconsistencies), potentially rendering the system indeterministic.
  3. To prove the thesis of a causally deterministic universe, one would have to prove (why) the universe would never encounter any such paradoxes breaking the determinism – and prove why the universe itself would never encounter Turing’s Halting Problem when deciding any effect ever in the universe.

52 Comments

  1. @Geir: the major flaw I see in your reasoning is point #3 – Godel’s Proof is valid only for mathematical formal systems and is not generally applicable to the superset of “all systems of rules”.

      • Alan Turing’s notion that the Universe itself is some sort of Automaton is nothing but conjecture. No proof of this idea has ever yet been made. Equivalence of the Universe to a Mathematical Formal System has not been established, therefore Godel’s Proof does not support your contention.

        • One could argue that a universe containing a Turing machine or containing the notions of or indeed itself mathematics is then a superset of these groups and therefore must also abide by both Turing’s proofs and Gödel’s Incompleteness Theorems. Or further: If the universe has a set of algorithms (laws) that fully describes it, and the universe is a completeness, then it must also be inconsistent and therefore not deterministic.

  2. That is an interesting exercise.

    Determinism is (1) the doctrine that all facts and events exemplify natural laws, or (2) the doctrine that all events, including human choices and decisions, have sufficient causes. Determinism does not explain where the first cause came from. It also precludes the possibility of original thought.

    To me a universe may be deterministic or consistent in itself, but, ultimately it is an arbitrary system. It is something like a soap bubble that may be consistent in itself but it floats around without any anchor.

    So, just by plain looking, I can say that however deterministic a universe may appear, its presence is totally arbitrary.

    .

    • I don’t see that Gödel was speaking of determinism in such a simplistic sense.

      IMO, he meant all LIVING things, all things in physical existence, are in co-operative relationships with a set of rules for the living, moving universe experience.

      One might infer that a non “living” existence may also have its set of rules, of truths.

      We can do what we want while alive – within limits. Abraham Lincoln, Mother Theresa, Jesus Christ, Genghis Khan, Forrest Gump, Buddha or Hitler – they all had limitations. Health, age, pain, intelligence, sanity or insanity, prosperity or poverty, the universe set its limitations. They died. They may or may not have taken another form, but they entered and left our world the same way we will leave it and didn’t live any remarkable lifespan, either.

      • All determinism and non-determinism lie in the “bubble”.
        All living and non-living that you are talking about are within that “bubble.”
        All things physical and spiritual are within that bubble.
        All that exists and the co-operative relationships are within that bubble.
        All rules and truths are within that bubble.
        All complexity and simplicity are within that bubble.
        Everything that you can think of is within that bubble.
        And that bubble itself is without anchor and cannot be anchored by itself.

        That is not simplistic. That is the broad overview of Godel’s theorem.

        .

  3. There are formal systems which are complete and consistent. Gödels incompleteness theorem only says that any formal system that is *expressive* enough to make statements about first order arithmetic (and thus permits statements about itself by codifying the language of the system in numbers) must be either incomplete or inconsistent.
    An example for a nontrivial formal system which is complete and consistent is the first order theory of the field of real numbers (R,+,*,0,1) (as opposed to the first order theory of the natural numbers (N,+,*,0,1) , which is incomplete according to Gödel).

      • It’s not clear how the universe itself could be considered as a system in the sense of Gödel.

        —-short explanation of representation systems—-
        The most general formulation of Gödel’s theorem that i know, which is by Raymond Smullyan
        (see R. Smullyan “theory of formal systems” chapter 3
        download link:
        http://depositfiles.com/de/files/1gworfvkq
        )
        , is concerned with so called “representation systems”.

        A representation system consists of
        – a set of expressions E
        – a subset S of E (sentences)
        – a subset T of S (“true”/”provable” sentences)
        – a subset R of S (“refutable” sentences)
        – a subset P of E (unary predicates)
        – a mapping *:PxE->S , this means it is possible to combine any predicate p with any expression e to get a sentence p*e

        If you have such a representation system, then any predicate p from the set P is said to “represent” a subset of the expressions E.
        The set that is represented by a predicate p by definition consists of all expressions e in E , such that
        p*e is an element of T.

        Any formal system in Gödel’s original sense is also a representation system.
        So for example the first order arithmetic is a representation system for sets of natural numbers.
        But also for example the english language itself could be seen as a rep. system, if one defines exactly what are the expressions, what are the sentences etc.
        Maybe one could even think of representation systems that are not even language based.

        Then Gödel’s incompleteness theorem says, that under certain conditions
        either this system is inconsistent, which means that the sets T and R are not disjoint
        or
        it is incomplete, which means that the union of T and R is not the full set of sentences S or in other words that there is a sentence which is neither provable(not in T) nor refutable(not in R).
        This is for example the case when the system is expressive enough, that it can represent the set sqrt(R)={ H : H*H \in R}.
        Because if p is a predicate that represents sqrt(R) , then we have
        p*p \in R p \in sqrt(R) p*p \in T
        so either p*p is in R and T => the system is inconsistent
        or
        p*p is neither in R nor in T => the system is incomplete

        Note that in this abstract formulation of Gödel’s theorem the system doesn’t even have to be countable, as it would have to be in a “formal” system.
        The sets E,S,R,T can be chosen arbitrarily.
        —–
        Now, if you want to think of the universe somehow as a representation system, so that Gödel’s theorem can be applied, then you would have to find meaningful interpretations of the sets E,S,T,R.

  4. Soderqvist1: Isene, I take it for granted at the outset that you are wrong, not because of personality flaws or things like that, no, but because of that I have read some books as you probably have seen in my messages, and I as far as I know nobody have done such a thing in example; disproved determinism, so it is not very likely that you are right. As a matter of fact Schrödinger ’s wave equation develop deterministically, but the collapse of wave-function is not part of the equation which give room to different interpretations, some is based on determinism, some on indeterminism. Godel’ s incompleteness theorem concerns empty symbols, and how they are deduced from axioms, but you seem to suggest to us it in theories with structural content, which describe reality by induction. Up to date I have only seen metaphorical uses in example by doctor Richard Hofstadter!

    Hofstadter, Gödel, Escher, Bach
    All consistent axiomatic formulations of number theory include undecidable propositions … Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved …

    How can you figure out if you are sane? … Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is “peculiar’ or not, given that you have only your own logic to judge itself? I don’t see any answer. I am reminded of Gödel’s second theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent.

    The other metaphorical analogue to Gödel’s Theorem which I find provocative suggests that ultimately, we cannot understand our own mind/brains … Just as we cannot see our faces with our own eyes, is it not inconceivable to expect that we cannot mirror our complete mental structures in the symbols which carry them out? All the limitative theorems of mathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally.
    http://www.miskatonic.org/godel.html

    The Fabric of Reality
    David Deutsch laid the foundations of the quantum theory of computation. The Universal Turing machine is replaced by Deutsch’s universal quantum computer. (“The theory of computation is now the quantum theory of computation.”)
    http://en.wikipedia.org/wiki/David_Deutsch

    Soderqvist1: I have read this book some time ago, his Theory of Everything is deterministic, and I suspect that his Virtual Reality Machine, and Quantum Computer is only a subset of the Universal Turing Machine, but if they are equivalent then Godel’ s Incompleteness Theory should hold sway!

  5. Three three questions just came to me and I thought I would jot them down:

    A. “Can a self create mathematics and space/time?”
    B. “Can mathematics create a self and space/time?”
    C. “Can space/time create a self and mathematics?

    Let’s presume there are multiple answers that are “not wrong.”

    Q1: What kind of Self does A = “Yes” and B = “No” and C = “No” create?
    Q2: What kind of Self does A = “Yes” and B = “No” and C = “Yes” create?
    Q3: What kind of Self does A = “No” and B = “Yes” and C = “No” create?
    Q4: What kind of Self does A = “No” and B = “Yes” and C = “Yes” create?
    Q5: What kind of Self does A = “Yes” and B = “Yes” and C = “No” create?
    Q6: What kind of Self does A = “Yes” and B = “Yes” and C = “Yes” create?
    Q7: What kind of Self does A = “No” and B = “No” and C = “No” create?
    Q8: What kind of Self does A = “No” and B = “No” and C = “No” create?

    And how would free will and determinism fit in each of these?

    Could there be versions of me that are wholly deterministic?
    Could there be versions of me that have complete free will?
    Could there be versions of me that are a mixture of determinism AND free will?

    Are there versions that have each of these and feel EXACTLY the same even though one has free will and another does not?

    And what happens when they all discover the other?

    Who would that be?

      • Okay. Let’s Go With Number 2.

        The Pythagoreans initially only believed in whole numbers and ratios of numbers. At this time they didn’t know about irrational numbers. They thought their system was complete but it wasn’t.

        Addition, subtraction, multiplication and division all worked (and still work) in a deterministic manner as a system before and after they acquired that knowledge. Hence, a deterministic system (Early Pythagorarianism) worked with the false belief that the whole numbers were sacred and that every number was either a whole number or a ratio.

        They also like the Babylonians ignored Zero as a real value. Therefore, their system was incomplete even in real numbers.

        Pick any ancient number system. They all work with the basics and they all are not complete.

        Determinism does not need to be complete to work. In fact, the less complete something is, the more deterministic it becomes.

        Software that is not complete is very deterministic. It just crashes. The same way. Every time.

  6. Isene, what do you mean with ; “deterministic”?
    In what sense do you think Godel’ s Incompleteness Theorem inconsistent with determinism? How do you think said theorem disturb a real deterministic phenomena? How can it be possible to make prediction if nothing is deterministic?

  7. Geir, beware the deadly, “Amalgamated Union of Philosophers, Sages, Luminaries and Other Thinking Persons.”

    Adams works isn’t just fiction you know.

    You may not know this, but Deep Thought actually KNEW MORE than the Ultimate Answer “42.” But DT out of fear of losing funding to pay his enormous power bill kept it under wraps.

    Just like Lao Tsu sold out …

    “Those who know, don’t say. Those who say, don’t know.”
    – Lao Tsu (TAUOPSLAOTP Member #12002)

    Need more proof? Lao Tsu makes it even clearer.

    “The Tao that CAN BE TOLD is not the eternal Tao. The way that CAN BE NAMED is not the eternal name.” – Lao Tsu

    TRANSLATION: “Can Be Named/Told” is union speak meaning “Use only Union Approved Proofs, Syllogisms, Reasons and Symbols.”

    Fugging unions. . .
    :-)

      • JOKE EXPLAINED:

        I cut the entry short. I guess in doing so, I took out the introduction to the core idea. So here goes. Courtesy laugh appreciated.

        The “Amalgamated Union of Philosophers, Sages, Luminaries and Other Thinking Persons” is the fictitious Philosopher’s Union in “The Hitchhikers Guide to the Galaxy” that complains to Deep Thought that if He answers the ultimate question, that they would be out of a jobs.

        Deep Thought tells them how long it will take to figure it out and that they can make a bundle trying to guess his answer.

        You actually providing a solution to the “Free Will and Determinism” problem would effectively do the same thing – making this Union come after you.

        Lao Tsu’s statements were twisted to show that he was a “member’ of this fictitious group and that they were actually holding this data from Earth as part of their Union Bylaws. His core truths reduced to petty Union Rules.

        Unions are sometimes associated with violence. Hence the faux warning.

        ((INSERT CANNED SIT-COM LAUGHTER HERE BECAUSE ANY JOKE REGARDLESS OF QUALITY LOSES ALL HUMOR WHEN EXPLAINED.))

          • Agreed. Gödel’s work in IMHO is more relevant to reality than Einstein’s. He’s finally getting the respect he deserves.

            If his theorems extend to the physical universe, then God cannot exist because one being cannot know everything.

            Which points directly to #2. Because for God to exist, he/she must know EVERYTHING. Ending determinism by a BEING.

            Therefore, if this is correct in physical reality, then Gödel may have disproved an ultimate God.

            Doesn’t mean a being cannot be infinite. It just cannot be everything.

            • Interesting take on it. It may be though that a being escapes Gödel’s theorems just like Gödel seemed to have a hunch about, because a being has intuition. But still you are right as intuition is not the same as Knowing Everything.

  8. In the sense that Gödel uses in the theorem, consistent means that that there are no contradictions/paradoxes in the system. Complete means that every theorem must be provable. In simple terms his incompleteness theorem states that there will always be some theorems that are true, but that are not provable. A deterministic system, however, does not necessarily need to be provable, and so does not need to be complete in the Gödelian sense. Example: we create a system (universe) which consists of a single mechanical cat walking in a circle. We can envision this as a universe separate from the physical or any other universe. The fundamental, and axiomatic, rule is that the cat walks in a circle. It does not lose energy, it does not deteriorate with time, it simply does just that. The system is consistent, as the rule constantly applies and there is no contradition inherent in the system. The system is also complete in that it describes everything hapening in it, but it is NOT complete in Gödel’s sense, because the single rule (that the cat walks in a circle) can not be used to prove itself, thus there is a rule that is true, but not provable. I’m perhaps oversimplifying a bit, but the point is that a deterministic system does not have to be complete in Gödel’s sense, and therefore the Gödel theorem can not be used to prove that a deterministic system is impossible.

    • I have seen this. It could perhaps be possible to devise a deterministic system presumably complete. But as far as a deterministic universe goes, it encompasses mathematics and any and all ideas of any systems envisioned by any being therein. It is then a superset of the systems Gödel described. Hence, the universe would contain paradoxes in itself, possible rendering it indeterministic…?

  9. Soderqvist1: Isene, have you ever heard about Ilya Prigogine, he won the Nobel Laureate 1977? Is his work of an use in you endeavour for free will in the universe?

    The End of Certainty by Ilya Prigogine
    In his 1997 book, The End of Certainty, Prigogine contends that determinism is no longer a viable scientific belief. “The more we know about our universe, the more difficult it becomes to believe in determinism.” This is a major departure from the approach of Newton, Einstein and Schrödinger, all of whom expressed their theories in terms of deterministic equations. According to Prigogine, determinism loses its explanatory power in the face of irreversibility and instability.
    http://en.wikipedia.org/wiki/Ilya_Prigogine

    Soderqvist1: there are also effects in the universe without causes, in example: Radio-Isotopic decay. Isn’t that strange a effect without a cause?

  10. Soderqvist1: your premise number 1 seems suspect to me!
    It is a mixing of dimension, where the abstract is confused with the concrete.
    It is a description of a finite set, thus where the counting of its members terminates somewhere, but somehow infinite sets are members there. In example; phi is approximately 3,14 but its decimals are infinitely many, how can your finite set contain infinitely many of something?

    Soderqvist1: you have said earlier that you want to drink a cup of coffee without touching the cup with your hands; it would violate the law of gravitation, and the laws of thermodynamics by doing work without burning energy. Or the Heinleinian dictum; there is no such thing as free lunch.
    http://en.wikipedia.org/wiki/No_free_lunch

    Soderqvist1: btw, L. Ron Hubbard discusses by letter correspondence with his friend Robert Heinlein in 1949, about the construction of his Dianetics Organization.
    http://www.heinleinarchives.net/upload/index.php?_a=viewProd&productId=882

    Soderqvist1: then we have the old conundrum how can “nothingness” interact with “somethingness”? In example; a hammer and a nail can interact because they both are solid, and thus a balloon and a nail can interact less because the balloon in less solid, and it is obvious that you can not hammer a nail with a balloon, and it is impossible to hammer a nail with a hammer made of nothing. So how can static or nothingness grasp or impinge upon the physical world? More about the conundrum here!
    http://skepdic.com/refuge/hubbard.html

    • I don’t understand your focus on Hubbard in this thread.

      That aside, addressing your points:

      “It is a mixing of dimension, where the abstract is confused with the concrete.” Well, the underlying rules of the universe is abstract while the universe remains concrete. What is your point there?
      “It is a description of a finite set,…” Is it finite? Who says? Apart from that, you can surely have a finite set of items with an infinity of properties. But what is your point?

      “you have said earlier that you want to drink a cup of coffee without touching the cup with your hands”. Have I? Where?

      “then we have the old conundrum how can “nothingness” interact with “somethingness”” This is amply tackled in the Standard Model of particle physics.

  11. Have you never heard of the Bell’s Theorem (aka “Bell’s Inequality,” once called “the most profound theorem in science”)?

    Scores of tests have been done verifying the theorem. These tests indicate, conclusively, that the universe is inherently non-deterministic. In fact, these tests confirm that, even if quantum mechanics is altogether incorrect, the universe cannot *possibly* be physically deterministic.

    It’s not worth even trying to explain the experiments here. Just research Bell’s theorem.

    This is probably a decent starting place: http://en.wikipedia.org/wiki/Bell%27s_theorem

    • I know. It is a very important theorem. But it may not be conclusive/absolute like the one I propose – here as it hinges on experimentation (whereas a mathematical or philosophical proof like that of Gödel’s does not). Quoting from the WP article: “To date, Bell’s theorem is supported by an overwhelming body of evidence and is treated as a fundamental principle of physics in mainstream quantum mechanics textbooks.”

  12. Tangaku, I have read your link, but it is nothing there about “determinism is disproved”. But in stead, what I saw was pretty much in according with what I have seen some time ago that Albert Einstein’s worldview; “local realism” is not sustainable anymore, either his locality or reality has to go! Copenhagen interpretation claim that his locality is truth in the sense no signals travel faster than light, but his reality has to go, because quantum objects doesn’t have real values until they are measured, but John Bell has proposed his Aether Theory which permit signals travel faster than light through said aether which reject his locality, and is thus a non-local theory which agree with Einstein’s reality that the universe is real all the time. I have read Paul Davies “The Ghost in the Atom” there 9 quantum physicists give their explanation in including David Deutsch as have been mentioned earlier here by me and John Bell. There are at least 15 interpretations some is local and some is non-local!
    http://en.wikipedia.org/wiki/Aether_theories

  13. Comments on the Essay “On Will” by Geir:

    To me, the bottom line is NOTHINGNESS, and that means an absence of all considerations. Thus, there exists infinite choice at the outset. Any limitation on choice will then come from a prior consideration.

    Geir defines “will” as “exercise of choices.” Thus, one would start with an infinite “will” and that “will” shall decrease inversely proportional to the number of choices that are made and kept.

    “Potential free will” shall be recoverable by as-ising one’s existing choices.

    “No free will” shall exist when one has chosen to agree completely with the status quo.

    There is free will in this physical universe to the degree one is aware of the laws and principles that are keeping the physical structure there, and one can move within that structure. Ignorance of those laws and principles would limit that free will.

    As-isness of physical laws and principles is not essential to exercise free will. But knowledge of them is essential. Choices may be made only when there are options. No option will exist when either nothing has been agreed upon, or everything has been agreed upon.

    If everything can be calculated/predicted as per Steven Hawking, then one’s “free willed actions” may be predicted too, putting them in the category of “bound will.” Thus, for “free will” to exist there must be a balance between KNOWABLE and UNKNOWABLE.

    Randomness exists not in the universe but in the very idea of free will. The determinic part is one’s agreements expressed as the universe; the random part is one’s free will.

    .

  14. Geir: Thesis: The universe contains all there is and all there ever will be, it is a complete and closed system and causally deterministic.

    Chris: For the universe to be a complete and closed system, it would include all space-time. Is the universe complete or closed? Its accelerating expansion points toward it being an open system receiving continuous injections of space-time for unknown reasons. Therefore, the universe as it is compared to how it will be is incomplete, thus falsifying the major premise.

    Does this speak to the OP?

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