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Free will, Determinism and Banach-Tarsky paradox

  1. Sep 14, 2009 #1
    I read many threads in this forum (in QM section), and in many cases I witnessed the same logical flaw over and over again. People conclude that, for example, if world is deterministic, than there is no free will, because our consciousness is deterministic too. Why? Because our brain is nothing more than a huge collection of atoms and molecules. Hence all properties of the brain can be reduced to the properties/configuration of the parts it consists of.

    This is not true for purely mathematical reasons. Let be begin from some hand-waving. Remember HUP? When we measure position we affect the impulse. So we can’t make 2 accurate position measurements of the same particle. This limitation is not instrumental. What if our consciousness has the same property? If you try to explain the consciousness based on how the molecules function in a brain, you destroy the brain in a process no matter how small is your scalpel. If you start measuring the positions of the particles in the brain you add random momentum to them, heating the brain and destroy it as it does not support even slight overheating.

    Now, if that limitation is not instrumental than our consciousness is a very interesting object. Let say we have a predicate [itex]IsCons(x)[/itex] which is true when system [itex]x[/itex] is conscious. What we know is that for any system we know all the details about, isCons is false: [itex]isCons(\overline{x})=0[/itex] Note that [itex]\overline{x}[/itex] is constant, not a variable. In another words:

    [itex]\forall X \vdash IsCons(X) = 0[/itex]
    (for every constant X we can prove that isCons(x) is false). At the same time, conscious objects do exist:

    [itex]\vdash \exists Y : IsCons(Y)=1[/itex]

    Is it a contradiction? NO, even it is conter-intuitive. In mathematics many object possess the same weird property. Look for example at famous http://en.wikipedia.org/wiki/Banach–Tarski_paradox
    There exists a decomposition of a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. That sounds really impossible because we can’t imagine such decomposition. But can we provide any examples of Banach-Tarsky decomposition?

    No, this is absolutely impossible! Banach-Tarsky theorem is a consequence of AC (Axiom Of Choice). If it would be possible to provide an example of such decomposition, then the example itself would be a proof, so no AC would be needed. Hence providing an example is not possible.

    The weird property I described is called [itex]\omega[/itex]-inconsistency, which is weaker then inconsistency. Formal arithmetic is [itex]\omega[/itex]-consistent. There is undecidable Goedel statement G, however. You can add G or [itex]\neg G[/itex] as a new axiom, but only one choice will be [itex]\omega[/itex]-consistent

    Unfortunately, more complicated theories (Set Theory) are [itex]\omega[/itex]-inconsistent from the very beginning (for other examples check Continuum Hypothesis for example). We see how these weird properties emerge on some level of complexity. Of course, it is not a proof that human consciousness actually has weird property I explained above. However, it shows that the knowledge about the structure of a system and about how that system works might ‘non commute’ with some properties of that system. You can know all the details about the dead brain, not about the alive one.

    Hence many ‘proofs’ in the QM section fundamentally flawed.
  2. jcsd
  3. Sep 14, 2009 #2

    Doc Al

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    Staff: Mentor

    I won't pretend to understand your Banach-Tarsky theorem argument, but can you at least give some kind of definition of "free will"? I see many arguments going round in circles because folks are assigning different meanings to the term.
  4. Sep 14, 2009 #3
    'Free will exists' is just a way of saying 'consiousness is non-deterministic'

    System is deterministic if the next state of the system is defined by it's previous state:

    [itex]\exists_1 X' = next(X)[/itex]

    However, as I explained above, X cant be found. So the statement above is undecidable. Hence the statement 'In the determinsitic world, consciousness is also deterministic' is neither true or false, it is undecidable.
  5. Sep 14, 2009 #4

    Doc Al

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    Basically you're saying that quantum mechanics (in the standard interpretation) adds a random factor, thus the next state cannot be determined from the previous. Is that an accurate summary?

    (On the side: A randomly changing brain state is not what most folks want when they say they want "free will".)
  6. Sep 14, 2009 #5
    No. My argument is also applicable to even very complicated mechanical systems, or, say, the imaginary universe of the Conway Game Of Life. I meantioned HUP just as an example.
    But I was mostly thinking about the BM and MWI as deterministic interpretations. 'Standard' CI is a total piece of crap (I dont say it in QM but here I can do it I guess?)
  7. Sep 14, 2009 #6
    Isn't that the definition of a contradiction?

    1. There is no such thing as a conscious object. (your 1)
    2. Y is an object.
    3. Y is not conscious. (from 1 and 2)
    4. Y is conscious. (your 2)

    Contradiction: Y is conscious and Y is not conscious.
  8. Sep 14, 2009 #7

    Doc Al

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    I'm still not getting it (or perhaps just not buying it). Just because you cannot predict what's next, doesn't mean it's not determined. Of course for many practical purposes, you can treat the systems as if they were "undetermined". But this is the philosophy section, after all.
    I'd certainly agree that what's called the CI (above and beyond a minimalist standard QM) creates more "mystery" than it solves. :wink:
  9. Sep 14, 2009 #8
    no. check the position of the |- sign :)
    yes, I know it is tricky

    If you can prove A(0), A(1), A(2), ...
    so if you can prove A(x) for any constant x
    DOES NOT MEAN that you can prove [itex]\forall X : A(x)[/itex] (*)
    If the last statement is undecidable then you can assume (*) or not (*) as an additional axiom.
    Assuming (*) extends formal arithmetic keeping it [itex]\omega[/itex] consistent
    Assuming not (*) making it [itex]\omega[/itex] inconsistent but consistent
  10. Sep 14, 2009 #9
    In my example only systems with UNKNOWN structure can be conscious. To check if system is deterministic we need to compare 2 states: before and after. But we cant obtain even a single state. Hence the question "is mind deterministic" does not make any sense.
  11. Sep 14, 2009 #10
    Do you mean unknowable?
  12. Sep 15, 2009 #11
    Yes, correct.
  13. Sep 15, 2009 #12
    Does that mean all systems with unknowable structure are conscious?
  14. Sep 15, 2009 #13
    No of course....
  15. Sep 15, 2009 #14
    Define "system" please.
  16. Sep 15, 2009 #15
    So what makes it conscious?
    And how does a non-determinist, non-conscious, system decide what to do?
  17. Sep 15, 2009 #16


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    Dealing with just the OP's attempt to abuse the BT theorem here, BT is a challenge to the axiom of choice rather than some kind of reason to believe the impossible is true.

    BT depends entirely on the magic of an infinity of points and so you have to ask in reality whether physical locations (which are what zero-D points are meant to model) are conserved or freely produced.

    So yes, axiom of choice is useful in modelling reality. But no, it has limits to its utility. It breaks down when pushed to an extreme - as vividly demonstrated via BT theorem.

    Consciousness is a real world phenomenon and so unaffected by paradoxes that arise in mathematical models. BT in itself would be irrelevant to it.

    Of course I agree that consciousness does not reduce just to neural firing patterns or any other local actions. You can only start to model consciousness using systems approaches which include both bottom-up and top-down causality in interaction. A complex causality.

    A nice word for the naive reductionist approach is isomerism. Take a living creature and break it down into a test tube brew of its constituent chemicals. Surprise, surprise, the secret of life seems to have eluded you. Break down the molecules to QM level entities - wavefunctions measured to be at locations - and you have lost view of even the chemistry.

    The OP is worried that it is impossible to measure the whereabouts of every last particle in the brain due to QM measurement issues. But how is this even relevant if it is the organisation, the form, the pattern of relationships, that is causal in systems?

    The thermal jostle in real brains means that the constituent atoms are a statistical blur anyway. Nothing has to be located in exactly some place, just fairly roughly in a useful place. A little research into the topic of molecular turnover will show just how coarse-grained things actually are. The half life of microtubules is around 4 seconds for example! And those are structural macro-molecules.
  18. Sep 15, 2009 #17
    I wish I knew! :)
    I don't know any solutions to the 'Hard problem of consciousness'
  19. Sep 15, 2009 #18
    It is not a fault of AC. There are other examples: GCH or unreachable sets.

    When we prove a statement saying that [itex]\exists x : A(x)[/itex] we say: aha, x actually exists! But in fact, all we did is we played with character strings based on some rules, and as a result of such game we got a string which begins from a character [itex]\exists[/itex]. That is all! This a nightmare and damnation of the whole mathematics!

    And that whole crazy mathematical stuff leaks into physics, because we can not avoid using sets and real numbers in physics.

    I also wanted to attract an attention to another interesting fact. You definitely know the Conway Game of Life? Simple and deterministic Universe with integer time. 'Law of physics' in that Universe are effectively computable, deterministic and again very simple.

    It is quite shoking that still there are undecidable statements about the configurations of dots in that universe!
  20. Sep 15, 2009 #19
    You can only prove a logical statement about existence by in some way assuming the existence of something. It's always a conditional proof. Math has nothing to do with existence. Math is the game. Or was that your point?
  21. Sep 15, 2009 #20
    Exactly. Math is a game.

    Like in QM, in math there is a Formalism (character [itex]\exists[/itex]) and Interpretation ([itex]\exists[/itex] means 'exists') (well, it is called a 'model')

    Like different Interpretaations in QM, there are many different and mutually inconsistent mathematics: with AC or without AC, with CCH or without it, with classes or jsut pure ZF.
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