Free will, Determinism and Banach-Tarsky paradox

**Dmitry67** · Sep14-09, 08:34 AM

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

which is true when system

is conscious. What we know is that for any system we know all the details about, isCons is false: $LaTeX Code: isCons(\\overline{x})=0$ Note that $LaTeX Code: \\overline{x}$ is constant, not a variable. In another words:

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

$LaTeX Code: \\vdash \\exists Y : IsCons(Y)=1$

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 $LaTeX Code: \\omega$ -inconsistency, which is weaker then inconsistency. Formal arithmetic is $LaTeX Code: \\omega$ -consistent. There is undecidable Goedel statement G, however. You can add G or $LaTeX Code: \\neg G$ as a new axiom, but only one choice will be $LaTeX Code: \\omega$ -consistent

Unfortunately, more complicated theories (Set Theory) are $LaTeX Code: \\omega$ -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.

**Doc Al** · Sep14-09, 08:47 AM

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.

**Dmitry67** · Sep14-09, 08:53 AM

'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:

$LaTeX Code: \\exists_1 Xsingle-quote = next(X)$

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.

**Doc Al** · Sep14-09, 09:00 AM

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".)

**Dmitry67** · Sep14-09, 09:04 AM

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?)

**kote** · Sep14-09, 11:20 AM

Originally Posted by Dmitry67

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

$LaTeX Code: \\vdash \\exists Y : IsCons(Y)=1$

Is it a contradiction? NO

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.

**Doc Al** · Sep14-09, 11:30 AM

Originally Posted by Dmitry67

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.

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.

'Standard' CI is a total piece of crap (I dont say it in QM but here I can do it I guess?)

I'd certainly agree that what's called the CI (above and beyond a minimalist standard QM) creates more "mystery" than it solves.

**Dmitry67** · Sep14-09, 11:34 AM

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 $LaTeX Code: \\forall X : A(x)$ (*)
If the last statement is undecidable then you can assume (*) or not (*) as an additional axiom.
Assuming (*) extends formal arithmetic keeping it $LaTeX Code: \\omega$ consistent
Assuming not (*) making it $LaTeX Code: \\omega$ inconsistent but consistent

**Dmitry67** · Sep14-09, 12:06 PM

Originally Posted by Doc Al

Just because you cannot predict what's next, doesn't mean it's not determined.

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.

**JoeDawg** · Sep14-09, 11:11 PM

Originally Posted by Dmitry67

In my example only systems with UNKNOWN structure can be conscious.

Do you mean unknowable?

**Dmitry67** · Sep15-09, 02:26 AM

Yes, correct.

**JoeDawg** · Sep15-09, 02:52 AM

Originally Posted by Dmitry67

Yes, correct.

Does that mean all systems with unknowable structure are conscious?

**Dmitry67** · Sep15-09, 04:46 AM

No of course....

**Willowz** · Sep15-09, 05:16 AM

Originally Posted by JoeDawg

Does that mean all systems with unknowable structure are conscious?

Define "system" please.

**JoeDawg** · Sep15-09, 05:50 AM

Originally Posted by Dmitry67

No of course....

So what makes it conscious?
And how does a non-determinist, non-conscious, system decide what to do?

**apeiron** · Sep15-09, 07:43 AM

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.