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Part of the reason is that the math is in fact horribly complicated, and there is very little we can do about that.
Most of the math that we have for these kinds of systems only allow for a very small number of parameters and a very small number of variables. Our mathematical understanding of dynamical systems is really not that great; we haven't even solved 3-dimensional non-compressible fluid dynamics.
Since we don't have math that can take a lot of parameters and variables and simplify them down to something that humans can make sense of, we run into two obstacles when we try to apply the methods we do have to these problems: chaos, and emergent effects.

Chaos is when very small changes in the starting parameters create really large differences down the road. This means that our inaccurate, imprecise measurement tools render any modelling we do useless. Unless we know exactly what the parameters are, even at a single moment, we can't say where the system will end up.

Emergent effects are complicated patterns resulting from a lot of things interacting according to very simple rules. For example, ants are very simple creatures, but when you get a lot of them, you get the almost intelligence that anthills exhibit. Unfortunately, we don't quite understand how that simple behavior leads to the complicated behavior.

Both of these obstacles straddle comp-sci and mathematics and, coupled with our still budding understanding of multivariable non-linear diff-eqs and other non-perturbative systems, make it almost impossible to make such a notion as the brain as a wave function collapser verifiable.

Ick. I hope it won't go like Penrose vs. the world again.


How would one even start going about 'proving' such a claim experimentally, though? What does it mean to be conscious and what constitutes a thought?

Personally (and this view is not generally accepted or backed by hard evidence), I have lot of sympathy for the view that there is simply no such thing as wavefunction collapse. Decoherence already explains how classicially-additive behavior can arise out of large quantum systems, although the proposal is not strong enough to resolve the measurement problem. I suppose all I can reasonably say about it is that I "hope" it will be solved without collapse.


The most basic assumption about the universe is that it is self-consistent. As long as that holds, it's definable by some mathematics. Sometimes we have the wrong math Other times, we have a third option: we have the right math, but not enough of it--or simply said, we aren't good enough at math. Layra-chan has a perfect example: the Navier-Stokes equations describe the behavior of fluids, but the specific behaviors of fluids aren't well-understood at all. Defining a system mathamatically is different from being able to solve it. For example, Einstein thought no one would ever find an exact solution in GTR (interestingly, he derived the deviation of light without the Schwarzschild solution as it is normally done in textbooks). Happily, he turned out to be incorrect, but there are well-formulated systems that still resist solution.

But your question is more interesting than that. I have a nagging suspicion that yes, we have quite the wrong math, and that part of the reason why measurement and collapse present such difficulties is that we're not even framing the problems correctly. The reason for this suspicion on my part is the correspondence between commutative C*-algebras and locally compact Hausdorff spaces (which form the sprectra of the algebras); the C*-algebras are in turn representable as Hilbert space together with a certain *-homomorphism. So, in classical field theory, we have something like:
[Hilbert space] ←→ commutative C*-algebra of observables ←→ ordinary space
whereas we can strongly interpret quantum field theory to have (this also occurs in certain limits of string theories):
Hilbert space ←→ noncommutative C*-algebra of observables ←→ "noncommutative space"
Noncommutative C*-algebras can have very complicated structure but no characters at all (i.e., empty spectrum). Thus, they 'correspond' to spaces with no points at all and yet be very richly structured. Perhaps the "solution" to the "measurement problem" of why things are measured to be in definite states is that they simply aren't--if the space has no points, the very concept may be incoherent.

The above paragraph is speculative and represents just one of the classes of QFTs out there, and thus should not be taken too seriously. Yet.

Layra-chan, IIRC (correct me if this is wrong), you disliked point-set topology but had no problem with algebras. The relatively new development of pointless topology might be right up your alley. And if it turns out that you dislike algebras as well... well, insert a bad pun on Heyting algebras here.


But we do have separate theories! No engineer would use quantum mechanics to build a bridge. The laws of thermodynamics are a good example of 'chunking' very complicated microscopic bevahior into a very simple (er, simpler) macroscoping description. Or do you mean to propose that nature may force a continuum of qualitatively different theories each valid at particular scale with nearly no commonality between them? I suppose it might be possible in the abstract, but having nature be so schizophrenic would certainly be highly surprising.


If "infinitesimal error" means "exact" and "prediction" means "prediction abitrarily far away", then yes, but only if the system is deterministic in the first place. If it means something a bit looser, like "can be determined with arbitrary accuracy", then it's always enough, e.g., if one is unlucky enough to be around a bifurcation. Although the we can say that the set of such problematic points has measure zero, and so if you're only interested in making your predictions in some fixed interval, it's (in principle) possible to pin down the starting point enough to make arbitrarily precise predictions. Chaos theory doesn't say that it's impossible to make accurate predictions. It just says that if you're trying to do so far away, you need a lot of precision in your measurements and model--this quickly grows more than is feasible for any just about any "interesting" situation.