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> The brain has no such constraint. It is analog, and therefore infinite in State representation.

This is a common misconception.

I'm sure you are aware that analog signals can be approximated by digital values -- a 10 bit ADC will read a channel to one part in 1024, etc.

You might say that even a 64 bit representation is a poor approximation of a real life signal, which is a real number with infinite precision... But it isn't.

The brain operates at about 300 Kelvin, and so there's a noise floor to all analog signals of about that times Boltzmann's constant, or 10^-20 J. If a neuron impedance is 1 ohm, and at a bandwidth of just 10 kHz, the thermal noise is about 1 nV. For a membrane potential of 100 mV, that's a maximum possible noise to signal ratio of one part in 100 million, which is 26 bits.

Now the brain could depend on the signal below the noise floor, but if so those would be extremely fragile operations, and you could get the same thing on a computer by padding your numbers with random data.

Given how robust a brain is against noise, I'd be surprised if any brain signals are more precise than an equivalent of 3-4 bits.

I agree, and I think in practice the brain's noise floor is also much higher than the theoretical thermal-noise minimum. But I guess the main point is that once we acknowledge that even 32 bits is more than enough, the difference between an analog and digital machine loses a lot of its philosophical weight.

> The brain cannot simulate a turing machine. It does not have infinite memory, which is a requirement for a turing machine.

In practice we call modern computers turing-complete even though they don't have infinite memory. The brain can simulate such a machine.

> The brain has no such constraint. It is analog, and therefore infinite in State representation.

If this mattered, then it would mean analog computers are more powerful than digital computers and therefore the Church-Turing thesis is wrong

Regarding the Church-Turing thesis, it is exactly that, just a thesis. Again, akin to P vs NP. It seems to hold for most cases, but is not proven.

The reason that it's difficult to apply in regards to the brain is that we don't exactly know how the brain is computing... or if it "computes" at all! To my knowledge, we don't have a model of computation for consciousness, emotion, free will, Etc.

Perhaps these are better classified as emergent Behavior rather than computation, but if that is the case I still don't know of a model explaining what computations or rules give rise to the emergent Behavior.

Perhaps the problem is in our definition of computation and what it means to compute.

We do know that the cardinality of the set of possible computational problems is larger than the cardinality of the set of all possible Turing machines. This is provable by simple diagonalization proofs.

The question, then, is whether or not the computations of the brain fall Within the set of Turing recognizable languages (computational problems). To my knowledge, this has not been shown.

As far as I understand, the prevailing opinion is that the brain is a physical object and that its operation does not involve currently-unknown laws of physics (because we have a good understanding of what happens at the scale of an entire atom or above).

A Turing machine can run a simulation based on such physical laws to any desired level of precision (which is enough, because as mentioned in TFA, processes in the brain aren't individually very precise). This is true because of the nature of these laws, which are mostly just asking you to integrate differential equations. If you accept this, then it should follow that a Turing machine can in fact simulate a brain: just run a physics sim on a brain's initial state.

(I do realize that this is far outside the realm of what's doable today, but it seems to provide a solid justification for why it's conceptually possible).

"any desired level of precision" is actually the issue. The moment you choose a level of precision, you cease being accurate (at that level). If you make the argument that a TM has infinite memory, and can therefore represent an infinite precision, then I would counter that our current defintion of a TM requires a finite tape alphabet (and finite number of states), which is part of the TM's known computational limitations. And, of course, the moment that you use any finite set of symbols to represent an infinitely precise value, you fall into the problem that the set of real numbers has a larger cardinality than the set of possible turing machines (again, simple proof via diagonalization).

It is possible that the brain's imprecision (I would argue that "inconsistency" might be a better word) is a requirement of it's computational ability. Again, we haven't defined how the brain computes, nor do we have a model for explaining its computation, encoding or representation of knowledge, or emergent behavior. We have observed phenomena related to some of these things, but we are far from understanding it. It may be that the computational processes are dependent on the surrounding environment. We know that the biological processes are influenceable by the physical world, but we do not know much about how these external forces affect, limit, or are required for, the process of brain computation.

The quantum world may play a part in consciousness (or no, we don't know). Non-determinism may play a part. It is possible that, in order to simulate a brain, one would have to simulate the entire universe around it in order to predict the behavior... meaning that it may well require a universe to perform the simulation.

Which brings us to a related theory of whether or not we are living in a simulation, but I digress... :)

> It is possible that the brain's imprecision (I would argue that "inconsistency" might be a better word) is a requirement of it's computational ability

Is it possible that brain is in fact a quantum computer? I can imagine that under all those neural networks there is a small part where, trapped in some complex protein structure, some qbits exist and are crucial to most advanced brain functions, such as consciousness.

"Is it possible that brain is in fact a quantum computer?"

It's an interesting thing to ponder.

Quantum computing is still just another computational model, and it's main Advantage is that it involves non determinism. But non determinism, in and of itself, can be modeled by deterministic computer.

I think the biggest problem is that we don't understand what computation is taking place in the brain, or even if it is "computation" according to our current definition of the word. I think that this issue is the biggest problem in reconciling whether or not it is possible to accurately model the human brain.

> that its operation does not involve currently-unknown laws of physics (because we have a good understanding of what happens at the scale of an entire atom or above)

Well, we know certain approximations of those laws. Purely theoretically, it is possible that the exact laws at some level of detail that we have not yet been able to observe involve functions that are not computable by a Turing machine, and then it is theoretically possible that the brain itself is computing functions which are not computable by a Turing machine (this would of course assume that the Church-Turing thesis is actually wrong).

As long as the Church-Turing thesis is not proven, we can't say with absolute certainty that the physical world can be simulated to any level of detail by a Turing machine.

Furthermore, even if the Church-Turing thesis was proven, is it possible that the physical world involves transformations that are not even computable at all (even if they can be approximated by computable functions)?

Just to be clear, I do not believe these things. But it is fun to think about the limits of our knowledge.

Isn't the recent Google quantum "supremacy" experiment evidence against the extended Church-Turing thesis?

No, quantum computers as we understand them can be simulated by a turing machine

The extended Church-Turing thesis which I specifically referred to concerns efficient simulation, not just whether it can be simulated.

Google has not proved quantum supremacy, it is a scam. They have proved the truism that running a physical system is faster than running a simulation of a physical system...

https://www.nature.com/articles/s41586-019-1666-5

What part of the experiment in the paper released did you feel like was inadequate?

I mostly agree with your post but:

> The brain has no such constraint. It is analog, and therefore infinite in State

Not necessarily infinite. A lot of people believe that nothing in the world is truly infinite (just very large/small). Infinite quantities in mathematics are just approximations that simplify calculations.

If you go to a sufficiently high precision, neurons and their communication are discrete - the number of neurotransmitter molecules and ions transferred across any synapse is countable, so the number of states (even if we ignore noise and noise tolerance, which we shouldn't ignore) is finite.

The big question is whether a CPU can emulate a brain with the same or better efficiency.