Hypercomputation is not known to be possible and our brain has not been observed to exhibit it. If it did, we would be able to solve much harder problems than we currently can. Here are some models for hypercomputers that clearly outperform a brain (often dealing with infinite precision or an infinite number of calculations in a finite time), and are also not possible: http://en.wikipedia.org/wiki/Hypercomputation#Hypercomputer_proposals
Exactly. Hence the halting problem remains relevant in reality. However, even if hypercomputation were possible, the problem of undecidability is still present
From the wiki on hypercomputation
He used this device to prove that even in those more powerful systems, undecidability is still present. Turing’s writing made it clear that oracle machines were only mathematical abstractions, and were not thought of as physically realizable.
Also, from this article on the Halting Problem
Oracles have their own halting problem:
What’s interesting, though, is that these hypercomputers, computers with the black box, have a Halting Problem of their own: Given a hypercomputer program (i.e., a sequence of 1′s and 0′s, which might include special instructions to invoke the Oracle), and an input, determine whether or not the hyperprogram will freeze when run on the input.
A hierarchy of computability ad infinitum:
You can keep going like this forever, and it gives rise to a hierarchy of computability, with different tasks falling into different levels of computability.
This is reminiscent of Cantor’s hierarchy of infinities.
You’re not saying it’s less than turing complete, are you?
What I am saying is, we don’t know exactly what the brain is. All we know is that it is has fractality like other biological organisms and natural objects in nature. OTOH, a computer is not fractal at all.
I’m glad you are doing some research, kkwan. Unfortunately, you have some misconceptions. The halting problem is deciding whether or not a program will halt. This can be undecidable for some programs. The halting problem has no relation to turing completeness. However, given a hypercomputer, the halting problem can be decided for all programs. The brain cannot solve the halting problem for all programs, so there is some evidence right there that it is not a hypercomputer (although it should be obvious enough that it is not given that it cannot compute an infinite amount of things in a finite amount of time).
A Turing complete machine requires an explicit HALT state. If there is a halting problem, there is no explicit HALT state. OTOH, a hypercomputer (if it is possible) will have its own halting problem as shown above. Computing an infinite amount of things in a finite period of time is called a supertask.
Fractals have nothing to do with the halting problem. In nature, fractals are not infinitely detailed anyway. They only are in math.
From this paper HERE
We show that even under the classical theory of computation over the rational numbers, in which the Turing machine is the model of computation, one can prove some questions about fractals to be undecidable.
Is the brain fractal? (Not that it matters for this argument)
From this article
Researchers from the University of Cambridge took a big step forward this year in understanding how our brains work. It seems that the brain has a fractal organization.
It does matter because computers are not fractal. Thus, the organization of the brain is fundamentally different to that of the computer and as such it cannot be compared to a computer at all.
According to the laws of physics, yes, it is possible. It’s all computation, and it’s not infinite, so it’s possible.
But, the brain is a living biological organism and cannot be purely described by the laws of physics. Also, it is not just “all computation” as you put it so simply.
Since it is not, the distinction between analogue and digital is not relevant for the question of turing equivalence between computers and brains.
The issue is, analog and digital computers work differently. Each have their strengths and weaknesses. Thus, the analog fractal brain has tremendous advantages over digital computers in situations where computations are astronomical, tedious, time consuming and there could also be a halting problem as well.
Why do you keep bringing this up? You know it is only a matter of time until a Go champion is beaten by a computer, just like Jeopardy before it, and Chess, and Checkers before that.
No, I don’t think it is only just a matter of time. Why not now, or is it that the most powerful supercomputer available now is not capable to do that at all?
Would you like to bet on the computer beating a Go master?
Go is entirely different in nature and complexity to the other games mentioned by you.
From the wiki on Go
Decisions in one part of the board may be influenced by an apparently unrelated situation in a distant part of the board. Plays made early in the game can shape the nature of conflict a hundred moves later.
Since <this device that was made by aliens> is not designed/constructed by humans, it follows that <this device that was made by aliens> is not a machine and cannot be conceived/characterized as such.
It is that simple. “This device that was made by aliens” IS an alien machine. As such, it is just as inane, inappropriate and absurd as putting “an alien machine” into the sentence.
In the context of the sentence I wrote, it was specifically “the brain” and the meaning of the sentence should only be interpreted as such. Putting anything else in place of that is preposterous and a travesty of my intention in composing the sentence.