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Why Roger Penrose thinks computers can't
21–30 of 37 posts
Re: Why Roger Penrose thinks computers can't
#22I bought and read "The Emperor's New Mind" in the early 90s, mostly for "knowing your enemy." I have slowly but surely come to mostly agree with Penrose: I think there is something magical and "quantum mechanical" about brain consciousness (including animals). I believe in eventual real AI, but I would guess that it will not be on current computer hardware.
Why is quantum physics magical? Or is that not what you wanted to say?
As far as I know we have a pretty clear understanding of quantum physics.
Re: Why Roger Penrose thinks computers can't
#23Earlier quoted context omitted.
From Wikipedia: The true but unprovable statement referred to by the theorem is often referred to as “the Gödel sentence” for the theory. It is not unique; there are infinitely many statements in the language of the theory that share the property of being true but unprovable.
Don't the existence of thing like the Gödel sentence show that there are things that are true but which humans can't prove, contradicting the claim that this is an important distinction between humans and computers?
I'll try to give an overview of what's going on. In the late 1800s Peano came up with his set of axioms for the integers. This set is second order which means it isn't computable. Hilbert and others wanted a computable set of axioms so that one could, in theory, remove humans from the discovery of mathematical truth.
So the first order axioms were invented. Godel showed that the first order axioms are not strong enough to prove all true statements about the integers. There are infinitely distinct models of the first order integers. There is only one model of the second order integers.
In the proof of the Incompleteness Theorem Godel proves a statement about an integer which is not provable in the first order system.
Humans can work in the second order system but not computers. So, from my perspective there is something different about human thought patterns. I'm not an expert and would like to know what is wrong with my reasoning.
Re: Why Roger Penrose thinks computers can't
#24I was annoyed at how the article just blithely asserted that wavefunctions collapse, as if that wasn't a matter of ongoing debate. In fact, Penrose is remarkable among big name physicists for being sure that wavefrom collapse occurs.
http://www.hedweb.com/manworld.htm#believes
Re: Why Roger Penrose thinks computers can't
#25Earlier quoted context omitted.
From Wikipedia: The true but unprovable statement referred to by the theorem is often referred to as “the Gödel sentence” for the theory. It is not unique; there are infinitely many statements in the language of the theory that share the property of being true but unprovable.
How does that prove in any way that the human mind can find all true statements in Mathematics and thus go beyond the limits of computability? Do you have any proof of this? Also, do you have any proof that the human mind is in fact consistent (meaning it can't possibly reach A and ~A at the same time)?
I'm not claiming that humans can discover all true statements in mathematics. I'm saying the Incompleteness Theorem demonstrates that such a task is not computable and in the proof of the theorem a statement that can't be proven in a computable system is proved. So....can a computer ever reach the same level of reasoning? I'm skeptical.
Re: Why Roger Penrose thinks computers can't
#26The argument that computers can't think derives from the idea that there are noncomputational processes at work in the brain. Essentially, we don't know how certain thoughts arrive in our mind. We can't create an algorithm to mimic our chain-of-thought generator. But that doesn't mean we can't create a computer that can have similar noncomputational "thoughts". People must choose, when a computer need not choose. Wha…
"People must choose, when a computer need not choose."
Perhaps I'm not understanding something, but that seems to be obviously false. A human always has the option of being unsure in the face of non-computable statements like "You can't know that this sentence is true" or such.
Re: Why Roger Penrose thinks computers can't
#27Earlier quoted context omitted.
From Wikipedia: The true but unprovable statement referred to by the theorem is often referred to as “the Gödel sentence” for the theory. It is not unique; there are infinitely many statements in the language of the theory that share the property of being true but unprovable.
How does that prove in any way that the human mind can find all true statements in Mathematics and thus go beyond the limits of computability? Do you have any proof of this? Also, do you have any proof that the human mind is in fact consistent (meaning it can't possibly reach A and ~A at the same time)?
[deleted]
Re: Why Roger Penrose thinks computers can't
#28Earlier quoted context omitted.
How does that prove in any way that the human mind can find all true statements in Mathematics and thus go beyond the limits of computability? Do you have any proof of this? Also, do you have any proof that the human mind is in fact consistent (meaning it can't possibly reach A and ~A at the same time)?
I'm not claiming that humans can discover all true statements in mathematics. I'm saying the Incompleteness Theorem demonstrates that such a task is not computable and in the proof of the theorem a statement that can't be proven in a computable system is proved. So....can a computer ever reach the same level of reasoning? I'm skeptical.
Solving a particular instance of a noncomputable problem is not the same than solving the problem itself. Eg, the busy beaver problem is noncomputable, but it's trivial to make a machine output solutions for simple cases in the same way that it's trivial for a human to do so. In fact, you cannot prove that what we are doing is essentially programming ourselves to solve ad-hoc cases of the general problem, which is computable, as opposed to somehow having in our heads a general way of solving the problem, which we know it's not computable.
Basically, what you need to prove to show that humans are intrinsically more powerful than machines is that we have the ability to generally solve noncomputable problems in a general way, ie, that we are hypercomputers.
Re: Why Roger Penrose thinks computers can't
#29I bought and read "The Emperor's New Mind" in the early 90s, mostly for "knowing your enemy." I have slowly but surely come to mostly agree with Penrose: I think there is something magical and "quantum mechanical" about brain consciousness (including animals). I believe in eventual real AI, but I would guess that it will not be on current computer hardware.
Why is quantum physics magical? Or is that not what you wanted to say? As far as I know we have a pretty clear understanding of quantum physics.
I only have a BS in Physics (UCSB) so this may be rough: about 10 years ago I went to a Quantum Mechanics and Consciousness conference and the gist is that there is a quantum effect that is required consciousness. My dad taught physics at Berkeley and his friend from Berkeley Henry Stapp presented an interesting paper on this theory. The philosopher David Chalmers was also there and he talked about the difficult question of consciousness: why did evolution favor the development of consciousness and qualia (an inner mental life)?
Re: Why Roger Penrose thinks computers can't
#30Earlier quoted context omitted.
Don't the existence of thing like the Gödel sentence show that there are things that are true but which humans can't prove, contradicting the claim that this is an important distinction between humans and computers?
I'll try to give an overview of what's going on. In the late 1800s Peano came up with his set of axioms for the integers. This set is second order which means it isn't computable. Hilbert and others wanted a computable set of axioms so that one could, in theory, remove humans from the discovery of mathematical truth. So the first order axioms were invented. Godel showed that the first order axioms are not strong enou…
Ok, I'll confess that I'm not sure what you're talking about when you mention second order systems. However, Godel's incompleteness theorems (plural) were a lot more general than you're suggesting.
First Theorem: "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory."
Second Theorem: "For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent."
It wasn't that Godel proved that Peano arithmetic was incomplete that shocked mathematicians, it was that he showed that _no formal system_ could ever be complete as long as remained consistent.
Of course maybe this second order system isn't consistent, in which case there's no problem. After all, I'm fairly convinced that human reason isn't always perfectly consistent, so it makes sense that we can avoid Godel's theorems. Of course, I see no reason we couldn't make a slightly inconsistent computer either.