[Chronos]

Ugh, this again.

1. Let's suppose for sake of argument that humans really can see the inherent truth of "Peano Arithmetic is consistent". That doesn't mean humans violate Gödel's Incompleteness Theorem: it could just mean that humans use axioms stronger than PA.

2. Gödel's Incompleteness Theorem only applies to systems that are perfectly logically consistent. Not sure how Penrose didn't notice, but humans... aren't.

3. When scientists proposed Quantum Mechanics as a replacement for Classical Mechanics, it was on them to explain how Quantum Mechanics simplified to Classical Mechanics in the common case. "Penrose Mechanics" is an even more radical departure — especially from a physics of computation standpoint, as Penrose Mechanics by definition would allow solving at least some of the problems in (ALL - R) in ~polynomial time. Penrose needs to explain how Penrose Mechanics reduces to Quantum Mechanics in the common case.

4. Penrose proposes that (a) there exist new physics, (b) that evolution has learned to computationally exploit the new physics via microtubules, and yet (c) that humans are the only lineage to make use of this feature of microtubules, even though microtubules are found in all eukaryotic cells (from mushrooms to amoebae). From a predator-prey standpoint alone, it would seemingly be a huge evolutionary advantage to be able to compute NP or R functions in polynomial time. (That ability is not _strictly_ implied by Penrose Mechanics, but it's a very likely consequence.) Penrose needs to explain why only humans are taking advantage of the computational power of microtubules, when microtubules have existed for billions of years and across millions of species. (TL;DR: It's the pineal gland all over again.)

[manyoso]

#1 Please show how to program a Turing machine to use "axioms stronger than PA"

#2 Please show how to program a Turing machine so that it is not logically consistent at a fundamental level?

#3 Are you questioning the viability of objective collapse theories in general? To date we have no experimental evidence of said theories, but I don't think anyone has suggested they are inconsistent with QM. Are you? If so, please show how...

#4 Show where Penrose has said humans are only species to make use of microtubules and your claims regarding Penrose' theory of objective collapse and having staggering new computational speedups are ... without any evidence. Please provide some.

The tenor of your comment suggests you find Penrose to be an idiot and missing obvious problems. Consider that perhaps you are the one misunderstanding what he is proposing and that he is not, in fact, an idiot.

[pdkl95]

> how to program a Turing machine to use "axioms stronger than PA"

http://www.scottaaronson.com/blog/?p=2725

"... a one-tape, two-symbol Turing machine with 7,918 states, whose behavior (when run on a blank tape) can never be proven from the usual axioms of set theory, under reasonable consistency hypotheses."

[Chronos]

1. Easy. Program a computer with the axioms of ZFC. Not powerful enough? Program it with ZFC+Con(ZFC). Repeat as necessary.

2. The Turing machine itself is logically consistent. The semantic interpretation of its data need not be logically consistent: it's possible to write a computer program that prints "2+2=5" to the screen. Likewise, a human brain may be made of physics and physics may be logically consistent, but the semantics of the data in the brain (the "beliefs") need not be consistent.

3. No, no need to reject collapse itself (though I'm more of an Everett guy). Penrose postulates the existence of a new level of physics that is not Quantum Mechanics. People who aren't careful call it an extension of QM, but QM's computing model is limited to solving a small subset of NP problems (BQP) in polynomial time, and BQP is theorized to be a proper superset of P. However, Penrose's proposed physics would allow solving problems outside of NP in polynomial time. Hence, Penrose's physics IS NOT quantum physics.

4. It is physically impossible for computers (as we understand them today) to solve problems outside of the set we call "Turing complete". Penrose claims that there's at least one problem outside that set that the human brain can nonetheless solve: checking the consistency of formal systems. If Penrose's assumption were true, computers (as we understand them today) would consequently be strictly less powerful than the human brain. The crux of Penrose's position is accepting this consequence as true, then asking how that's possible. I reject that hypercomputation of problems beyond Turing-completeness is what's happening, i.e. that humans simply hold unjustified beliefs about the answers to such problems.

I do not doubt that Penrose is an intelligent and educated person. However, his expertise is General Relativity. It's bad enough when he tries dipping his toes into Quantum Physics, but when he wades into Physics of Computation, he is no longer acting as an expert, but as an interested layman. It's only because of the authority he holds as a GR expert that his argument is treated seriously, but that authority is not actually relevant to his argument.

[manyoso]

#1 Not easy. Prove it. Show a Turing machine programmed with ZFC that can not be modeled by a Turing machine programmed with PE. You can not do so because Turing machines are universal. The fact that you claim this is easy tells me you do not understand what a Turing machine is.

#2 See fundamental level. Penrose claims that what sets us apart from Turing machines is fundamental. But whatever, you've refuted whatever point you wanted to make by saying human's are not logically consistent. Your snark about Penrose not understanding is as empty as whatever point you were trying to make.

#3 Cite a paper showing that objective collapse gives these supposed algorithmic speedups or go home. NOTE: objective collapse theories are an active area of research and are not limited to Penrose by any means. Your claim that they are inconsistent with QM begs for evidence. Cite some or stop spreading nonsense.

#4 Your answer here is completely void of any context to the question: where Penrose says that microtubules are limited to humans. I take it you concede that he did not say any such thing?

You treat Penrose as an idiot missing obvious problems. I think it far more likely that you've misunderstood. Pity your lack of humility might make it impossible for you to understand what he actually argues rather than your strawmen.

[Chronos]

Also: before I bother building a Turing machine that implements a proof-generator for statements in ZFC, you should do me the courtesy of showing your investment by building me a Turing machine that multiplies two integers.

Turing machines suck. Building a Turing machine that implements ZFC proof-generation is a project appropriate to a graduate-level paper, not something to toss off in an Internet pissing contest.

[manyoso]

You were the one who used the phrase "easy" with regard to Turing machines and feasibility of showing them.

[Chronos]

Fine. "Straightforward, if menial and tedious".

[pdkl95]

> You can not do so because Turing machines are universal.

Just because Turing machines are universal making it possible to write a program that asks a question about another type of Turning machine doesn't mean that question is decidable.

[manyoso]

Show a Turing machine programmed with ZFC axioms that can do something that a Turing machine programmed with PE can not do. If a Turing machine programmed with PE can simulate the Turing machine with ZFC and thus give the same answer, then I state that the program with ZFC is nothing of the sort.

[pdkl95]

The paper[1] I linked to in my other post shows that you cannot perform such a simulation because it is undecidable. (Specifically, BB(N >= 1919) is provably undecidable in ZFC).

[1] http://www.scottaaronson.com/blog/?p=2725

[Chronos]

Show me a paper proving that what Penrose argues for and what quantum physicists call "objective collapse" are the same thing.

[manyoso]

https://link.springer.com/article/10.1007%2FBF02105068 https://link.springer.com/article/10.1007%2Fs10701-013-9770-... https://en.wikipedia.org/wiki/Objective_collapse_theory -- See Penrose' as canonical example of objective collapse

There is just tons of reading on this if you care to scour the literature. Penrose' objective collapse theory is usually given as the prime example of the whole genre.

[Chronos]

I already read through https://en.wikipedia.org/wiki/Penrose_interpretation when I was trying to figure out what nonsense you were spewing. The article does not mention the Church-Turing thesis, hypercomputation, Gödelian incompleteness, or the Lucas-Penrose argument.

People can be known for more than one thing. Those two things can be unrelated, even if they're in the same field.