[manyoso]

Let's try this again...

Yes, you can make a computer spit out nonsense, but this is a far cry from the computer actually behaving in an illogical way. A Turing machine's entire operation is logical. It is built with logic. It is a logical machine.

Penrose claims the human mind is not a Turing machine. That it is not logical. That it is not built with logic. That it is not a logical machine.

Pointing out that the human mind is not logical is thus restating Penrose' point and not refuting it.

[Chronos]

I can put it another way:

It's easy to write a computer program that outputs every statement provable from the Peano axioms: start with the most primitive possible statements, then progressively output more complex ones. This is because the Peano axiom system is "recursively enumerable", to use Computer Science terminology. The program never terminates, but any provable statement you name will be output at some finite time.

Additionally, it's easy to write a computer program that outputs every possible combination of symbols. Most of the combinations aren't statements at all. Of the ones that are statements, most of them are not consistent with the Peano axioms. Of the ones that are consistent with the Peano axioms, most of them are unprovable. But if the set of truths is countable, this scheme guarantees that the program will output every true statement that exists.

It is possible to write a computer program that checks if a statement is provable from the Peano axioms. The naïve way is to run the program that outputs every statement as a subroutine, then halt if the subroutine prints out the statement which we wish to verify. This program will halt iff the statement is provable.

It is NOT possible to write a computer program that checks if a statement is unprovable from the Peano axioms. Such a program may be able to detect a subset of unprovable statements. The subset it can detect may even be countably infinite. But there are some statements which will cause the program to run forever.

Let statement X represent a formulation of "the Goldbach conjecture is true" written in the language of Peano arithmetic. Does the program run forever if asked whether X is provable? If you are a non-Gödelian being, you will be able to answer that question with no error, because Gödel's Incompleteness Theorem only applies to formal systems which can prove all true statements.

[Chronos]

Penrose is not arguing that the human mind is illogical. He is arguing that it is uncomputable, which is a big difference.

In particular, he's saying that the human mind is capable of outputting true statements that cannot be proven to be true via any bounded number of proof steps. I don't disagree with that.

However, given that the human mind is capable of outputting false statements -- witness this conversation, wherein at least one of us is outputting false statements -- Penrose has failed to prove that the human mind is non-Gödelian, i.e. can output all the true statements that exist, while outputting no false ones.