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Comment by leephillips | Hacker News Reader

There is no algorithm that can determine the truth of any proposition in arithmetic (for example). It is impossible, in principle, to construct such an algorithm. But given any proposition in arithmetic, the mind can decide its truth value. It is in this sense that Penrose claims the mind can not be modeled by a computer program. It may seem odd now, but at the time there was still a school of artificial intelligence that assumed that a big computer running a really sophisticated program could be equivalent to a human mind. Penrose was trying to show that this idea is a non-starter.

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    at the time there was still a school of artificial 
    intelligence that assumed that a big computer running a
    really sophisticated program could be equivalent to a
    human mind

We're talking theory, so imagine we have extremely powerful hardware, gobs of memory, understand the brain really well, and have some sort of extremely high-resolution scanning device. We could build an "emulator" that would simulate the brain's "hardware" and then load a scan of someone's brain onto it as "software". The system as a whole is just a big computer running a sophisticated program but it's equivalent to a human mind. This mind running on a computer would have the same strengths and weaknesses as the mind had when running on a biological brain when it comes to determining the truth values of propositions in arithmetic (and everywhere else).

To say that a computer program can not, in principle, do what the mind does is not to say that the mind can not exist on a substrate other than the familiar biological one. In your thought experiment the "software" would not be a computer program as we understand it; it would not be a set of algorithms. Also, it is problematic to separate "hardware" and "software", and to define what a "scan" of the brain might be.

>There is no algorithm that can determine the truth of any proposition in arithmetic (for example).

Currently, yes. But I don't see that this is necessarily true. Why do you think it is?

Simply because there are some propositions that are unprovable in a consistent system with finite axioms. Having an algorithm that solves any proposition would be equivalent to having an algorithm that solves the halting problem.

>There is no algorithm that can determine the truth of any proposition in arithmetic (for example). It is impossible, in principle, to construct such an algorithm. But given any proposition in arithmetic, the mind can decide its truth value.

These two statements seem to contradict, unless you grant that the mind either fudges the proof or constructs new axioms such that the proof can be valid. Why do you assume a program can never do this?

Because the program that does it would itself be an algorithm with its own finite set of axioms. There would be undecidable propositions unreachable by the program.

Solving the halting problem wouldn't be sufficient to tell you whether a proposition was true, but it could tell you whether a proposition is independent and if it wasn't whether it was true or false.

"Given any proposition in arithmetic, the mind can decide its truth value."

Really? I don't see what reason we have to believe that this is true.

I guess I didn't really mean that. What I should have said was that in any system of axioms (or within any algorithmic structure) there will exist undecidable propositions (unreachable by the algorithm). The mind can determine the truth value of some of these, so the mind can not be encompassed or modeled by an algorithm.

> The mind can determine the truth value of some of these, so the mind can not be encompassed or modeled by an algorithm.

Do you have an example of this? I have an extremely difficult time comprehending it. Also, I was under the impression that there weren't necessarily propositions that are not algorithmically possible to reach, but rather, there are always propositions a given algorithm cannot solve.

> not algorithmically possible to reach

I just meant undecidable propositions.

> Do you have an example

Maybe check out the conclusion on p.76 of this book[1] by Penrose (and the argument supporting it that begins on p.72.): "Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth".

His example is our knowledge that a certain computation will not halt, and the impossibility of constructing an algorithm that can prove that fact.

More flippantly, you know that you can examine the soundness of your own reasoning, but no algorithm can reach conclusions about its own correctness.

[1]http://www.amazon.com/dp/0198539789

Okay, so is the twin prime constant transcendental?

Yes.

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