As a part of my self-learning about Gödel's Incompleness Theorems, I try to dive a little bit into their different interpretations and (possible) philosophical implications. One that immediately catches the eye relates to artifical intelligence. It seems like some people believe Gödel's theorems indicate that the mind is somehow transcendent to machines. As J. R. Lucas, puts it in his book (which I did not read) "Minds, Machines and Gödel": there is "some elusive and ineffable quality to human intelligence, which makes it unattainable by machines".
In short, Lucas' argument goes as follows:
However complicated a machine we construct, it will, if it is a machine, correspond to a formal system, which in turn will be liable to the Gödel procedure for finding a formula unprovable-in-that- system. This formula the machine will be unable to produce as true, although a mind can see that it is true. And so the machine will not be an adequate model of the mind.
Following this line of reasoning, it seems like the "elusive quality" Lucas mentions lies not within human consciousness, feelings or imagination but rather within the short textbook-proof that Gödel's sentence is true.
To me that sounds quite ridiculous (as you might have guessed). However, when I do try to point out exactly where Lucas' argument breaks down, I mainly get confused. So I'd appriciate your help with clearing things up.
It seems like the key here is to carefully follow the mathematical details of that textbook proof and try to see what's so "human" about it and why. Recall that given an arithmetical formal system $S$, its Godel sentence $G$ satisfies $S\vdash G\leftrightarrow \neg{\rm Prov}(\left\lceil G \right\rceil )$, where $${\rm Prov}(\left\lceil \varphi \right\rceil):=\exists x ({\rm Prf} (x, \left\lceil \varphi \right\rceil))$$ is the "provability" formula (and $\left\lceil \cdot \right\rceil$ means Gödel numbering).
Now, the textbook proof goes like this. In the proof of Gödel's first incompleteness theorem it is shown that if $S$ is consistent then it cannot prove $G$. Therefore, in that case, for every natural $n\in\mathbb{N}$, $\neg {\rm Prf} (n, \left\lceil G \right\rceil)$ is true (in the standard model $\mathbb{N}$). Thus $\forall x(\neg {\rm Prf} (x, \left\lceil G \right\rceil))$ is true (in $\mathbb N$) which is exactly $\neg{\rm Prov}(\left\lceil G \right\rceil )$. But it is known to be equivalent to $G$ by definition. Therefore $G$ is true.
How much of that does $S$ "know" (able to prove)? As shown in the proof of the second theorem, the system $S$ knows that if it is consistent it cannot prove $G$. It can also be shown that $S$ knows, for every $n\in \mathbb{N}$, that $\neg {\rm Prf} (n, \left\lceil G \right\rceil)$. So it seems like the only thing that $S$ is missing is the knowledge that if $\psi(n)$ is true for every $n\in\mathbb{N}$ then $\forall x (\psi(x))$ is true. This is where I get confused:
- I was able to say "true in the standard model $\mathbb{N}$" but $S$ doesn't have that privilege; when it asserts that something is true, it has to be the case in all of its models (which might be non-standard). Wouldn't that be an "unfair advantage" that I have?
- If "concluding that $\forall x (A(x))$ from $A(n)$ for every $n\in\mathbb{N}$ "is really the thing that seperates my mind from $S$ (regarding the truth of $G$), what is it about my human mind that causes that? What's so "human" here?
- I know that $G$ is true only assuming that I know $S$ is consistent. So maybe Lucas' argument doesn't prove that the human mind transcends all machines, but only those it "knows" their consistence. In that case, the human mind might still reducable to a formal system, but simply one we will never be aware of its consistence. Perhaps that's where the Lucas' argument fails?
