What is a computation? According to Alan Turing, writing in 1936/7, it is this:
A computation occurs when the human mind carries out a mental action according to a rule..
Thus computability, at least as far as minds are concerned, is:
Computability arises from a human mind carrying out rules.
Now this doesn't mean that the mind, or person, knows it's following a rule. And therefore it must mean that the mind doesn't know what that rule is. Human grammar, after all, is rule-governed; though not all children or even adults can formulate grammatical rules. Nonetheless, we follow them. (This is like the philosopher of language's “tacit knowledge”.)
I said that computability is the mind's following a rule. Well, that's usually the case; though not always. Does that mean that there can be computability, or computations, without following a rule?
When machines, or computers, modify their own behaviour, is that an example of not following a rule? Well, it's certainly the case that computers can do things which weren't predicted by their designers or programmers. Does that mean that such computers aren't following any rules? After all, they could be following new rules which they have formulated themselves. Not following the programmers' rules doesn't automatically mean that computers aren't following rules per se.
We can even say that such computers have genuinely learned something which wasn't fed into them by their designers or programmers. They may still be following rules. In fact the new rules may be the logical consequence of the programmer's older rules.
How does all this stuff about rule-following connect with the human brain-mind?
Turing did think that the brain is a machine. Or at least he thought that the function of the brain is that of a machine. Nonetheless, he also believed that the brain is so complex that it could have the appearance of not following a rule.
Now it's clear that it's the complexity of the brain that generates only an “appearance” of the brain not following a rule. So that basically means that even though the brain appears not to be following a rule, it still may be doing so. It's just that the brain is so complex that the investigator, or even the owner of the brain (as it were), couldn't know all the rules that the brain is following. Similarly, the complexity of the brain may also generate the belief that it is an indeterministic “machine”. (I suppose, if the brain were truly indeterministic, one could question its status as a true machine.)
Following on from this, does that mean that if the computer has learned something, or has created its own rules, that it's displaying or showing genuine intelligence? After all, it has gone beyond what the programmer programmed. I suppose this all depends on the semantics of the word “intelligence”. If not following the rules of the programmer is a case of genuine intelligence, then the computer is displaying genuine intelligence. Nonetheless, is not following the rules of the programmer genuine intelligence or is it simply something else? In that case, what is it?
What is an intuition? It depends on how the word is used and in which context it's being used. In Turing's case, we, or a mathematician, use our intuition when seeing the truth of a formally unprovable Gödel sentence. Gödel sentences can't be proved. Nonetheless they're true and they're taken to be true.
So how do we know they are true without mathematical proof? According to Turing, and Gödel himself, through the use of human intuition.
Another way of looking at intuition is with another of Turing's ideas: the“oracle”.
In the case of a Gödel sentence, the mathematician, or the oracle, simply “has an idea” that the Gödel sentence is true. That is, he doesn't use a mechanical method to establish its truth. He has an idea or an intuition that it's true.
You may now ask how something can be establish as true, especially in maths, without proof. You may also ask how truths can be established, especially in maths, only on the flimsy basis of a mathematician's -or even on hundreds of mathematicians' - intuition or his simply“having an idea”.
Computer scientists - and the philosophers of mind who focus on computer-brain comparisons (or who even see the brain-mind as a literal computer) - will like Turing's conclusion (of 1945) that algorithms are enough to account for all mental activity. Bearing in mind the previous comments about intuition, Turing also thought that algorithms also encompassed non-mechanical intuition.
Just as intuition followed algorithms (therefore rules?), Turing believed that “initiative” didn't need un-computable steps. In other words, human and computer initiative is also a mechanical process. (That would make the idea of computer's showing initiative, or intuition, less problematic for the simple reason that what it's doing is still a computable or a mechanical process.) However, as I mentioned earlier when I mention the fact that computers may go beyond the rules (or programmes) laid down by the programmer, even if a computer departs from the computations which were programmed by the programmer, it would still be following a (new) rule, or indulging in computations, or following mechanical processes. (Indeed what else could it be doing?)
Another way of looking at a computer's, or a Turing machine's, ability to follow its own rules (or to show initiative) is for the programmer to engineer an element of randomness into the computer (or into the programme?). That was what Turing tried to do with his Manchester computer (1948/50). That seems to mean that such randomness as it were brings about intuition or initiative. However, it would still be intuition or initiative that is grounded in computation, or rules, or mechanical processes. The randomness, therefore, would simply be a result of the computer not abiding by the programmer's rules (or programmes). It doesn't mean that the computer has gone beyond rules or computations.