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“In order for A to apply to computations generally, we shall need a way of coding all the different computations C(n) so that A can use this coding for its action. All the possible different computations C can in fact be listed, say as

C0, C1, C2, C3, C4, C5,...,

and we can refer to Cq as the qth computation. When such a computation is applied to a particular number n, we shall write

C0(n), C1(n), C2(n), C3(n), C4(n), C5(n),....

We can take this ordering as being given, say, as some kind of numerical ordering of computer programs. (To be explicit, we could, if desired, take this ordering as being provided by the Turing-machine numbering given in ENM, so that then the computation Cq(n) is the action of the qth Turing machine Tq acting on n.) One technical thing that is important here is that this listing is computable, i.e. there is a single computation Cx that gives us Cq when it is presented with q, or, more precisely, the computation Cx acts on the pair of numbers q, n (i.e. q followed by n) to give Cq(n).

The procedure A can now be thought of as a particular computation that, when presented with the pair of numbers q,n, tries to ascertain that the computation Cq(n) will never ultimately halt. Thus, when the computation A terminates, we shall have a demonstration that Cq(n) does not halt. Although, as stated earlier, we are shortly going to try to imagine that A might be a formalization of all the procedures that are available to human mathematicians for validly deciding that computations never will halt, it is not at all necessary for us to think of A in this way just now. A is just any sound set of computational rules for ascertaining that some computations Cq(n) do not ever halt. Being dependent upon the two numbers q and n, the computation that A performs can be written A(q,n), and we have:

(H) If A(q,n) stops, then Cq(n) does not stop.

Now let us consider the particular statements (H) for which q is put equal to n. This may seem an odd thing to do, but it is perfectly legitimate. (This is the first step in the powerful 'diagonal slash', a procedure discovered by the highly original and influential nineteenth-century Danish/Russian/German mathematician Georg Cantor, central to the arguments of both Godel and Turing.)
With q equal to n, we now have:

(I) If A(n,n) stops, then Cn(n) does not stop.

We now notice that A(n,n) depends upon just one number n, not two, so it must be one of the computations C0,C1,C2,C3,...(as applied to n), since this was supposed to be a listing of all the computations that can be performed on a single natural number n. Let us suppose that it is in fact Ck, so we have:

(J) A(n,n) = Ck(n)

Now examine the particular value n=k. (This is the second part of Cantor's diagonal slash!) We have, from (J),

(K) A(k,k) = Ck(k)

and, from (I), with n=k:

(L) If A(k,k) stops, then Ck(k) does not stop.

Substituting (K) in (L), we find:

(M) If Ck(k) stops, then Ck(k) does not stop.

From this, we must deduce that the computation Ck(k) does not in fact stop. (For if it did then it does not, according to (M)! But A(k,k) cannot stop either, since by (K), it is the same as Ck(k). Thus, our procedure A is incapable of ascertaining that this particular computation Ck(k) does not stop even though it does not.

Moreover, if we know that A is sound, then we know that Ck(k) does not stop. Thus, we know something that A is unable to ascertain. It follows that A cannot encapsulate our understanding.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“Nothing – I repeat – nothing can become so big an obstruction in your path that you cannot overcome it, for your brain is the most divine and most godly Turing Machine of all times, capable of deeds that humanity is yet to encounter.”
― Abhijit Naskar, Saint of The Sapiens

“With Bayesian networks, we had taught machines to think in shades of grey, and this was an important step toward humanlike thinking. But we still couldn't teach machines to understand causes and effects. We couldn't explain to a computer why turning the dial of a barometer won't cause rain.... Without the ability to envision alternate realities and contrast them with the currently existing reality, a machine...cannot answer the most basic question that makes us human: "Why?”
― Judea Pearl, The Book of Why: The New Science of Cause and Effect

“I would say that fair play must be given to the machine. Instead of it sometimes giving no answer we could arrange that it gives occasional wrong answers. But the human mathematician would likewise make blunders when trying out new techniques. It is easy for us to regard these blunders as not counting and give him another chance, but the machine would probably be allowed no mercy. In other words then, if a machine is expected to be infallible, it cannot also be intelligent.

If Turing did not yet understand what it meant to be shown “no mercy,” he would, alas, learn that lesson all too soon. For the present, he was content to make a “plea” that his machines be treated with greater tolerance than he, as a homosexual man, was destined to experience:

To continue my plea for “fair play for the machines” when testing their I.Q. A human mathematician has always undergone an extensive training. This training may be regarded as not unlike putting instruction tables into a machine. One must therefore not expect a machine to do a very great deal of building up of instruction tables on its own. No man adds very much to the body of knowledge, why should we expect more of a machine? Putting the same point differently, the machine must be allowed to have contact with human beings in order that it may adapt itself to their standards.

It all came down to loneliness.”
― David Leavitt, The Man Who Knew Too Much: Alan Turing and the Invention of the Computer