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4.4 Argument from Godel's Theorem
We will now attempt to give a rigorous mathematical argument against the hypothesis that computations can simulate intelligence. This argument is based on the Godel's theorem, and appears in Sir Roger Penrose's book, "Shadows of the Mind." Below, we present this argument in a slightly modified form.Suppose that all thinking were computation—that we actually follow an algorithm in our thinking process to solve mathematical problems. Then, let's take the collection of all algorithms we use to solve mathematical problems and represent it by a Turing machine, Y. Any problem that we can solve, the Turing machine Y can also solve, and there is no problem that we can solve, but Ycannot. This is our hypothesis.Now consider the well known halting problem: decide whether a given computation C on a given input n, represented as C(n), will ever halt. For an arbitrary computation, we may not be able to decide whether it will halt or not,but for some computations we can prove that they will never halt, by using our intelligence. For example, consider the following computation:
C: Find an even number that is the sum of n odd numbers.
This computation proceeds by testing for each even number, starting from 2, whether it is a sum of n odd numbers. For testing this, it finds the sum of all possible combinations of n odd numbers less than the even number, and testing if die sum equals the even number. We can easily tell that this computation will never halt for odd n.
The Turing machine that represents you, Y, takes as input a computation C, and its input, n, and tells us whether the computation will not halt. Y has the property that if Y halts, it means that the computation C(n) never halts, but if Y does not halt then we can say nothing about the halting of C(n). Notice that Y is falsifiable. We can always run the actual computation C(n) on a Turing machine and check if it halts. If it actually halts, it will do so in a finite time, and we will know that Y gave us the incorrect answer. So we can always check whether the answer given by Y is correct or not.
Now let us enumerate all possible single input computations as follows,
Co, Ci, C2,...
Note that this is possible since each computation can be encoded as a string of characters. Then we can lexicographically sort all computations with less than 1 character, 2 characters, and so on ad infinitum. This gives us an enumeration.
For any computation Ck, the following holds by our hypothesis,
If Y(k,n) halts then Gc(n) does not halt.
Now, fix n, and set k=n,
If Y(n,n) halts then Cn(n) does not halt.
Y(n,n) is now a single input computation, therefore it must be equal to some Ck.
Let Y(n,n) = Ck (n). Replace n by k in the above conditional,
If Y(k,k) halts then Ck(k) does not halt.
But Y(k,k) = Ck(k). So the above conditional reduces to,
IfCk(k) halts then Ck(k) does not halt.
This statement can only be true if Ck(k) does not halt. Therefore, we know that Ck(k) does not halt. But remember that Ck(k)=Y(k,k), and this means that Y(k,k) does not halt. In other words, the Turing machine that represents our thinking, Y, is not able to determine if Ck(k) will halt or not, but we know that Ck(k) does, indeed, not halt. This directly contradicts our hypothesis that all your thinking could be simulated computationally. Therefore, we have successfully argued that thinking is non-computational. there are clear indications from the mathematical logic, particularly from the theorem ofGodel that understanding is not a process which you can formalize. It is also not a computational process.Sir Roger Penrose If thinking is non-computational, and within the laws of current science at the same time, then there must exist a non-computational ingredient to the existing laws of physics, chemistry and biology. If we cannot find such a non-computational theory within current science, then we must conclude that the current scientific theories must be extended to incorporate an explanation for consciousness.