Enm Quotes

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.

Every thing in life a been for a reason and because god what that thing to happen

A computational procedure is said to have a top-down organization if it has been constructed according to some well-defined and clearly understood fixed computational procedure (which may include some preassigned store of knowledge), where this procedure specifically provides a clear-cut solution to some problem at hand. (Euclid's algorithm for finding the highest common factor of two natural numbers, as described in ENM, p. 31, is a simple example of a top-down algorithm.) This is to be contrasted with a bottom-up organization, where such clearly defined rules of operation and knowledge store are not specified in advance, but instead there is a procedure laid down for the way that the system is to 'learn' and to improve its performance according to its 'experience'. Thus, with a bottom-up system, these rules of operation are subject to continual modification. One must allow that the system is to be run many times, performing its actions upon a continuing input of data. On each run, an assessment is made-perhaps by the system itself-and it modifies its operations, in the lifht of this assessment, with a view to improving this quality of output. For example, the input data for the system might be a number of photographs of human faces, appropriately digitized, and the system's task is to decide which photographs represent the same individuals and which do not. After each run, the system's performance is compared with the correct answers. Its rules of operation are then modified in such a way as to lead to a probable improvement in its performance on the next run.

There are strong reasons for suspecting that the modification of quantum theory that will be needed, if some form of R is to be made into a real physical process, must involve the effects of gravity in a serious way. Some of these reasons have to do with the fact that the very framework of standard quantum theory fits most uncomfortably with the curved-space notions that Einstein's theory of gravity demands. Even such concepts as energy and time-basic to the very procedures of quantum theory-cannot, in a completely general gravitational context, be precisely defined consistently with the normal requirements of standard quantum theory. Recall, also, the light-cone 'tilting' effect (4.4) that is unique the physical phenomenon of gravity. One might expect, accordingly, that some modification of the basic principles of quantum theory might arise as a feature of its (eventual) appropriate union with Einstein's general relativity. Yet most physicists seem reluctant to accept the possibility that it might be the quantum theory that requires modification for such a union to be successful. Instead, they argue, Einstein's theory itself should be modified. They may point, quite correctly, to the fact that classical general relativity has its own problems, since it leads to space-time singularities, such as are encountered in black holes and the big bang, where curvatures mount to infinity and the very notions of space and time cease to have validity (see ENM, Chapter 7). I do not myself doubt that general relativity must itself be modified when it is appropriately unified with quantum theory. And this will indeed be important for the understanding of what actually takes place in those regions that we presently describe as 'singularities'. But it does not absolve quantum theory from a need for change. We saw in 4.5 taht general relativity is an extraordinarily accurate theory-no less accurate than is quantum theory itself. Most of the physical insights that underlie Einstein's theory will surely survive, not less than will most of those of quantum theory, when the appropriate union that moulds these two great theories together is finally found.