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**Datum narození:** 8. srpen 1931

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„We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it.“
The Road to Reality: A Complete Guide to the Laws of the Universe

„No doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the face of a stern parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence--a duty and a duty alone--and no hint of magic or beauty of the subject might be allowed to come through.“
The Road to Reality: A Complete Guide to the Laws of the Universe

„To make this condition mathematically clearer, it is convenient to assert it in the form that the space-time can be continued smoothly, as a conformal manifold, a little way prior to the hypersurface. To before the Big Bang? Surely not: the Big Bang is supposed to represent the beginning of all things, so there can be no ‘before’. Never fear—this is just a mathematical trick. The extension is not supposed to have any physical meaning! Or might it …?“
Cycles of Time: An Extraordinary New View of the Universe

„There are considerable mysteries surrounding the strange values that Nature's actual particles have for their mass and charge. For example, there is the unexplained 'fine structure constant'... governing the strength of electromagnetic interactions,....“
The Road to Reality: A Complete Guide to the Laws of the Universe

„It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-we can keep adjoining new axioms, according to our whim, and obtain all kinds of alternative consistent systems within which we may choose to work. The comparison is sometimes made with the situation that occurred with Euclidean geometry. For some 21 centuries it was believed that Euclidean geometry was the only geometry possible. But when, in the eighteenth century, mathematicians such as Gauss, Lobachevsky, and Bolyai showed that indeed there are alternatives that are equally possible, the matter of geometry was seemingly removed from the absolute to the arbitrary. Likewise, it is often argued, Godel showed that arithmetic, also, is a matter of arbitrary choice, any one set of consistent axioms being as good as any other.

This, however, is a completely misleading interpretation of what Godel has demonstrated for us. He has taught us that the very notion of a formal axiomatic system is inadequate for capturing even the most basic of mathematical concepts. When we use the term 'arithmetic' without further qualification, we indeed mean the ordinary arithmetic which operates with the ordinary natural numbers 0,1,2,3,4,...(and perhaps their negatives) and not with some kind of 'supernatural' numbers. We may choose, if we wish, to explore the properties of formal systems, and this is certainly a valuable part of mathematical endeavour. But it is something different from exploring the ordinary properties of the ordinary natural numbers. The situation is, in some ways, perhaps not so very unlike that which occurs with geometry. The study of non-Euclidean geometries is something mathematically interesting, with important applications (such as in physics, see ENM Chapter 5 especially Figs 5.1 and 5.2, and also 4.4), but when the term 'geometry' is used in ordinary language (as distinct from when a mathematician or theoretical physicist might use that term), we do indeed mean the ordinary geometry of Euclid. There is a difference, however, in that what a logician might refer to as 'Euclidean geometry' can indeed be specified (with some reservations) in terms of a particular formal system, whereas, as Godel has shown, ordinary 'arithmetic' cannot be so specified.

Rather than showing that mathematics (most particularly arithmetic) is an arbitrary pursuit, whose direction is governed by the whim of Man, Godel demonstrated that it is something absolute, there to be discovered rather than invented (cf. 1.17). We discover for ourselves what the natural numbers are, and we do not have trouble in distinguishing them from any sort of supernatural numbers. Godel showed that no system of 'man-made' rules can, by themselves, achieve this for us. Such a Platonic viewpoint was important to Godel, and it will be important also for us in the later considerations of this book (8.7).“ Shadows of the Mind: A Search for the Missing Science of Consciousness

This, however, is a completely misleading interpretation of what Godel has demonstrated for us. He has taught us that the very notion of a formal axiomatic system is inadequate for capturing even the most basic of mathematical concepts. When we use the term 'arithmetic' without further qualification, we indeed mean the ordinary arithmetic which operates with the ordinary natural numbers 0,1,2,3,4,...(and perhaps their negatives) and not with some kind of 'supernatural' numbers. We may choose, if we wish, to explore the properties of formal systems, and this is certainly a valuable part of mathematical endeavour. But it is something different from exploring the ordinary properties of the ordinary natural numbers. The situation is, in some ways, perhaps not so very unlike that which occurs with geometry. The study of non-Euclidean geometries is something mathematically interesting, with important applications (such as in physics, see ENM Chapter 5 especially Figs 5.1 and 5.2, and also 4.4), but when the term 'geometry' is used in ordinary language (as distinct from when a mathematician or theoretical physicist might use that term), we do indeed mean the ordinary geometry of Euclid. There is a difference, however, in that what a logician might refer to as 'Euclidean geometry' can indeed be specified (with some reservations) in terms of a particular formal system, whereas, as Godel has shown, ordinary 'arithmetic' cannot be so specified.

Rather than showing that mathematics (most particularly arithmetic) is an arbitrary pursuit, whose direction is governed by the whim of Man, Godel demonstrated that it is something absolute, there to be discovered rather than invented (cf. 1.17). We discover for ourselves what the natural numbers are, and we do not have trouble in distinguishing them from any sort of supernatural numbers. Godel showed that no system of 'man-made' rules can, by themselves, achieve this for us. Such a Platonic viewpoint was important to Godel, and it will be important also for us in the later considerations of this book (8.7).“ Shadows of the Mind: A Search for the Missing Science of Consciousness

„However, in 1930 (published in 1931), Godel produced his bombshell, which eventually showed that the formalists' dream was unattainable! He demonstrated that there could be no formal system F, whatever, that is both consistent (in a certain 'strong' sense that I shall describe in the next section) and complete-so long as F is taken to be powerful enough to contain a formulation of the statements of ordinary arithmetic together with standard logic. Thus, Godel's theorem would apply to systems F for which arithmetical statements such as Lagrange's theorem and Goldbach's conjecture, as described in 2.3, could be formulated as mathematical statements.“

„Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature, and lies beyond any such system of specifiable rules. Support for the Platonic viewpoint... was an important part of Godel's initial motivations.“
Shadows of the Mind: A Search for the Missing Science of Consciousness

„It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-we can keep adjoining new axioms, according to our whim, and obtain all kinds of alternative consistent systems within which we may choose to work. The comparison is sometimes made with the situation that occurred with Euclidean geometry. For some 21 centuries it was believed that Euclidean geometry was the only geometry possible. But when, in the eighteenth century, mathematicians such as Gauss, Lobachevsky, and Bolyai showed that indeed there are alternatives that are equally possible, the matter of geometry was seemingly removed from the absolute to the arbitrary. Likewise, it is often argued, Godel showed that arithmetic, also, is a matter of arbitrary choice, any one set of consistent axioms being as good as any other.“
Shadows of the Mind: A Search for the Missing Science of Consciousness

„Q5. Have not I merely shown that it is possible to outdo just a particular algorithmic procedure, A, by defeating it with the computation Cq(n)? Why does this show that I can do better than any A whatsoever?

The argument certainly does show that we can do better than any algorithm. This is the whole point of a reductio ad absurdum argument of this kind that I have used here. I think that an analogy might be helpful here. Some readers will know of Euclid's argument that there is no largest prime number. This, also, is a reductio ad absurdum. Euclid's argument is as follows. Suppose, on the contrary, that there is a largest prime; call it p. Now consider the product N of all the primes up to p and add 1:

N=2*3*5*...*p+1.

N is certainly larger than p, but it cannot be divisible by any of the prime numbers 2,3,5..., p (since it leaves the remainder 1 on division); so either N is the required prime itself or it is composite-in which case it is divisible by a prime larger than p. Either way, there would have to be a prime larger than p, which contradicts the initial assumption that p is the largest prime. Hence there is no largest prime. The argument, being a reductio ad absurdum, does not merely show that a particular prime p can be defeated by finding a larger one; it shows that there cannot be any largest prime at all. Likewise, the Godel-Turing argument above does not merely show that a particular algorithm A can be defeated, it shows that there cannot be any (knowably sound) algorithm at all that is equivalent to the insights that we use to ascertain that certain computations do not stop.“ Shadows of the Mind: A Search for the Missing Science of Consciousness

The argument certainly does show that we can do better than any algorithm. This is the whole point of a reductio ad absurdum argument of this kind that I have used here. I think that an analogy might be helpful here. Some readers will know of Euclid's argument that there is no largest prime number. This, also, is a reductio ad absurdum. Euclid's argument is as follows. Suppose, on the contrary, that there is a largest prime; call it p. Now consider the product N of all the primes up to p and add 1:

N=2*3*5*...*p+1.

N is certainly larger than p, but it cannot be divisible by any of the prime numbers 2,3,5..., p (since it leaves the remainder 1 on division); so either N is the required prime itself or it is composite-in which case it is divisible by a prime larger than p. Either way, there would have to be a prime larger than p, which contradicts the initial assumption that p is the largest prime. Hence there is no largest prime. The argument, being a reductio ad absurdum, does not merely show that a particular prime p can be defeated by finding a larger one; it shows that there cannot be any largest prime at all. Likewise, the Godel-Turing argument above does not merely show that a particular algorithm A can be defeated, it shows that there cannot be any (knowably sound) algorithm at all that is equivalent to the insights that we use to ascertain that certain computations do not stop.“ Shadows of the Mind: A Search for the Missing Science of Consciousness

„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.“ Shadows of the Mind: A Search for the Missing Science of Consciousness

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.“ Shadows of the Mind: A Search for the Missing Science of Consciousness

„Thus, Godel appears to have taken it as evident that the physical brain must itself behave computationally, but that the mind is something beyond the brain, so that the mind's action is not constrained to behave according to the computational laws that he believed must control the physical brain's behavior.“
Shadows of the Mind: A Search for the Missing Science of Consciousness

„The thrust of Godel's argument for our purposes is that it shows us how to go beyond any given set of computational rules that we believe to be sound, and obtain a further rule, not contained in those rules, that we must believe to be sound also, namely the rule asserting the consistency of the original rules. The essential point, for our purposes, is:

belief in soundness implies belief in consistency.

We have no right to use the rules of a formal system F, and to believe that the results that we derive from it are actually true, unless we also believe in the consistency of that formal system. (For example, if F were inconsistent, then we could deduce, as TRUE, the statement '1=2', which is certainly not true!) Thus, if we believe that we are actually doing mathematics when we use some formal system F, then we must also be prepared to accept reasoning that goes beyond the limitations of the system F, whatever that system F may be.“ Shadows of the Mind: A Search for the Missing Science of Consciousness

belief in soundness implies belief in consistency.

We have no right to use the rules of a formal system F, and to believe that the results that we derive from it are actually true, unless we also believe in the consistency of that formal system. (For example, if F were inconsistent, then we could deduce, as TRUE, the statement '1=2', which is certainly not true!) Thus, if we believe that we are actually doing mathematics when we use some formal system F, then we must also be prepared to accept reasoning that goes beyond the limitations of the system F, whatever that system F may be.“ Shadows of the Mind: A Search for the Missing Science of Consciousness

„The reason that I have concentrated on non-computability, in my arguments, rather than on complexity, is simply that it is only with the former that I have been able to see how to make the necessary strong statements. It may well be that in the working lives of most mathematicians, non-computability issues play, if anything, only a very small part. But that is not the point at issue. I am trying to show that (mathematical) understanding is something that lies beyond computation, and the Godel (-Turing) argument is one of the few handles that we have on that issue. It is quite probable that our mathematical insights and understandings are often used to achieve things that could in principle also be achieved computationally-but where blind computation without much insight may turn out to be so inefficient that it is unworkable (cf. 3.26). However, these matters are much more difficult to address than the non-computability issue.“
Shadows of the Mind: A Search for the Missing Science of Consciousness

„If, as I believe, the Godel argument is consequently forcing us into an acceptance of some form of viewpoint C, the we shall also have to come to terms with some of its other implications. We shall find ourselves driven towards a Platonic viewpoint of things. According to Plato, mathematical concepts and mathematical truths inhabit an actual world of their own that is timeless and without physical location. Plato's world is an ideal world of perfect forms, distinct from the physical world, but in terms of which the physical world must be understood. It also lies beyond our imperfect mental constructions; yet, our minds do have some direct access to this Platonic realm through an 'awareness' of mathematical forms, and our ability to reason about them. We shall find that whilst our Platonic perceptions can be aided on occasion by computation, they are not limited by computation. It is this potential for the 'awareness' of mathematical concepts involved in this Platonic access that gives the mind a power beyond what can ever be achieved by a device dependent solely upon computation for its action.“
Shadows of the Mind: A Search for the Missing Science of Consciousness