Roger Penrose citáty

Roger Penrose foto
1  0

Roger Penrose

Datum narození: 8. srpen 1931


Roger Penrose je anglický matematický fyzik. Je znám především svými příspěvky k obecné relativitě a kosmologii. Mezi jeho přínosy patří např. spolupráce na objevu Hawkingova záření. Jedná se také o rekreačního matematika a filosofa, jeho Erdősovo číslo je 3.

Citáty Roger Penrose


„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).“

— Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness


„Q7. The total output of all the mathematicians who have ever lived, together with the output of all the human mathematicians of the next (say) thousand years is finite and could be contained in the memory banks of an appropriate computer. Surely this particular computer could, therefore, simulate this output and thus behave (externally) in the same way as a human mathematician-whatever the Godel argument might appear to tell us to the contrary?

While this is presumably true, it ignores the essential issue, which is how we (or computers) know which mathematical statements are true and which are false. (In any case, the mere storage of mathematical statements is something that could be achieved by a system much less sophisticated than a general purpose computer, e. g. photographically.) The way that the computer is being employed in Q7 totally ignores the critical issue of truth judgment. One could equally well envisage computers that contain nothing but lists of totally false mathematical 'theorems', or lists containing random jumbles of truths and falsehoods. How are we to tell which computer to trust? The arguments that I am trying to make here do not say that an effective simulation of the output of conscious human activity (here mathematics) is impossible, since purely by chance the computer might 'happen' to get it right-even without any understanding whatsoever. But the odds against this are absurdly enormous, and the issues that are being addressed here, namely how one decides which mathematical statements are true and which are false, are not even being touched by Q7.

There is, on the other hand, a more serious point that is indeed being touched upon in Q7. This is the question as to whether discussions about infinite structures (e. g. all natural numbers or all computations) are really relevant to our considerations here, when the outputs of humans and computers are finite.“

— Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

„At this point, the cautious reader might wish to read over the whole argument again, as presented above, just to make sure that I have not indulged in any 'sleight of hand'! Admittedly there is an air of the conjuring trick about the argument, but it is perfectly legitimate, and it only gains in strength the more minutely it is examined. We have found a computation Ck(k) that we know does not stop; yet the given computational procedure A is not powerful enough to ascertain that facet. This is the Godel(-Turing) theorem in the form that I require. It applies to any computational procedure A whatever for ascertaining that computations do not stop, so long as we know it to be sound. We deduce that no knowably sound set of computational rules (such as A) can ever suffice for ascertaining that computations do not stop, since there are some non-stopping computations (such as Ck(k)) that must elude these rules. Moreover, since from the knowledge of A and of its soundness, we can actually construct a computation Ck(k) that we can see does not ever stop, we deduce that A cannot be a formalization of the procedures available to mathematicians for ascertaining that computations do not stop, no matter what A is.


(G) Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth.

It seems to me that this conclusion is inescapable. However, many people have tried to argue against it-bringing in objections like those summarized in the queries Q1-Q20 of 2.6 and 2.10 below-and certainly many would argue against the stronger deduction that there must be something fundamentally non-computational in our thought processes. The reader may indeed wonder what on earth mathematical reasoning like this, concerning the abstract nature of computations, can have to say about the workings of the human mind. What, after all, does any of this have to do with the issue of conscious awareness? The answer is that the argument indeed says something very significant about the mental quality of understanding-in relation to the general issue of computation-and, as was argued in 1.12, the quality of understanding is something dependent upon conscious awareness. It is true that, for the most part, the foregoing reasoning has been presented as just a piece of mathematics, but there is the essential point that the algorithm A enters the argument at two quite different levels. At the one level, it is being treated as just some algorithm that has certain properties, but at the other, we attempt to regard A as being actually 'the algorithm that we ourselves use' in coming to believe that a computation will not stop. The argument is not simply about computations. It is also about how we use our conscious understanding in order to infer the validity of some mathematical claim-here the non-stopping character of Ck(k). It is the interplay between the two different levels at which the algorithm A is being considered-as a putative instance of conscious activity and as a computation itself-that allows us to arrive at a conclusion expressing a fundamental conflict between such conscious activity and mere computation.“

— Roger Penrose, 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.“

— Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

Dnešní výročí
 Voltaire foto
francouzský spisovatel a filozof 1694 - 1778
Isaac Bashevis Singer foto
Isaac Bashevis Singer6
polsko-židovsko-americký autor 1902 - 1991
Heinrich Von Kleist foto
Heinrich Von Kleist5
německý básník, dramatik, romanopisec a spisovatel novel 1777 - 1811
Josef Lux foto
Josef Lux3
český ministr zemědělství ČR, místopředseda vlády ČR, čle... 1956 - 1999
Another 29 today anniversaries
Podobní autoři
Carl Friedrich Gauss foto
Carl Friedrich Gauss1
německý matematik a fyzikální vědec
Denis Diderot foto
Denis Diderot47
francouzský osvícenský filozof a encyklopedista
René Descartes foto
René Descartes32
francouzský filozof
Hermann Weyl foto
Hermann Weyl1
německý matematik
Christian Doppler foto
Christian Doppler1
rakouský fyzik a matematik
Georg Christoph Lichtenberg foto
Georg Christoph Lichtenberg171
německý vědec, satirik
Blaise Pascal foto
Blaise Pascal112
francouzský filozof, matematik, fyzik, spisovatel
Galileo Galilei foto
Galileo Galilei18
italský matematik, fyzik, filozof a astronom
 Archimedes foto
řecký matematik, fyzik, inženýr, vynálezce a astronom
Jules Henri Poincaré foto
Jules Henri Poincaré4
prominentní francouzský matematik, fyzik, inženýr a filoz...