Roger Penrose. Shadows of the mind. Oxford University Press., New York, Oxford, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Thus, due to the connectedness of the universes, the empty set or the whole universe R k are the only sets having computable characteristic functions. The lack of a generally acceptable concept of decidability (recursivity) for subsets of the Euclidean space R k was emphasized by R. Penrose [17]. In particular, he argued if the Mandelbrot set could be regarded as undecidable. L. Blum, M. Shub and S. Smale [1] immediately proposed an answer by showing that the Mandelbrot set is indeed undecidable in their setting. Unfortunately, the BSS model of computation has some disadvantages with ....

....in such a strong sense as TTE recursivity is. If a closed set A R k is TTE recursive, then its complement A is TTErecursive too. However, the interiors A ffi and A ffi = A are quite differently dealt with in this context. This is in contrast to the intuitive point of view also taken in [17], according to which the interiors A ffi and A ffi of recursive sets A should be easily separable from each other, even in a certain symmetric way. So it is convenient to require that, for a recursive set A R satisfying N A ffi [ A ffi , the set N A = N A ffi is (classically) ....

R. Penrose: The emporer's new mind. Oxford University Press, New York 1989

....group. However, George Boolos [2, p. 295] correctly notes that the arguments of these writers have as yet obtained little credence , and there is an extensive literature attacking the Lucas Penrose position. In a recent article, Lucas [19] holds his ground, or tries to. About 200 pages of Penrose [22] are devoted to responses to various criticisms of the argument and an intriguing new version of it (see 3 below and Penrose [23] The second sphere considered here takes o# from the observation that G odel s incompleteness theorem is completely constructive. Given any # consistent formal ....

, Shadows of the mind: A search for the missing science of consciousness, Oxford University Press, Oxford, 1994.

....provides the reader with a proof of a form of the G odel Theorem due to Alan Turing, the father of the modern digital computer and the creator of the mathematical subject recursion theory, which analyzes what computers can and cannot in principle accomplish. Hilary Putnam, review of Penrose 1994. 12 Physicist Penrose, like most scientists, never mentions the term recursive, but he has an extensive discussion of Turing and Turing machines covering a whole chapter. Penrose [p. 66] writes, by a computation (or algorithm) I indeed mean the action of some Turing machine, i.e. in e#ect, ....

R. Penrose, Shadows of the mind, Oxford University Press, Oxford.

....The first problem in the research on consciousness is to clearly define what the real problem is. What should we explain 2 Understanding Consciousness Some physicist think that a unifiedTheory of Everything (TOE) will explain consciousness together with everything else (cf. Penrose [14] writing on consciousness, quantum gravity and unified field theories) but since they do not define what they want to explain (Penrose writes that consciousness is indeed something ) it is not clear IV National Conference on Models of Biological Systems, Krakw 1 2.06.1995 what do they mean. Of ....

R. Penrose, The shadows of mind (Oxford University Press 1994)

.... and the Quantum Zeno (sometimes also called watched dog) e#ect discussed by Misra and Sudarshan [15] With interaction free measurements we can detect a live bomb or a piece of film without even sending a photon to trigger the bomb or to expose the film (see for example Bennett [2] or Penrose [16]) To illustrate this let us suppose that we are given a bomb which is either live or dead, and we want to determine, with some reasonable probability, this fact without detonating it. This problem classically has no solution (see Penrose photon detectors A B mirror mirror splitter beam ....

....are given a bomb which is either live or dead, and we want to determine, with some reasonable probability, this fact without detonating it. This problem classically has no solution (see Penrose photon detectors A B mirror mirror splitter beam splitter beam Figure 1: An interferometer. [16]) However, there is a quantum mechanics solution Consider a MachZehnder interferometer, as illustrated in Figure 1, which is composed of two perfect mirrors and two 50 50 beam splitters. The upper and lower path lengths are set to be equal. Two detectors A and B determine the output ....

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R. Penrose. Shadows of the Mind, Oxford University Press, Oxford, 1994.

....Largely unexplored, but of some interest, is the extension of [5] to TxtFex a b identification. We originally suggested in [16] on the basis of our main corollary (Corollary 3.7 to Theorem 3. 3) that Gold s model be extended to embrace the success criteria TxtFex a b 29 Sources such as [73, 74], sadly, seem to have overlooked the important result in [32] that the expected input output behavior of a Turing machine with random oracle subject to a computable probability distribution is computable (and constructively so) THE POWER OF VACILLATION IN LANGUAGE LEARNING 1965 for small ....

R. Penrose, Shadows of the Mind, Oxford University Press, New York, 1994.

....it or not. Thus, the theorem can be viewed, crudely, as saying that no consistent algorithm can produce a proof of its 2 1 For example: If our mathematical reasoning were indeed fundamentally unsound, then the whole edifice of scientific understanding would come crashing to the ground [8]. own consistency. This gives at least some indication why this theorem is often thought to be relevant to the computationalist hypothesis. In fact, Penrose does not actually use Gdel s theorem, but rather an easier result inspired by Gdel, namely, Turing s theorem that the halting problem is ....

....to assumption, have Roger s power to prove computations not to halt. Thus the assumption that Roger s knowledge is computably enumerable leads to a contradiction; and so the computationalist hypothesis is false. QED. This is what Penrose calls the bare Gdel argument, used in [7] Later, in [8], it was modified to refer not to any particular human s ability, but to the accumulated wisdom of the entire human race, i.e. the set of all humanly accessible methods; and his conclusion then is that no algorithm can fully encompass the set of all humanly accessible methods for ascertaining ....

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R. Penrose, Shadows of the Mind (Oxford University Press, New York, 1994).

....that are akin to brain processes, such as the notion of entanglement. Furthermore, in the quantum model of brain behavior the infinite variety of behavior emerges elegantly out of the superposition property. No wonder then that several quantum models of brain behavior have been proposed (e.g. [14, 20, 21, 23, 24, 25]) These models have been criticized on the grounds that the thermal noise inside the brain renders it impossible for the establishment of a quantum process. In the proposal that water molecules within cytoskeletal microtubules support a coherent process [14, 20] the question has been asked as ....

....proposed (e.g. 14, 20, 21, 23, 24, 25] These models have been criticized on the grounds that the thermal noise inside the brain renders it impossible for the establishment of a quantum process. In the proposal that water molecules within cytoskeletal microtubules support a coherent process [14, 20], the question has been asked as to how such a coherent state can exist across synaptic junctions. Furthermore, simple models of quantum function of the brain do not capture the essential characteristic of adaptation of the system to a changing environment. Neither do these models explain how the ....

R. Penrose, Shadows of the Mind. (Oxford University Press, London, 1994).

....the back (dotted line) before completing its cycle. weighted sum which in the case of two possible locations equals w jA x jB , where w and x are complex number weighting factors of the particle being in locations A and B, respectively, and where w jA and x jB are themselves state vectors [8]. For n possible states, there will be n different complex number weighting factors, each describing the weighted probability of the particle being at that location. 1 j Psi represents a linear superposition of the particle given individual quantum state vectors: If there are n locations as ....

R. Penrose. Shadows of the Mind. Oxford University Press, 1994.

....of questions: Does the kth Turing machine halt on input n A familiar diagonal argument shows that there is no Turing machine which can answer all n, k instantiations of this question 6 Supposedly we humans can. 7 Thus, following Godel and Lucas, it is still argued (e.g. by Penrose [8] [9]) that there are things which we can do which no Turing machine can, so computationalism is false. There are many ways to respond to this argument, but the transparent computationalist approach seems novel: reject the last step as a non sequitur. Even if Turing machines can t do everything humans ....

R. Penrose. The Shadows of the Mind. Oxford University Press, Oxford, 1994.

.... [22] while others argue for their equivalence and cite performance issues as the main concern [23] 10] Still others are more pessimistic as how intelligent a Connectionist Symbolic hybrid will ever be [24] and some claim that the entire idea of modeling cognition as computation is hopeless [25]. Many people agree that cognition can and should be described at different levels and that care should be taken to compare and contrast only those explanations models theories that exist on the same level [23] 14] 16] However, others feel that an explanation of cognition should not or can not ....

Penrose, Roger, Shadows of the Mind, Oxford University Press, 1994.

....are beyond the scope of this paper. Third, analog computation can be utilized to test possible theoretical limitations of the physical Church Turing thesis [19] that states that the computational capabilities of any physical device should not exceed (in idealization) that of a Turing machine [20]. If a device that computes problems that cannot be computed by the Turing machine (and therefore digital computers) is found, it will challenge the physical Church Turing thesis and will therefore be of great interest. Some theoretical analog models of computation have the capability of ....

R. Penrose. Shadows of the Mind. Oxford University Press, Oxford, England, 1994.

....show that he can do so only by adopting rather extraordinary positions in philosophy of mathematics and in psychology. 2 1 For example: If our mathematical reasoning were indeed fundamentally unsound, then the whole edifice of scientific understanding would come crashing to the ground [8] 2.1 The Bare G del Argument G del s Second Incompleteness Theorem can be stated informally as the claim that no reasonable set of axioms for arithmetic which is strong enough to prove certain basic facts about numbers is also strong enough to prove its own consistency, unless it is actually ....

....to assumption, have Roger s power to prove computations not to halt. Thus the assumption that Roger s knowledge is computably enumerable leads to a contradiction; and so the computationalist hypothesis is false. QED. This is what Penrose calls the bare G del argument, used in [7] Later, in [8] it was modified to refer not to any particular human s ability, but to the accumulated wisdom of the entire human race, i.e. the set of all humanly accessible methods; and his conclusion then is that no algorithm can fully encompass the set of all humanly accessible methods for ascertaining ....

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R. Penrose, Shadows of the Mind (Oxford University Press, New York, 1994).

....trained by back propagation using simple heuristics as an oracle. 1 Introduction The rapid growth in Artificial Life research brings us faster than we might ever have thought to the mystery of consciousness itself. There are still informed opinions from authors such as Searle [11] and Penrose [9, 10] who believe that the workings of the human mind are fundamentally different to Turing computation. Yet the diversity and complexity of behaviour we see in ALife models does little to reassure us that they are correct. Penrose has an illustrative argument [10, p46] where he exhibits a chess ....

R. Penrose. Shadows of the Mind. Oxford University Press, 1994.

....variants of the argument in the just previous sentence can be made. In another direction, we note that the proof of Theorem 1 permits a modification so that the relative density of output of all the final programs grammars but one is as small as we like. Hence, the 25 Sources such as [Pen89, Pen94] sadly seem to have overlooked the important result in [dMSS56] that the expected I O behavior of a Turing machine with random oracle subject to a computable probability distribution is computable (and constructively so) performance vacillation may exist, but significant degradations in ....

R. Penrose. Shadows of the Mind. Oxford University Press, NY, 1994.

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Roger Penrose. Shadows of the mind. Oxford University Press., New York, Oxford, 1994.

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R. Penrose, Shadows of the Mind, Oxford University Press, Oxford, 1994.

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R. Penrose, Shadows of the Mind , Oxford University Press. (1994).

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R. Penrose. Shadows of the mind. Oxford University Press, 1994.

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R. Penrose. Shadows of the Mind, Oxford University Press, Oxford, 1994.

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: R. Penrose, Shadows of the Mind. Oxford University Press, 1994, p.237.

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Penrose, R., Shadows of the Mind, Oxford University Press, Oxford, 1994.

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R. Penrose. Shadows of the mind. Oxford University Press, Oxford, New-York, 1994.

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Penrose, R., Shadows of the Mind, Oxford University Press, Oxford, 1994.

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R. Penrose. Shadows of the Mind. Oxford University Press, Oxford, 1994.

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