Whether an unreasonable algorithm is capable of modeling mathematical understanding in a knowable way.
Roger Penrose
Shadows of the mind. In search of a science of consciousness
Foreword
This book can be considered, in a sense, a continuation of " The new mind of the king(hereinafter referred to as NRK). That is, I really intend to continue the development of the theme begun in the NQF, however, the material presented here can be considered completely independently of the previous book. Part of the need for revisiting the subject originally arose from a desire to provide as thorough an answer as possible to the many questions and criticisms with which a wide variety of people responded to the reasoning and evidence presented in the NQF. However, the topic of the new book is a completely independent study, and the ideas proposed here are by no means limited to the framework established in the NQF. One of the main themes of NQF was my belief that by using consciousness we are able to perform actions that have nothing to do with any computational processes. However, in the NQF this idea was presented only as a cautious hypothesis; there was also some uncertainty as to which types of procedures should be included in the category of "computing processes". In the pages of this book, it seems to me, the reader will find a much more consistent and rigorous justification for the above general statement, and the justification presented turns out to be applicable to all types of computational processes that one can imagine. In addition, there is a much more plausible (than was possible at the time of the NRK) suggestion regarding the mechanism of cerebral activity, whereby our mind-controlled behavior can be based on some kind of physical activity of a non-computational nature.
This justification is carried out in two different directions. One of them is essentially negative; here I strongly oppose the widely held opinion that our conscious mental activity - in all its various manifestations - can, in principle, be adequately described within the framework of certain computational models. Another line of my reasoning can be considered positive - in the sense that it involves a genuine search (of course, within the framework of the need to adhere to strict and irrefutable scientific facts) for tools that allow the brain described in scientific terms to apply subtle and mostly us not yet known physical principles.
In accordance with this dichotomy, the argument presented in the book is divided into two parts. The first part contains a comprehensive and thorough study, the results of which most strongly support my thesis that consciousness, in its concrete manifestation of human "understanding", does something that simple calculations cannot reproduce. Moreover, the term "computing" here means both processes implemented by systems of the "top-down" type, acting in accordance with specific and transparent algorithmic procedures, and processes implemented by systems of the "upstream" type, which are programmed not so rigidly and, as a result, are capable of learning. based on the experience gained. Gödel's famous theorem occupies a central place in the reasoning of the first part; a most detailed consideration of the consequences of this theorem, which are relevant to our case, is also given. Such a presentation significantly expands the argument presented first by Gödel himself, and later by Nagel, Newman and Lucas; moreover, here I have tried to answer as thoroughly as possible all the objections known to me. In this regard, detailed proofs are also given of the impossibility of reaching systems of the ascending (as well as descending) type of genuine intelligence. In conclusion, it is concluded that conscious thinking must indeed include processes that cannot even be adequately modeled using computational methods alone; even less capable of calculation, taken by itself, to cause any conscious sensation or desire. In other words, the mind seems to be an entity that can in no way be described by any kind of calculation.
In the second part, we turn to physics and biology. Although the individual links in the chain of our inferences are clearly more conjectural here than the rigorous proofs of the first part, we will nevertheless try to figure out exactly how, within the limits of the action of scientifically comprehensible physical laws such non-computable activity may occur. The necessary fundamental principles of quantum mechanics are stated from the very beginning, so that the reader is not required to have any prior knowledge of quantum theory. Enough is given deep Scan some of the mysteries and paradoxes of quantum theory, with a number of new examples graphically illustrating the role of nonlocality and counterfactuality, as well as some very complex problems associated with quantum entanglement. I am deeply convinced - and I am ready to substantiate my conviction - of the need for a fundamental revision (at a certain, clearly defined level) of our current quantum mechanical views. (The considerations expressed here are very close to the ideas recently published by Ghirardi, Diosi, and others.) It should be noted that significant changes have occurred in this respect since the time of the NQF.
I believe that it is at this level that physical incomputability must come into play - a condition necessary to explain the incomputability of the activity of consciousness. In accordance with this assumption, I must demand that the level at which the said physical incomputability becomes significant plays a special role in the functioning of the brain. It is at this point that my current assumptions diverge most significantly from those expressed in the NQF. I argue that although neuron signals may behave like events that are determined in the classical sense, the control of synaptic connections between neurons occurs at a deeper level, i.e. where we can expect the presence of significant physical activity on the border between quantum and classical processes. The specific assumptions I put forward require the emergence of macroscopic quantum-coherent behavior inside the microtubules of the cytoskeleton of neurons (in exact accordance with the assumptions of Fröhlich). In other words, I believe that the mentioned quantum activity must be related in some non-computable way to the computable process that Hameroff and his colleagues claim takes place inside these very microtubules.
The evidence I have presented indicates that the views held today in certain areas of science do nothing to advance any scientific understanding of the human mind. And yet this does not mean that the phenomenon of consciousness will never find its own scientific explanation. I am deeply convinced - in this respect my views have not changed since the time of the NQF - that the scientific path to understanding the phenomena of the mind undoubtedly exists, and this path must begin with a deeper knowledge of the nature of physical reality proper. I think it is extremely important that any serious reader who intends to understand how such an outstanding phenomenon as the mind can be explained in terms of the material physical world, should first have a fairly clear idea of \u200b\u200bhow strange laws can turn out to be. in reality controlling this very “material” of which our physical world consists.
Ultimately, it is for the sake of understanding that we started all of science, and science is still something more than just mindless calculation.
April 1994
Thanks
For helping me write this book, I am deeply indebted to many people—too many to thank any one of them, even if I could remember all the names. However, I would like to express my special thanks to Guido Baccagaluppi and Jeremy Butterfield for their criticisms of certain parts of the draft book, finding in particular a serious error in my then reasoning (the corrected text was included in the third chapter of the final version books). In addition, I am grateful to Dan Isaacson, Abhay Ashtekar, Mary Bell, Brian Birch, Jeff Brooker, Susan Greenfield, Robin Gandy, Roger James, David Deutsch, Ezio Insinna, Richard Joje, Francis Crick, John Lucas, Bill McCall, Graham Michison, Klaus Moser, Ted Newman, Jonathan Penrose, Oliver Penrose, Stanley Rosen, Ray Sachs, Graham Segal, Aaron Sloman, Lee Smolin, Ray Streeter, Valerie Willoughby, Solomon Feferman, Andrew Hodges, Deepankar Home, David Chalmers, Anton Zeilinger and especially Arthur Eckert for all kinds of information and help. Since the publication of my previous book (“ The new mind of the king”) I received a lot of oral and written feedback about her. Taking this opportunity, I want to thank everyone who expressed their opinion - it was not in vain, although most I never got around to answering the emails. If I had not benefited from all these very different comments on my previous book, it is unlikely that I would have embarked on such an intimidating enterprise as writing the next one.
(hereinafter - NRK). That is, I really intend to continue the development of the theme begun in the NQF, however, the material presented here can be considered completely independently of the previous book. Part of the need for revisiting the subject originally arose from a desire to provide as thorough an answer as possible to the many questions and criticisms with which a wide variety of people responded to the reasoning and evidence presented in the NQF. However, the topic of the new book is a completely independent study, and the ideas proposed here are by no means limited to the framework established in the NQF. One of the main themes of NQF was my belief that by using consciousness we are able to perform actions that have nothing to do with any computational processes. However, in the NQF this idea was presented only as a cautious hypothesis; there was also some uncertainty as to which types of procedures should be included in the category of "computing processes". In the pages of this book, it seems to me, the reader will find a much more consistent and rigorous justification for the above general statement, and the justification presented turns out to be applicable to all types of computational processes that one can imagine. In addition, there is a much more plausible (than was possible at the time of the NRK) suggestion regarding the mechanism of cerebral activity, whereby our mind-controlled behavior can be based on some kind of physical activity of a non-computational nature.
This justification is carried out in two different directions. One of them is essentially negative; here I strongly oppose the widely held opinion that our conscious mental activity - in all its various manifestations - can, in principle, be adequately described within the framework of certain computational models. Another line of my reasoning can be considered positive - in the sense that it involves a genuine search (of course, within the framework of the need to adhere to strict and irrefutable scientific facts) for tools that allow the brain described in scientific terms to apply subtle and mostly us not yet known physical principles.
In accordance with this dichotomy, the argument presented in the book is divided into two parts. The first part contains a comprehensive and thorough study, the results of which most strongly support my thesis that consciousness, in its concrete manifestation of human "understanding", does something that simple calculations cannot reproduce. Moreover, the term "computing" here means both processes implemented by systems of the "top-down" type, acting in accordance with specific and transparent algorithmic procedures, and processes implemented by systems of the "upstream" type, which are programmed not so rigidly and, as a result, are capable of learning. based on the experience gained. Gödel's famous theorem occupies a central place in the reasoning of the first part; a most detailed consideration of the consequences of this theorem, which are relevant to our case, is also given. Such a presentation significantly expands the argument presented first by Gödel himself, and later by Nagel, Newman and Lucas; moreover, here I have tried to answer as thoroughly as possible all the objections known to me. In this regard, detailed proofs are also given of the impossibility of reaching systems of the ascending (as well as descending) type of genuine intelligence. In conclusion, it is concluded that conscious thinking must indeed include processes that cannot even be adequately modeled using computational methods alone; even less capable of calculation, taken by itself, to cause any conscious sensation or desire. In other words, the mind seems to be an entity that can in no way be described by any kind of calculation.
In the second part, we turn to physics and biology. Although the individual links in the chain of our reasoning are here clearly more conjectural than the rigorous proofs of the first part, we will nevertheless try to figure out exactly how such incomputable activity can arise within the limits of the action of scientifically comprehensible physical laws. The necessary fundamental principles of quantum mechanics are stated from the very beginning, so that the reader is not required to have any prior knowledge of quantum theory. A fairly deep analysis of some of the mysteries and paradoxes of quantum theory is given with the involvement of a number of new examples graphically illustrating the role of nonlocality and counterfactuality, as well as some very complex problems associated with quantum entanglement. I am deeply convinced - and ready to substantiate my conviction - of the need for a fundamental revision (at a certain, clearly defined level) of our current quantum mechanical views. (The considerations expressed here are very close to the ideas recently published by Ghirardi, Diosi, and others.) It should be noted that significant changes have occurred in this respect since the time of the NQF.
I believe that it is at this level that physical incomputability must come into play - a condition necessary to explain the incomputability of the activity of consciousness. In accordance with this assumption, I must demand that the level at which the said physical incomputability becomes significant plays a special role in the functioning of the brain. It is at this point that my current assumptions diverge most significantly from those expressed in the NQF. I argue that although neuron signals may behave like events that are determined in the classical sense, the control of synaptic connections between neurons occurs at a deeper level, i.e. where we can expect the presence of significant physical activity on the border between quantum and classical processes. The specific assumptions I put forward require the emergence of macroscopic quantum-coherent behavior inside the microtubules of the cytoskeleton of neurons (in exact accordance with the assumptions of Fröhlich). In other words, I believe that the mentioned quantum activity must be related in some non-computable way to the computable process that Hameroff and his colleagues claim takes place inside these very microtubules.
The evidence I have presented indicates that the views held today in certain areas of science do nothing to advance any scientific understanding of the human mind. And yet this does not mean that the phenomenon of consciousness will never find its scientific explanation. I am deeply convinced - in this respect my views have not changed since the time of the NQF - that the scientific path to understanding the phenomena of the mind undoubtedly exists, and this path must begin with a deeper knowledge of the nature of physical reality itself. I think it is extremely important that any serious reader who intends to understand how such an outstanding phenomenon as the mind can be explained in terms of the material physical world, should first have a fairly clear idea of \u200b\u200bhow strange laws can turn out to be. in reality controlling this very “material” of which our physical world consists.
Ultimately, it is for the sake of understanding that we started all of science, and science is still something more than just mindless calculation.
In the following sections (especially c), I will present to your attention several detailed justifications (some of which are quite complex, confusing or ad hoc), the purpose of which is to demonstrate the inability of computational models A and B to act as a probable basis for the study of the phenomenon of mathematical understanding. If the reader does not need such persuasion or is not inclined to dive into details, then I would recommend that he (or she) still start reading, and then, when he is completely tired, go straight to the final imaginary dialogue (). If you then have a desire to return to the missed arguments, I will only be glad, but if not, forget about them and read on.
3.2. Is an unreasonable algorithm able to model mathematical understanding in a knowable way?
According to the conclusion G, in order for mathematical understanding to be the result of the execution of a certain algorithm, this algorithm must be unreasonable or unknowable, but if it is justified and cognizable in itself, then it should be fundamentally impossible to know for sure about its validity (we call such an algorithm unknowably reasonable); moreover, it is possible that various mathematicians "work" on various types such algorithms. By "algorithm" here we mean simply some kind of computational procedure (see), i.e. any set of operations that can, in principle, be simulated on a general purpose computer with an unlimited amount of memory. (As we know from the discussion of the objection Q8, "unlimited" memory size in this idealized case does not affect the results of the reasoning in any way.) Such a concept of an algorithm includes top-down procedures, bottom-up self-learning systems, as well as various combinations of them. This, for example, includes any procedures that can be implemented using artificial neural networks (see). Other types of bottom-up mechanisms also meet this definition - for example, the so-called "genetic algorithms" that increase their efficiency with the help of some built-in procedure similar to Darwinian evolution (see).
About the specifics of the application of the argumentation presented in this section (as well as the arguments put forward in) to ascending procedures, I will speak later in ( summary they can be found in imaginary dialogue,). In the meantime, for greater clarity of presentation, we will reason on the assumption that only one type of algorithmic procedures is involved in the process, namely, descending ones. Such an algorithmic procedure can be attributed both to an individual mathematician and to the mathematical community as a whole. Comments on objections Q11 and Q12, the assumption was considered that different reasonable and known algorithms may be inherent in different people, and we came to the conclusion that this possibility does not affect the results of the reasoning in any significant way. It is also possible that different people perceive the truth through different unfounded and unknowable algorithms; we will return to this issue a little later (see). In the meantime, I repeat, we will assume that mathematical understanding is based on a single algorithmic procedure. It is also possible to restrict the area under consideration to that part of the mathematical understanding that is responsible for the proof of Π-statements (i.e., definitions of those operations of the Turing machine that do not terminate; see the commentary on the objection Q10). In what follows, it is quite sufficient to interpret the combination "mathematical understanding" in just such a limited sense (see the wording G**).
Depending on the cognition, presumably
underlying mathematical understanding of the algorithmic procedure F(whether justified or not), three quite distinct cases should be clearly distinguished. Procedure F may be:
......I consciously knowable, and we also know the fact that it is this algorithmic procedure that is responsible for mathematical understanding;
...II consciously knowable, but the fact that mathematical understanding is based precisely on this algorithmic procedure remains both unconscious and unknowable;
III unconscious and unknowable.
Consider first the fully conscious case I. Since both the algorithm itself and its role are knowable, we may well assume that we are talking about them. already we know. Indeed, nothing prevents us from imagining that all our reasoning takes place after we have received the appropriate knowledge at our disposal - for the word "knowable" just implies that such time, at least in principle, someday it will come. So the algorithm F we know, and at the same time we know its fundamental role in mathematical understanding. As we have already seen (), such an algorithm is effectively equivalent to the formal system F. In other words, it turns out that mathematical understanding - or at least understanding of mathematics by some individual mathematician - is equivalent to deducibility within some formal system F. If we are to retain any hope of satisfying the conclusion G, to which the considerations presented in the previous chapter so unexpectedly led us, we must assume that the system F is an unfounded. However, oddly enough, the groundlessness in this case does not change the situation at all, since, in accordance with I, the well-known formal system F is indeed famous, that is, any mathematician knows and as a consequence, believes that it is this system that underlies his (or her) mathematical understanding. And such faith automatically entails faith (albeit erroneous) in the validity of the system F. (You will agree that the point of view, according to which a mathematician allows himself not to believe in the most fundamental provisions of his own obviously irrefutable system of views, looks extremely unreasonable.) Regardless of whether the system is F really justified faith in its validity already contains the belief that the assertion G(F) (or, alternatively, Ω( F), see) true. However, since we now believe (based on faith in the validity of Gödel's theorem) that the truth of the statement G(F) within the system F unprovable, this contradicts the assumption that the system F is the basis any(essential for the case under consideration) mathematical understanding. (This consideration is equally true both for individual mathematicians and for the entire mathematical community as a whole; it can be applied individually to any of the various algorithms that supposedly form the basis of the thought processes of a particular mathematician. Moreover, according to a preliminary agreement, for us at this time At the moment, the applicability of this consideration only in that area of mathematical understanding that is relevant to the proof of Π-statements is important.) Thus, it is impossible to know for sure that some hypothetical known unfounded algorithm F, which supposedly underlies mathematical understanding, and indeed fulfills this role. Therefore, the case I excluded, regardless of whether the system is F justified or not. If the system F is itself knowable, one should consider the possibility II, the essence of which is that the system F may still form the basis of mathematical understanding, but we are not in a position to learn about its role. Remains in force and possibility III: the system itself F is both unconscious and unknowable.
Career
Penrose's most important scientific contribution can be considered the invention of spin networks ( spin networks) (), which were then actively used to describe the geometry in .
Bibliography
- Structure of space-time
Structure of space-time
The book outlines the basics of rigorous modern methods studies of the structure of space-time. The methods presented are typically mathematical and are distinguished by the necessary rigor. However, Penrose has an excellent physical intuition, and this makes the presentation lively and visual. The author shares with readers those ideas that help him find general and deep solutions to problems. The book is recommended primarily to astronomers, physicists and mathematicians interested in the problems of the general theory of relativity, as well as to students and postgraduates of physical and mathematical specialties.
- Spinors and space-time. Volume 1: Two-Spinor Calculus and Relativistic Fields. Volume 2: Spinor and twistor methods in space-time geometry (with Wolfgang Rindler)
Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields. Volume 2, Spinor and Twistor Methods in Space-Time Geometry
- The new mind of the king. About computers, thinking and the laws of physics
The Emperor's New Mind. Concerning Computers, Minds and The Laws of Physics
Dedicated to the study of the problem of artificial intelligence based on a comprehensive analysis of achievements modern sciences. Is mind modeling possible? To find the answer to this question, Penrose discusses a wide range of phenomena: the algorithmization of mathematical thinking, Turing machines, complexity theory, Godel's theorem, teleportation of matter, the paradoxes of quantum physics, entropy, the birth of the universe, black holes, the structure of the brain, and much more.
- Shadows of the Mind: In Search of a Science of Consciousness. Part 1: Understanding the mind and the new physics. Part 2: The New Physics Needed to Understand the Mind
Shadows of the Mind: A Search for the Missing Science of Consciousness
A book by a famous physicist about modern approaches to the study of brain activity, thought processes, etc. The foundations of the mathematical apparatus are outlined - from classical theory (Gödel's theorem) to the latest achievements related to quantum computing. The book consists of two parts: the first part discusses the thesis of the incomputability of consciousness, the second part deals with the physics and biology necessary to understand the functioning of the real brain. In particular, it is hypothesized that physical basis consciousness.
- The nature of space and time(shared with Stephen Hawking)
The Nature of Space and Time
The main content of the book consists of six lectures that precede the discussion, and a presentation of the actual discussion that took place between Hawking and Penrose on some of the most fundamental issues associated with the nature of the universe. These questions include the `arrow of time`, the initial conditions for the birth of the universe, the absorption of information by black holes, etc. The discussion is largely a continuation of the dispute between Bohr and Einstein on the foundations of quantum mechanics. For a wide audience of readers interested in the problems modern physics and cosmology.
- Big, small and human mind(shared with Abner Shimoni, Nancy Cartwright and Stephen Hawking)
The Large, the Small and the Human Mind
The book is based on the prestigious Tenner Lectures (delivered by Penrose in 1995) and the controversy that these lectures provoked. Therefore, it includes sections written by prominent English scientists Nancy Cartwright and Abner Shimoni, as well as the famous theoretical physicist. The book is distinguished by the originality of the author's ideas, the diversity of the problems discussed (the paradoxes of quantum mechanics, astrophysics, the theory of knowledge, problems of artistic perception) and the exceptionally high scientific and philosophical level of presentation.
- The path to reality, or the laws that govern the universe. Complete guide
The Road to Reality: A Complete Guide to the Laws of the Universe
Awards
- (1988)
- (2000)
Notes
Links
see also
| Laureates for | |
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