The Emperor's New Mind
Penrose (Roger)
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Foreword by Martin Gardner (Full Text)

  1. Many great mathematicians and physicists find it difficult, if not impossible, to write a book that non-professionals can understand. Until this year one might have supposed that Roger Penrose, one of the world's most knowledgeable and creative mathematical physicists, belonged to such a class. Those of us who had read his non-technical articles and lectures knew better. Even so, it came as a delightful surprise to find that Penrose had taken time off from his labours to produce a marvellous book for informed laymen. It is a book that I believe will become a classic.
  2. Although Penrose's chapters range widely over relativity theory, quantum mechanics1, and cosmology, their central concern is what philosophers call the ‘mind—body problem'. For decades now the proponents of ‘strong Al' (Artificial Intelligence)2 have tried to persuade us that it is only a matter of a century or two (some have lowered the time to fifty years!) until electronic computers will be doing everything a human mind can do. Stimulated by science fiction read in their youth, and convinced that our minds are simply ‘computers made of meat' (as Marvin Minsky once put it), they take for granted that pleasure and pain, the appreciation of beauty and humour, consciousness, and free will are capacities that will emerge naturally when electronic robots become sufficiently complex in their algorithmic behaviour.
  3. Some philosophers of science (notably John Searle, whose notorious Chinese room thought experiment3 is discussed in depth by Penrose), strongly disagree. To them a computer is not essentially different from mechanical calculators that operate with wheels, levers, or anything that transmits signals. (One can base a computer on rolling marbles or water moving through pipes.) Because electricity travels through wires faster than other forms of energy (except light) it can twiddle symbols more rapidly than mechanical calculators, and therefore handle tasks of enormous complexity. But does an electrical computer ‘understand' what it is doing in a way that is superior to the ‘understanding' of an abacus? Computers now play grandmaster chess. Do they 'understand' the game any better than a tick-tack-toe machine that a group of computer hackers once constructed with tinker toys?
  4. Penrose's book is the most powerful attack yet written on strong Al. Objections have been raised in past centuries to the reductionist claim that a mind is a machine operated by known laws of physics, but Penrose's offensive is more persuasive because it draws on information not available to earlier writers. The book reveals Penrose to be more than a mathematical physicist. He is also a philosopher of first rank, unafraid to grapple with problems that contemporary philosophers tend to dismiss as meaningless.
  5. Penrose also has the courage to affirm, contrary to a growing denial by a small group of physicists, a robust realism. Not only is the universe ‘out there', but mathematical truth also has its own mysterious independence and timelessness. Like Newton and Einstein, Penrose has a profound sense of humility and awe toward both the physical world and the Platonic realm of pure mathematics. The distinguished number theorist Paul Erdos likes to speak of ‘God's book' in which all the best proofs are recorded. Mathematicians are occasionally allowed to glimpse part of a page. When a physicist or a mathematician experiences a sudden ‘aha' insight, Penrose believes, it is more than just something 'conjured up by complicated calculation'. It is mind making contact for a moment with objective truth. Could it be, he wonders, that Plato's world and the physical world (which physicists have now dissolved into mathematics) are really one and the same?
  6. Many pages in Penrose's book are devoted to a famous fractal-like structure called the Mandelbrot set after Benoit Mandelbrot who discovered it. Although self-similar in a statistical sense as portions of it are enlarged, its infinitely convoluted pattern keeps changing in unpredictable ways. Penrose finds it incomprehensible (as do I) that anyone could suppose that this exotic structure is not as much ‘out there' as Mount Everest is, subject to exploration in the way a jungle is explored.
  7. Penrose is one of an increasingly large band of physicists who think Einstein was not being stubborn or muddle-headed when he said his ‘little finger' told him that quantum mechanics4 is incomplete. To support this contention, Penrose takes you on a dazzling tour that covers such topics as complex numbers, Turing machines, complexity theory, the bewildering paradoxes of quantum mechanics5, formal systems, Godel undecidability, phase spaces, Hilbert spaces, black holes, white holes, Hawking radiation, entropy, the structure of the brain, and scores of other topics at the heart of current speculations. Are dogs and cats ‘conscious' of themselves? Is it possible in theory for a matter-transmission machine to translocate a person from here to there the way astronauts are beamed up and down in television's Star Trek series? What is the survival value that evolution found in producing consciousness? Is there a level beyond quantum mechanics6 in which the direction of time and the distinction between right and left are firmly embedded? Are the laws of quantum mechanics7, perhaps even deeper laws, essential for the operation of a mind?
  8. To the last two questions Penrose answers yes. His famous theory of ‘twistors' — abstract geometrical objects which operate in a higher-dimensional complex space that underlies space-time — is too technical for inclusion in this book. They are Penrose's efforts over two decades to probe a region deeper than the fields and particles of quantum mechanics8. In his fourfold classification of theories as superb, useful, tentative, and misguided, Penrose modestly puts twistor theory in the tentative class, along with superstrings and other grand unification schemes now hotly debated.
  9. Since 1973 Penrose has been the Rouse Ball Professor of Mathematics at Oxford University. The title is appropriate because W. W. Rouse Ball not only was a noted mathematician, he was also an amateur magician with such an ardent interest in recreational mathematics that he wrote the classic English work on this field, Mathematical Recreations and Essays. Penrose shares Ball's enthusiasm for play. In his youth he discovered an ‘impossible object' called a ‘tribar'. (An impossible object is a drawing of a solid figure that cannot exist because it embodies self-contradictory elements.) He and his father Lionel, a geneticist, turned the tribar into the Penrose Staircase, a structure that Maurits Escher used in two well-known lithographs: Ascending and Descending, and Waterfall. One day when Penrose was lying in bed, in what he called a ‘fit of madness', he visualized an impossible object in four-dimensional space. It is something, he said, that a four-space creature, if it came upon it, would exclaim ‘My God, what's that?'
  10. During the 1960s, when Penrose worked on cosmology with his friend Stephen Hawking, he made what is perhaps his best known discovery. If relativity theory holds ‘all the way down', there must be a singularity in every black hole where the laws of physics no longer apply. Even this achievement has been eclipsed in recent years by Penrose's construction of two shapes that tile the plane, in the manner of an Escher tessellation, but which can tile it only in a non-periodic way. (You can read about these amazing shapes in my book Penrose Tiles to Trapdoor Ciphers.) Penrose invented them, or rather discovered them, without any expectation they would be useful. To everybody's astonishment it turned out that three-dimensional forms of his tiles may underlie a strange new kind of matter. Studying these 'quasicrystals' is now one of the most active research areas in crystallography. It is also the most dramatic instance in modern times of how playful mathematics can have unanticipated applications.
  11. Penrose's achievements in mathematics and physics — and I have touched on only a small fraction — spring from a lifelong sense of wonder toward the mystery and beauty of being. His little finger tells him that the human mind is more than just a collection of tiny wires and switches. The Adam of his prologue and epilogue is partly a symbol of the dawn of consciousness in the slow evolution of sentient life. To me he is also Penrose — the child sitting in the third row, a distance back from the leaders of Al - who dares to suggest that the emperors of strong AI have no clothes. Many of Penrose's opinions are infused with humour, but this one is no laughing matter.

    Prologue – 1
  1. Can A Computer Have A Mind? – 3
    • Introduction – 3
    • The Turing test – 6
    • Artificial intelligence9 – 14
    • An Al approach to ‘pleasure' and ‘pain' – 17
    • Strong Al and Searle's Chinese room – 21
    • Hardware and software – 30
  2. Algorithms And Turing Machines – 40
    • Background to the algorithm concept – 40
    • Turing's concept – 46
    • Binary coding of numerical data – 56
    • The Church-Turing Thesis – 61
    • Numbers other than natural numbers – 65
    • The universal Turing machine – 67
    • The insolubility of Hilbert's problem – 75
    • How to outdo an algorithm – 83
    • Church's lambda calculus – 86
  3. Mathematics And Reality – 98
    • The land of Tor'Bled-Nam – 98
    • Real numbers – 105
    • How many real numbers are there? – 108
    • ‘Reality' of real numbers – 112
    • Complex numbers – 114
    • Construction of the Mandelbrot set – 120
    • Platonic reality of mathematical concepts? – 123
  4. Truth, Proof, And Insight – 129
    • Hilbert's programme for mathematics – 129
    • Formal mathematical systems – 133
    • Godel's theorem – 138
    • Mathematical insight – 141
    • Platonism or intuitionism? – 146
    • Godel-type theorems from Turing's result – 151
    • Recursively enumerable sets – 155
    • Is the Mandelbrot set recursive? – 161
    • Some examples of non-recursive mathematics – 168
    • Is the Mandelbrot set like non-recursive mathematics? – 177
    • Complexity theory – 181
    • Complexity and computability in physical things – 188
  5. The Classical World – 193
    • The status of physical theory – 193
    • Euclidean geometry – 202
    • The dynamics of Galileo and Newton – 209
    • The mechanistic world of Newtonian dynamics – 217
    • Is life in the billiard-ball world computable? – 220
    • Hamiltonian mechanics – 225
    • Phase space – 228
    • Maxwell's electromagnetic theory – 238
    • Computability and the wave equation – 243
    • The Lorentz equation of motion; runaway particles – 244
    • The special relativity of Einstein and Poincare – 248
    • Einstein's general relativity – 261
    • Relativistic causality10 and determinism – 273
    • Computability in classical physics: where do we stand? – 278
    • Mass, matter, and reality – 280
  6. Quantum Magic And Quantum Mystery – 291
    • Do philosophers need quantum theory11? – 291
    • Problems with classical theory – 295
    • The beginnings of quantum theory12 – 297
    • The two-slit experiment – 299
    • Probability amplitudes – 306
    • The quantum state of a particle – 314
    • The uncertainty principle – 321
    • The evolution procedures U and R – 323
    • Particles in two places at once? – 325
    • Hilbert space – 332
    • Measurements – 336
    • Spin and the Riemann sphere of states – 341
    • Objectivity and measurability of quantum states – 346
    • Copying a quantum state – 348
    • Photon spin – 349
    • Objects with large spin – 353
    • Many-particle systems – 355
    • The ‘paradox' of Einstein, Podolsky, and Rosen – 361
    • Experiments with photons: a problem for relativity? – 369
    • Schrodinger's equation; Dirac's equation – 372
    • Quantum field theory – 374
    • Schrodinger's cat – 375
    • Various attitudes in existing quantum theory13 – 379
    • Where does all this leave us? – 383
  7. Cosmology And The Arrow Of Time – 391
    • The flow of time – 391
    • The inexorable increase of entropy – 394
    • What is entropy? – 400
    • The second law in action – 407
    • The origin of low entropy in the universe – 411
    • Cosmology and the big bang – 417
    • The primordial fireball – 423
    • Does the big bang explain the second law? – 426
    • Black holes – 427
    • The structure of space-time singularities – 435
    • How special was the big bang? – 440
  8. In Search Of Quantum Gravity – 450
    • Why quantum gravity? – 450
    • What lies behind the Weyl curvature hypothesis? – 453
    • Time-asymmetry in state-vector reduction – 458
    • Hawking's box: a link with the Weyl curvature hypothesis? – 465
    • When does the state-vector reduce? – 475
  9. Real Brains And Model Brains – 483
    • What are brains actually like? – 483
    • Where is the seat of consciousness? – 492
    • Split-brain experiments – 496
    • Blindsight – 499
    • Information processing in the visual cortex – 500
    • How do nerve signals work? – 502
    • Computer models – 507
    • Brain plasticity – 512
    • Parallel computers and the ‘oneness' of consciousness – 514
    • Is there a role for quantum mechanics14 in brain activity? – 516
    • Quantum computers – 518
    • Beyond quantum theory15? – 520
  10. Where Lies The Physics Of Mind? – 523
    • What are minds for? – 523
    • What does consciousness actually do? – 529
    • Natural selection of algorithms? – 534
    • The non-algorithmic nature of mathematical insight – 538
    • Inspiration, insight, and originality – 541
    • Non-verbality of thought – 548
    • Animal consciousness? – 550
    • Contact with Plato's world – 552
    • A view of physical reality – 555
    • Determinism and strong determinism – 558
    • The anthropic principle – 560
    • Tilings and quasicrystals – 562
    • Possible relevance to brain plasticity – 566
    • The time-delays of consciousness – 568
    • The strange role of time in conscious perception – 573
    • Conclusion: a child's view – 578
  11. Epilogue – 583

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  1. Blue: Text by me; © Theo Todman, 2021
  2. Mauve: Text by correspondent(s) or other author(s); © the author(s)

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