The Ghost in the Quantum Turing Machine Scott Aaronson∗ Abstract In honor of Alan Turing’s hundredth birthday, I unwisely set out some thoughts about one of Turing’s obsessions throughout his life, the question of physics and free will. I focus relatively narrowly on a notion that I call “Knightian freedom”: a certain kind of in-principle physical un- predictability that goes beyond probabilistic unpredictability. Other, more metaphysical aspects of free will I regard as possibly outside the scope of science. I examine a viewpoint, suggested independently by Carl Hoefer, Cristi Stoica, and even Turing himself, that tries to find scope for “freedom” in the universe’s boundary conditions rather than in the dynamical laws. Taking this viewpoint seriously leads to many interesting conceptual problems. I investigate how far one can go toward solving those problems, and along the way, encounter (among other things) the No-Cloning Theorem, the measurement problem, decoherence, chaos, the arrow of time, the holographic principle, Newcomb’s paradox, Boltzmann brains, algorithmic information theory, and the Common Prior Assumption. I also compare the viewpoint explored here to the more radical speculations of Roger Penrose. The result of all this is an unusual perspective on time, quantum mechanics, and causation, of which I myself remain skeptical, but which has several appealing features. Among other things, it suggests interesting empirical questions in neuroscience, physics, and cosmology; and takes a millennia-old philosophical debate into some underexplored territory. Contents 1 Introduction 4 1.1 “FreeWill”Versus“Freedom”. ........ 7 1.2 NoteontheTitle.................................. .... 9 1.3 Level........................................... 9 2 FAQ 10 2.1 NarrowScientism................................. ..... 10 2.2 Bait-and-Switch................................. ...... 11 2.3 Compatibilism ................................... 13 2.4 QuantumFlapdoodle............................... ..... 14 2.5 Brain-Uploading: WhoCares?. ........ 16 2.6 Determinism versus Predictability . ........... 19 2.7 Predictability in the Eye of the Beholder . ............ 20 2.8 Quantum Mechanics and Hidden Variables . .......... 21 ∗MIT. Email: [email protected]. This material is based upon work supported by the National Science Foundation under Grant No. 0844626, as well as by an NSF STC grant, a TIBCO Chair, a Sloan Fellowship, and an Alan T. Waterman Award. 1 2.9 TheConsequenceArgument. ...... 23 2.10 ParadoxesofPrediction . ........ 24 2.11 Singularitarianism . ........ 25 2.12 TheLibetExperiments . ...... 26 2.13 MindandMorality ................................ ..... 28 3 Knightian Uncertainty and Physics 29 3.1 KnightianUncertainty . ....... 30 3.2 Quantum Mechanics and the No-Cloning Theorem . .......... 33 3.3 TheFreebitPicture ............................... ..... 36 3.4 AmplificationandtheBrain. ....... 38 3.5 AgainstHomunculi................................ ..... 40 4 Freedom from the Inside Out 41 4.1 TheHarmonizationProblem . ...... 44 4.2 Microfacts and Macrofacts . ........ 46 5 Further Objections 47 5.1 TheAdvertiserObjection . ....... 47 5.2 TheWeatherObjection ............................. ..... 48 5.3 TheGerbilObjection .............................. ..... 49 5.4 TheInitial-StateObjection . ......... 51 5.5 TheWigner’s-FriendObjection . ......... 53 6 Comparison to Penrose’s Views 56 7 “Application” to Boltzmann Brains 61 7.1 WhatHappensWhenWeRunOutofFreebits? . ....... 62 8 Indexicality and Freebits 62 9 Is The Freebit Picture Falsifiable? 65 10 Conclusions 67 10.1 ReasonandMysticism . .. .. .. .. .. .. .. .. ...... 69 11 Acknowledgments 71 12 Appendix: Defining “Freedom” 76 13 Appendix: Prediction and Kolmogorov Complexity 81 14 Appendix: Knightian Quantum States 84 2 Postcard from Alan M. Turing to Robin Gandy, March 1954 (reprinted in Hodges [43]) It reads, in part: Messages from the Unseen World The Universe is the interior of the Light Cone of the Creation Science is a Differential Equation. Religion is a Boundary Condition “Arthur Stanley” refers to Arthur Stanley Eddington, whose books were a major early influence on Turing. 3 1 Introduction When I was a teenager, Alan Turing was at the top of my pantheon of scientific heroes, above even Darwin, Ramanujan, Einstein, and Feynman. Some of the reasons were obvious: the founding of computer science, the proof of the unsolvability of the Entscheidungsproblem, the breaking of the Nazi Enigma code, the unapologetic nerdiness and the near-martyrdom for human rights. But beyond the facts of his biography, I idolized Turing as an “¨uber-reductionist”: the scientist who had gone further than anyone before him to reveal the mechanistic nature of reality. Through his discovery of computational universality, as well as the Turing Test criterion for intelligence, Turing had finally unmasked the pretensions of anyone who claimed there was anything more to mind, brain, or the physical world than the unfolding of an immense computation. After Turing, it seemed to me, one could assert with confidence that all our hopes, fears, sensations, and choices were just evanescent patterns in some sort of cellular automaton: that is, a huge array of bits, different in detail but not in essence from Conway’s famous Game of Life,1 getting updated in time by simple, local, mechanistic rules. So it’s striking that Turing’s own views about these issues, as revealed in his lectures as well as private correspondence, were much more complicated than my early caricature. As a teenager, Turing devoured the popular books of Sir Arthur Eddington, who was one of the first (though not, of course, the last!) to speculate about the implications of the then-ongoing quantum revolution in physics for ancient questions about mind and free will. Later, as a prize from his high school in 1932, Turing selected John von Neumann’s just-published Mathematische Grundlagen der Quan- tenmechanik [65]: a treatise on quantum mechanics famous for its mathematical rigor, but also for its perspective that the collapse of the wavefunction ultimately involves the experimenter’s mental state. As detailed by Turing biographer Andrew Hodges [43], these early readings had a major impact on Turing’s intellectual preoccupations throughout his life, and probably even influenced his 1936 work on the theory of computing. Turing also had a more personal reason for worrying about these “deep” questions. In 1930, Christopher Morcom—Turing’s teenage best friend, scientific peer, and (probably) unrequited love—died from tuberculosis, sending a grief-stricken Turing into long ruminations about the na- ture of personal identity and consciousness. Let me quote from a remarkable disquisition, entitled “Nature of Spirit,” that the 19-year-old Turing sent in 1932 to Christopher Morcom’s mother. It used to be supposed in Science that if everything was known about the Universe at any particular moment then we can predict what it will be through all the future. This idea was really due to the great success of astronomical prediction. More modern science however has come to the conclusion that when we are dealing with atoms and electrons we are quite unable to know the exact state of them; our instruments being made of atoms and electrons themselves. The conception then of being able to know 1Invented by the mathematician John Conway in 1970, the Game of Life involves a large two-dimensional array of pixels, with each pixel either “live” or “dead.” At each (discrete) time step, the pixels get updated via a deterministic rule: each live pixel “dies” if less than 2 or more than 3 of its 8 neighbors were alive, and each dead pixel “comes alive” if exactly 3 of its 8 neighbors were alive. ‘Life’ is famous for the complicated, unpredictable patterns that typically arise from a simple starting configuration and repeated application of the rules. Conway (see [53]) has expressed certainty that, on a large enough Life board, living beings would arise, who would then start squabbling over territory and writing learned PhD theses! Note that, with an exponentially-large Life board (and, say, a uniformly-random initial configuration), Conway’s claim is vacuously true, in the sense that one could find essentially any regularity one wanted just by chance. But one assumes that Conway meant something stronger. 4 the exact state of the universe then really must break down on the small scale. This means then that the theory which held that as eclipses etc. are predestined so were all our actions breaks down too. We have a will which is able to determine the action of the atoms probably in a small portion of the brain, or possibly all over it. The rest of the body acts so as to amplify this. (Quoted in Hodges [43]) The rest of Turing’s letter discusses the prospects for the survival of the “spirit” after death, a topic with obvious relevance to Turing at that time. In later years, Turing would eschew that sort of mysticism. Yet even in a 1951 radio address defending the possibility of human-level artificial intelligence, Turing still brought up Eddington, and the possible limits on prediction of human brains imposed by the uncertainty principle: If it is accepted that real brains, as found in animals, and in particular in men, are a sort of machine it will follow that our digital computer suitably programmed, will behave like a brain. [But the argument for this conclusion] involves several assumptions which can quite reasonably be challenged. [It is] necessary that this machine should