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«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 ...»

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The Ghost in the Quantum Turing Machine

Scott Aaronson∗


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 unpredictability 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 “Free Will” Versus “Freedom”............................... 7

1.2 Note on the Title...................................... 9

1.3 Level............................................. 9 2 FAQ 10

2.1 Narrow Scientism...................................... 10

2.2 Bait-and-Switch....................................... 11

2.3 Compatibilism.

–  –  –

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.

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 “¨ber-reductionist”: the scientist who u 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 Quantenmechanik [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 nature 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 Invented 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.

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 be of the sort whose behaviour is in principle predictable by calculation. We certainly do not know how any such calculation should be done, and it was even argued by Sir Arthur Eddington that on account of the indeterminacy principle in quantum mechanics no such prediction is even theoretically possible.2 (Reprinted in Shieber [80]) Finally, two years after his sentencing for “homosexual indecency,” and a few months before his tragic death by self-poisoning, Turing wrote the striking aphorisms that I quoted earlier: “The universe is the interior of the light-cone of the Creation. Science is a differential equation. Religion is a boundary condition.” The reason I’m writing this essay is that I think I now understand what Turing could have meant by these remarks. Building on ideas of Hoefer [45], Stoica [85], and others, I’ll examine a perspective—which I call the “freebit perspective,” for reasons to be explained later—that locates a nontrivial sort of freedom in the universe’s boundary conditions, even while embracing the mechanical nature of the time-evolution laws. We’ll find that a central question, for this perspective, is how well complicated biological systems like human brains can actually be predicted: not by hypothetical Laplace demons, but by prediction devices compatible with the laws of physics. It’s in the discussion of this predictability question (and only there) that quantum mechanics enters the story.

Of course, the idea that quantum mechanics might have something to do with free will is not new; neither are the problems with that idea or the controversy surrounding it. While I chose Turing’s postcard for the opening text of this essay, I also could have chosen a striking claim by

Niels Bohr, from a 1932 lecture about the implications of Heisenberg’s uncertainty principle:

[W]e should doubtless kill an animal if we tried to carry the investigation of its organs so far that we could tell the part played by the single atoms in vital functions. In As Hodges (personal communication) points out, it’s interesting to contrast these remarks with a view Turing had expressed just a year earlier, in “Computing Machinery and Intelligence” [89]: “It is true that a discrete-state machine must be different from a continuous machine. But if we adhere to the conditions of the imitation game, the interrogator will not be able to take any advantage of this difference.” Note that there’s no actual contradiction between this statement and the one about the uncertainty principle, especially if we distinguish (as I will) between simulating a particular brain and simulating some brain-like entity able to pass the Turing test. However, I’m not aware of any place where Turing explicitly makes that distinction.

every experiment on living organisms there must remain some uncertainty as regards the physical conditions to which they are subjected, and the idea suggests itself that the minimal freedom we must allow the organism will be just large enough to permit it, so to say, to hide its ultimate secrets from us. (Reprinted in [17])

Or this, from the physicist Arthur Compton:

A set of known physical conditions is not adequate to specify precisely what a forthcoming event will be. These conditions, insofar as they can be known, define instead a range of possible events from among which some particular event will occur. When one exercises freedom, by his act of choice he is himself adding a factor not supplied by the physical conditions and is thus himself determining what will occur. That he does so is known only to the person himself. From the outside one can see in his act only the working of physical law. [24]

I want to know:

Were Bohr and Compton right or weren’t they? Does quantum mechanics (specifically, say, the No-Cloning Theorem or the uncertainty principle) put interesting limits on an external agent’s ability to scan, copy, and predict human brains and other complicated biological systems, or doesn’t it?

Of course, one needs to spell out carefully what one means by “interesting limits,” an “external agent,” the “ability to scan, copy, and predict,” and so forth.3 But once that’s done, I regard the above as an unsolved scientific question, and a big one. Many people seem to think the answer is obvious (though they disagree on what it is!), or else they reject the question as meaningless, unanswerable, or irrelevant. In this essay I’ll argue strongly for a different perspective: that we can easily imagine worlds consistent with quantum mechanics (and all other known physics and biology) where the answer to the question is yes, and other such worlds where the answer is no.

And we don’t yet know which kind we live in. The most we can say is that, like P versus NP or the nature of quantum gravity, the question is well beyond our current ability to answer.

Furthermore, the two kinds of world lead, not merely to different philosophical stances, but to different visions of the remote future. Will our descendants all choose to upload themselves into a digital hive-mind, after a “technological singularity” that makes such things possible? Will they then multiply themselves into trillions of perfect computer-simulated replicas, living in various simulated worlds of their own invention, inside of which there might be further simulated worlds with still more replicated minds? What will it be like to exist in so many manifestations: will each copy have its own awareness, or will they comprise a single awareness that experiences trillions of times more than we do? Supposing all this to be possible, is there any reason why our descendants might want to hold back on it?

Now, if it turned out that Bohr and Compton were wrong—that human brains were as probabilistically predictable by external agents as ordinary digital computers equipped with randomnumber generators—then the freebit picture that I explore in this essay would be falsified, to whatever extent it says anything interesting. It should go without saying that I see the freebit picture’s vulnerability to future empirical findings as a feature rather than a bug.

My own attempt to do so is in Appendix 12.

In summary, I’ll make no claim to show here that the freebit picture is true. I’ll confine myself

to two weaker claims:

(1) That the picture is sensible (or rather, not obviously much crazier than the alternatives):

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