One task that philosophy can perform for scientific researchers is to criticize the logical structure of their arguments and to explore the possibility spaces they may be missing, wrapped up as they generally are in details of the content of their theories, and self – limiting as they often are, in the candidate propostions they entertain to fufill different explanatory roles within them. Philosophy takes a broader and longer view of such things, and can entertain more speculative possibilities than are generally considered in such theorizing.
Hector Zenil had an interaction with Seth Lloyd on quantum computation that he attempts to summarize on his blog. Without imputing Hector’s characterizations to Lloyd himself, I propose to criticize the argument Hector describes. Whether it is a fair characterization of Lloyd is Hector’s affair. The subject being discussed is quantum computation as a model of the universe, and the argument as Hector presents it is as follows –
“His chain of reasoning is basically as follows : a and b imply c
a) the universe is completely describable by quantum mechanics
b) standard quantum computing completely captures quantum mechanics
c) therefore the universe is a quantum computer.”
I will call the claim “a and b together imply c” the overall claim or LZ (read, “big LZ”, for Lloyd according to Zenil), and retain Hector’s labeling of the subcomponents or separate propositions, just using capitals to distinguish them from surrounding text.
The very first thing to notice is that LZ is a directional claim, and not simply an independent statement of the conclusion C. C might hold while LZ is false. Or C might be unsupported while LZ is false, or C might be false while LZ is false. Secondary claims not in evidence that independently support C or claim to, are therefore out of the scope of LZ and of consideration of it. I need not show that C is false or dubitable to refute or render dubitable the claim LZ. I will accordingly direct none of my criticism at C as a substantive claim. That leaves 3 ways in which LZ can fail – A may be false or unsupported; B may be false or unsupported; or A and B combined may not imply or support C.
While Lloyd himself may be chiefly concerned with establishing B, specifically the claim that the Turing computable version of QC can fully describe any QM system, I will direct none of my criticism at that claim. I consider it largely sound, but my reasons for doing so are tangential to LZ or to Lloyd. That intermural debate between different camps of QC operatives, the Turing complete and more than Turing claims for QC, are not relevant to my criticism. My sympathies in that dispute are with the Turing complete version of QC.
Instead my first criticism is directed at A, which I take Lloyd to consider a piece of unremarkable allegiance to a highly successful theory and nothing more, or exactly the sort of faith in theories that have withstood rigorous tests and led to new discoveries that we all honor and support etc. But I do not see the claim A that way, as stated. It isn’t a statement about QM as a theory and its great virtues compared to other theories, it is a statement about the universe and about something called *complete description*. And I deny that QM is a complete description of the universe, or even wants to be. I deny further that the universe is completely describable, full stop. I do not deny that the concept “complete description” can be well formed and can have a real domain of application – there are things that can be completely describable, and even that have been completely described (the times table up to 10 e.g.). But the universe is not one of them, and it isn’t going to be. I will elaborate more on all of this below, but first my second main criticism.
My second criticism is directed at LZ proper, or the claim that A and B even if both true, together imply C. I detect an additional unstated minor premise in this deduction, which I will formulate below and label M. I consider that unstated minor M to itself be false. I also detect an equivocation in the statement C itself – the predication O is P can mean for the copula to be understood in one of two distinct senses. Either the predicate P can be truthfully predicated of the object O, whatever else might also be predicated of O (“accidental copula” or C1, “in addition to being blue and round and crowded on Sundays, the universe also has quantum computation going on somewhere, sometime, and can therefore truthfully be said to “do” quantum computation”) or the predicate P is meant to exhaust O essentially (“being in the strong sense” or C2, “the universe *is* a quantum computer”). I will claim that the second reading of C aka C2 is unsupported by the preceding argument, and the unstated minor premise cannot sustain it. But I believe C2 is the intended reading of C. I will argue, moreover, that even C1 does not strictly follow from A and B jointly, although it is made plausible by them.
In the end, therefore, I will claim that LZ is false, whatever the case may be with C in either sense; that C1 is plausible but does not forcefully follow from A and B jointly; that the unstated minor M is false; and that independently A is also false. Without in any way denigrating QM’s status as our most successful physical theory to date, by far, nor positing any revision to it as a theory. B is untouched, C is left unsupported by LZ but is not affirmed or denied absolutely. Such in preview are my claims. It remains to explain my reasons for each claim, including the formulation of M. In order, those will be (1) what M is and why it is false; (2) why A and B jointly fail to imply more than C1 and that only suggestively, and above all (3) why A is independently false.
What minor premise is required to think C logically follows from A and B jointly?
(M) If A is “completely describable” by B, and C “completely captures” B, then A *is* C.
With suitable equivocation on “complete”, and on what is meant by “capture” and “describe”, this is a claim about the *transitivity* of *models*. And I claim it is in general false. Translations may exist from some system into another that are truth preserving but not reversible. As a first example, a translation may contain many to one mappings. B may lump together some cases that A distinguishes, and C may do the same to B, leaving C underspecific and many-valued in its “inverse image” claims about A. But it might be objected that this would apply to “capture” and “describe”, but not to either modified by “complete” – a complete description might mean no many to one mappings are allowed.
This will not help, however. I need only pick some other aspect of the original A to distort every so slightly in B – say, the number of logical operations required in B to arrive at the modeled result in A. Then there could be behaviors in A that are in principle calcuable with the resources of the universe within the time to its heat-death, that are not calculable in B within that resource limit within that time limit. Such behaviors would then have the attribute “physically realizable, in principle” in A, but not in B. To avoid inverse asymmetries of that kind, not only would one require the complete description to reach the same outcomes, but to do so in the same steps, memory, and speed. If to avoid any “miss” or slipperiness between levels of that sort, one requires the complete description to be instantiated in the same manner as what it describes, then description of the universe becomes impossible. The universe is a set of size one – that is what its first syllable means. An instantiated full copy of it is a contradiction in terms.
In other words, descriptions are always reductions or they are not descriptions, but copies of the original. But there are no actual copies of the universe, and reductions are always many to one maps. And many to one maps are not in general reversible. The unstated minor wants to read “description” as an equality sign. But description as an operator is truth-extracting, but not in general fully preservative. A description cannot say things about what it describes that are untrue, without failing as a (faithful) description. But it can readily leave things out that the thing described, contains. The terms of A attempt to avoid this possibility by saying “complete description”. But if complete is meant in a strong enough sense, a complete description is a contradiction in terms. Description means primarily discourse intended to evoke a mental image of an experienced reality, and secondarily a reduction of anything to its salient characteristics. A description of something is a predication structure – a set of true statements – or occasionally a diagram or map, preserving some formal, relational “bones” of the intended thing. An exactly similar physical instance of the original thing is not a description but a copy. No description, as distinct from a copy, is complete in the sense of invertible. For example, “this is not a body of true statements, but a physical instance” is predictable of the original but not of its description.
Leaving that model-terminological issue aside, consider instead a chain of emulations between universal computational systems, having the structure M claims between three of them, UA, UB, and UC. Then M claims “if UA can be emulated by UB, and UB and be emulated by UC, then UA *is* UC”. This is equivalent to the claim that there is only one computationally universal system. And it is false. As there is one concept of computational unversality (at least for Turing computation), a fair substitute inference M* would claim only that UA can be emulated by UC, and this would be true. Where “emulated by” allows for the usual issues of requiring a translation function and the potential slow down it may involve. Worse for LZ, M* follows regardless of whether the systems involved are the universe, or quantum anything. It would follow if the universe is Turing computable and the UC at the end of the emulation chain were rule 110. But nobody thinks this shows that the universe *is* rule 110, nor that the only alternative is to deny that the universe is Turing computable. In other words, M* applies to too much, and M doesn’t apply at all, to anything. It is just false.
Next I turn to (2), the issue of the equivocation in the sense of the copula in C. In ordinary speech, we sometimes say “X is A” where we intend merely to predicate A of X accidentally – the moon is spherical. Here it is enough that the statement be true (even roughly true, the approximate sense involved in all predication being understood), and no claim is being advanced that the essence of the moon is its spherical shape, nor are any other predicates being denied to it. At best, a few implicit constrasting predicates are being denied by implication – the moon isn’t triangular or completely flat, since those predicates contradict the true one. Call C1 the reading of C that understands the copula in this sense. C1 claims that whatever else it happens to be, the universe also has the distinct characteristics of a quantum computer. Given the encompassing definition of “universe”, this reduces to the statement that quantum computation is physically realized, anywhere and at any time. For if quantum computation is ever and anywhere physically realized, C1 holds. What else is supposed to be realizing it? Unless QC were nothing but a mathematical abstraction, C1 would hold, independent of LZ. I submit this shows that the stronger claim, C2, is in fact intended by C.
But first, leave aside the truth and inherent plausibility of even just C1, and ask whether it actually follows from A and B jointly. I submit that it does not. For B could hold even if QC *were* a mathematical abstraction, and never realized anywhere in the actual universe. QC is a theory and a description. Call QC beyond a few bits, worked up to the level of a working universal computer, UQC (U for universal). And call QC not as a theory but actually instantiated in the history of the universe with physical components, IQC (I for instantiated). Call the joint case of instantiated and universal QC, IUQC. Assume that behaviors actually occur in the universe sufficiently complex to require universality to model them (this is a more heroic assumption than it may seem, given finite nature possibilities and cardinality issues with universality, but let that slide). Then I claim that B could hold even if IUQC did not. There could be IQC, as QM provides sufficient basis for it. And there could be UQC in the realm of mathematical abstraction. But IUQC might fail to appear – if, say, all the instances of universal computation the universe actually happened to instantiate used unentangled components, and all the instances of true QC physically realized were simple enough to fall below the threshold of universality. If this does *not* occur, we will know it does not occur because of empirical evidence from actual instances of IUQC. Not as a supposedly logical consequence of B, assuming B is true. Logical consequences of possibilities of theories like B do not tell us what actually happened in the instantiated history of the universe. Observation does. Personally I think C1 is true. But I don’t think C1 follows logically from B. And it won’t be theoretical attributes of QC as a formal theory that convince me C1 is true, it will be an actual, empirical instance of IUQC. Hence my statement that C1 is plausible given A and B (or even if not given A, see below), but does not follow from them with logical necessity.
The intended meaning of C, however, is C2, strong *is*, essence or exhaustion. And while a claim of C2 might be rendered more plausible by A and B, it does not follow from them logically. Again it would be possible that QC models QM but is a mathematical abstraction that captures QM, without actually being the real, external, instantiated process that the universe follows – even if A and B were true in a suitably strong sense of “completely”. For it is one thing to say that QC models either QM or physical reality, and another to say that QC is physical reality. It might be, but this would be an independent claim and does not follow from QC modeling physical reality. Any more than rule 110 is my laptop because it can emulate any computation performed by my laptop, albeit more slowly.
In addition, C2 can fail because its stated premises fail. Leaving aside M as already adequately addressed above, we arrive at a still more basic problem. The primary premise A is also false. The universe is not fully describable by QM. And this, for multiple reasons – because QM does not even try to fully describe even the systems and relations to which it properly applies (NotA1), because QM is not all of physics (NotA2), because all of physics does not exhaust description of the universe (NotA3), and because the universe is not fully describable, full stop (NotA4).
Let us start with NotA1. Quantum mechanics is a deliberately underspecific theory. It describes the evolution of a probability amplitude, and only predicts that the absolute square of this quantity will approximate the sample average of classes of events it regards as equivalent. Strictly speaking, it can’t make contact with empirical reality (meant in the precise, exact sense of that compound term – not in a loose sense) in any other manner.
Anything is a reality, in the scholastic definition of that term, if its truth and content do not depend on a specific observor or that observor’s internal state. Loosely, realities are not matters of opinion. Anything is empirical, in the early modern sense of that term as appropriated by empiricists, if it is given directly in sense experience. An empirical reality, therefore, must be given directly to sense experience yet remain invariant across experiencers. Single observations of events of a given class described by quantum mechanics do not have this character, and QM does not claim that they do (some observors see an electron go that-a-way, and some see an electron go this-a-way). Sample averages of events described by quantum mechanics and regarded by it as equivalent to one another, do have this character. And QM deliberately restricts itself to making claims about such averages.
The universe, on the other hand, does not consist of probabilistic claims about classes of events regarded as equivalent under some theory. Definite things happen. The universe has a sample size of one, as its first syllable emphasizes. If one adopts a many-worlds interpretation of QM and regards all possible outcomes as real, one still has not explained the specific experience of oneself as an observor, which is part of the universe. Stipulate that there are a plethora of instances of oneself experiencing every possible outcome of each successive quantum event. QM does not describe which of these “you” will subjectively experience. At most, the voluntary addition to QM of a many-worlds interpretation (which is not itself a part of QM) may wave hands about the orthogonality of each of the possible experiences QM suggests could have occurred. Leaving aside the problem of probability measures across possible worlds in a universe of sample size one, QM itself is not even attempting to fully describe any selection, objective subjective or illusory, of world-lines through that branching possibility space.
Now, a doctrinaire many-worlds advocate may wish to deny the reality of anything left unspecified by QM. But he is abusing the term “reality” in doing so. It is exactly the actual historical observations shared by all inhabitants of any of his hypothetical orthogonal slices, that fit the scholastic definition of “real” – invariance over existing observors, each fully capable of talking to each other and communicating their experiences. While it is exactly his hypothetical alternate realities to which neither he nor anyone he talks to has immediate experiential access, that fail to match the definition of “empirical”. But let us leave aside the complexities of many-worlds interpretations of QM and their claims of exhaustive determinism; it would take us too far afield.
The more basic point must be seen that all descriptions as reductions are sets of true statements about a described, but that such sets are not the original being described. It is one thing to provide a map and another to provide a fully instantiated copy of the original. Descriptions as such achieve their reliability, accuracy, and usefulness by lumping some things together and making statements about the lumps that are true. QM does not attempt to describe the space-time trajectories of every electron since the begining of the universe, but instead, precisely, lumps together lots of disparate space-time events as involving a simplifying unity called “electrons”. The usefulness of a description lies in its generalities, but generalities precisely underdescribe single instances. In scholastic language, theories are about reality, while history is an actuality.
Next, NotA2, QM is not the whole of physics. There is no consistent theory of quantum gravity, but gravity is physically real. Perhaps there will someday be a consistent theory of quantum gravity, or perhaps conventional gravity theory will be found to be incorrect and will be replaced by some superior (but independent) theory. But A is a claim about QM as it stands, and QM as it stands does not even attempt to describe the gravitational behavior of bodies. QM is therefore not an exhaustive description of even physical reality.
Next, NotA3, physics and physical reality, defined as the subset of consistent relations discoverable about the material universe, do not exhaust the universe. The second syllable of “universe” refers to truth as such, and truth as such is not limited to the material world. While cosmologists these days use the term quite loosely, its proper meaning encompasses all truth of whatever character. Mathematical abstraction describes truths which are not described by physical law, but are no less true for that. Mathematical truth is more general than any specific possible world, but is smaller than and properly contained by the concept “universe”.
For that matter, there are truths about conventional categorizations, specifics of actuality, and of individual instances, that physical science even in the most general sense does not even attempt to speak about, let alone claims to fully describe, let alone actually captures fully. “Julius Caesar was a Roman politician assassinated in the senate in the year 44 BC”, is not a piece of physics, but it is a truth about the universe. It differs from physical law in the other direction, so to speak, than the mathematical truth case, above. It is more particular than the laws of physics, and refers to a single definite actuality. But it is part of the universe, just the same. The point is, physical reality lies in a specific place in terms of its generality-relations with the universe as a whole. A map isn’t the territory, nor mapping in general, nor geometry, nor logic, nor the history of the territory, nor its specific configuration at a specific instant in time, etc.
One might imagine reducing each of the categories involved in that statement to underlying physical relations, along the lines of the positivist program of trying to reduce every statement to either a statement about empirical facts or to nonsense, but this is actually hopeless. Truth is not confined to one level of description, and you could not derive many readily accessible “overlying” truths from underlyings buried deep enough below them, within the lifetime of the universe, using physically realizable amounts of computational resources. We can’t solve N body problems analytically, and quantum field theory for single large molecules is already intractable. Minimum energy configurations e.g. in protein folding are NP complete, generalized Ising models are known to be formally undecidable by reduction to known tiling problems, etc.
The idea that reduction can solve all overlying problems “in principle” is computationally naive. Overlying relations may in fact be vastly simpler than the reduction and may already encompass all the computationally important substitutions, when a process is reducible enough to actually be solved. And an accurate reduction may simply land in a morass of intractable unsolvability. The idea that ultimate and simple answers to everything lie at the bottom is simply false. It cannot withstand rational scrutiny. Since computational intractability will arise in any attempt to reformulate all problems in “lowest level” terms, QM will not succeed in exhaustively describing the universe, and other higher level descriptions will frequently succeed where a full QM treatment would be utterly hopeless from the start. One might hypothesize that in all cases of useful higher order descriptions, those descriptions are ultimately compatible with QM. But that needn’t mean they are derivable from QM in the lifetime of the universe, and in fact the higher level descriptions could be logically independent of the microlevel (the Taj Mahal is made out of bricks; this does not make the artistic character of that building a property deducible from the nature of bricks) and yet still hold for every actuality anyone will ever encounter. Enough, infatuation with the bottom level is simply that, a piece of boasting self importance from certain physicists, and not a dictate of reason.
But I am still not done with the ways in which A is false. I claim NotA4, that the universe is not fully describable, full stop. First as to mathematical truth, you aren’t going to capture all of it in one model, see Godel. Second as to historical actuality, stipulate that you have a perfect theory of all physical truth, and can moreover guess the boundary conditions of history. You still won’t be able to work out what happens inside – the consequence of your own theory – with the computational resources of only a subset of the universe. Sure, in principle you might be able to “run faster” that many subsets of the universe, by clever substitutions say, avoiding the need to mimic the universe’s own, full, computational capacity. But you are still a flyspeck on a galaxy – beyond the lightcone of everything that has ever interacted with you, or will in the next 60 billion years, you have no idea what physical immensities remain. Stipulate that you can guess averages and tendencies, and that every law you heroically generalize beyond the range of your empirical experience holds up perfectly. Completely describing the universe still requires you to give an elaborate aesthetic commentary on the exquisite electromagnetic symphonies of Hordomozart the Twenththird, of an obscure planet around an unseen star buried in the heart of an unsensed galactic supercluster 400 gazillion parsecs east-south-east of the great attractor, 46 billion years from now.
Actuality is measure epsilon in possibility-space. The most a good theory can give you is a map of generalities that will hold across wide swathes of space and time. You will not find the specific actuality that actually obtains by staring at the possibility-space such a map describes forever. At some point you have to go look at the actual world. And when you do, you will find yourself at the bottom of a well looking out through a straw at an immensity. Honesty and rationality start with an elementary humility in the face of this inescapable fact. This does not mean you can’t know things; you can. It does mean you can’t know everything.
And it means the actual evolution of the universe, actual history, is continually and meaningfully accomplishing something and adding “more”. Even if the universe follows definite laws (deterministic ones or objectively stochastic ones, either way) and even if you can guess those laws; even if the Church-Turing thesis is true and the universe is formally within the space of the computables, and even if the physical universe happens to be finite. None of those things implies “small” or “simple” or “solved”. Even a finite, computable, deterministic or QC universe will be bigger than every attempt to describe it ever achieved by all finite rational beings combined. It is vastly bigger than we are and just as complicated. Knowledge is not about getting all of it inside our heads – our heads are not big enough and would split wide open – but about getting outside of our heads and out into the midst of it, discovering definite true things about it that are helpful, for us.