Archive for the ‘Metaphysical Spouting’ Category

Integrated Information Theory: Virgil Griffith opines

Wednesday, June 25th, 2014

Remember the two discussions about Integrated Information Theory that we had a month ago on this blog?  You know, the ones where I argued that IIT fails because “the brain might be an expander, but not every expander is a brain”; where IIT inventor Giulio Tononi wrote a 14-page response biting the bullet with mustard; and where famous philosopher of mind David Chalmers, and leading consciousness researcher (and IIT supporter) Christof Koch, also got involved in the comments section?

OK, so one more thing about that.  Virgil Griffith recently completed his PhD under Christof Koch at Caltech—as he puts it, “immersing [him]self in the nitty-gritty of IIT for the past 6.5 years.”  This morning, Virgil sent me two striking letters about his thoughts on the recent IIT exchanges on this blog.  He asked me to share them here, something that I’m more than happy to do:

Reading these letters, what jumped out at me—given Virgil’s long apprenticeship in the heart of IIT-land—was the amount of agreement between my views and his.  In particular, Virgil agrees with my central contention that Φ, as it stands, can at most be a necessary condition for consciousness, not a sufficient condition, and remarks that “[t]o move IIT from talked about to accepted among hard scientists, it may be necessary for [Tononi] to wash his hands of sufficiency claims.”  He agrees that a lack of mathematical clarity in the definition of Φ is a “major problem in the IIT literature,” commenting that “IIT needs more mathematically inclined people at its helm.”  He also says he agrees “110%” that the lack of a derivation of the form of Φ from IIT’s axioms is “a pothole in the theory,” and further agrees 110% that the current prescriptions for computing Φ contain many unjustified idiosyncrasies.

Indeed, given the level of agreement here, there’s not all that much for me to rebut, defend, or clarify!

I suppose there are a few things.

  1. Just as a clarifying remark, in a few places where it looks from the formatting like Virgil is responding to something I said (for example, “The conceptual structure is unified—it cannot be decomposed into independent components” and “Clearly, a theory of consciousness must be able to provide an adequate account for such seemingly disparate but largely uncontroversial facts”), he’s actually responding to something Giulio said (and that I, at most, quoted).
  2. Virgil says, correctly, that Giulio would respond to my central objection against IIT by challenging my “intuition for things being unconscious.”  (Indeed, because Giulio did respond, there’s no need to speculate about how he would respond!)  However, Virgil then goes on to explicate Giulio’s response using the analogy of temperature (interestingly, the same analogy I used for a different purpose).  He points out how counterintuitive it would be for Kelvin’s contemporaries to accept that “even the coldest thing you’ve touched actually has substantial heat in it,” and remarks: “I find this ‘Kelvin scale for C’ analogy makes the panpsychism much more palatable.”  The trouble is that I never objected to IIT’s panpsychism per se: I only objected to its seemingly arbitrary and selective panpsychism.  It’s one thing for a theory to ascribe some amount of consciousness to a 2D grid or an expander graph.  It’s quite another for a theory to ascribe vastly more consciousness to those things than it ascribes to a human brain—even while denying consciousness to things that are intuitively similar but organized a little differently (say, a 1D grid).  A better analogy here would be if Kelvin’s theory of temperature had predicted, not merely that all ordinary things had some heat in them, but that an ice cube was hotter than the Sun, even though a popsicle was, of course, colder than the Sun.  (The ice cube, you see, “integrates heat” in a way that the popsicle doesn’t…)
  3. Virgil imagines two ways that an IIT proponent could respond to my argument involving the cerebellum—the argument that accuses IIT proponents of changing the rules of the game according to convenience (a 2D grid has a large Φ?  suck it up and accept it; your intuitions about a grid’s lack of consciousness are irrelevant.  the human cerebellum has a small Φ?  ah, that’s a victory for IIT, since the cerebellum is intuitively unconscious).  The trouble is that both of Virgil’s imagined responses are by reference to the IIT axioms.  But I wasn’t talking about the axioms themselves, but about whether we’re allowed to validate the axioms, by checking their consequences against earlier, pre-theoretic intuitions.  And I was pointing out that Giulio seemed happy to do so when the results “went in IIT’s favor” (in the cerebellum example), even though he lectured me against doing so in the cases of the expander and the 2D grid (cases where IIT does less well, to put it mildly, at capturing our intuitions).
  4. Virgil chastises me for ridiculing Giulio’s phenomenological argument for the consciousness of a 2D grid by way of nursery rhymes: “Just because it feels like something to see a wall, doesn’t mean it feels like something to be a wall.  You can smell a rose, and the rose can smell good, but that doesn’t mean the rose can smell you.”  Virgil amusingly comments: “Even when both are inebriated, I’ve never heard [Giulio] nor [Christof] separately or collectively imply anything like this.  Moreover, they’re each far too clueful to fall for something so trivial.”  For my part, I agree that neither Giulio nor Christof would ever advocate something as transparently silly as, “if you have a rich inner experience when thinking about X, then that’s evidence X itself is conscious.”  And I apologize if I seemed to suggest they would.  To clarify, my point was not that Giulio was making such an absurd statement, but rather that, assuming he wasn’t, I didn’t know what he was trying to say in the passages of his that I’d just quoted at length.  The silly thing seemed like the “obvious” reading of his words, and my hermeneutic powers were unequal to the task of figuring out the non-silly, non-obvious reading that he surely intended.

Anyway, there’s much more to Virgil’s letters than the above—including answers to some of my subsidiary questions about the details of IIT (e.g., how to handle unbalanced partitions, and the mathematical meanings of terms like “mechanism” and “system of mechanisms”).  Also, in parts of the letters, Virgil’s main concern is neither to agree with me nor to agree with Giulio, but rather to offer his own ideas, developed in the course of his PhD work, for how to move forward and fix some of the problems with IIT.  All in all, these are recommended reads for anyone who’s been following this debate.

Giulio Tononi and Me: A Phi-nal Exchange

Friday, May 30th, 2014

You might recall that last week I wrote a post criticizing Integrated Information Theory (IIT), and its apparent implication that a simple Reed-Solomon decoding circuit would, if scaled to a large enough size, bring into being a consciousness vastly exceeding our own.  On Wednesday Giulio Tononi, the creator of IIT, was kind enough to send me a fascinating 14-page rebuttal, and to give me permission to share it here:

Why Scott should stare at a blank wall and reconsider (or, the conscious grid)

If you’re interested in this subject at all, then I strongly recommend reading Giulio’s response before continuing further.   But for those who want the tl;dr: Giulio, not one to battle strawmen, first restates my own argument against IIT with crystal clarity.  And while he has some minor quibbles (e.g., apparently my calculations of Φ didn’t use the most recent, “3.0″ version of IIT), he wisely sets those aside in order to focus on the core question: according to IIT, are all sorts of simple expander graphs conscious?

There, he doesn’t “bite the bullet” so much as devour a bullet hoagie with mustard.  He affirms that, yes, according to IIT, a large network of XOR gates arranged in a simple expander graph is conscious.  Indeed, he goes further, and says that the “expander” part is superfluous: even a network of XOR gates arranged in a 2D square grid is conscious.  In my language, Giulio is simply pointing out here that a √n×√n square grid has decent expansion: good enough to produce a Φ-value of about √n, if not the information-theoretic maximum of n (or n/2, etc.) that an expander graph could achieve.  And apparently, by Giulio’s lights, Φ=√n is sufficient for consciousness!

While Giulio never mentions this, it’s interesting to observe that logic gates arranged in a 1-dimensional line would produce a tiny Φ-value (Φ=O(1)).  So even by IIT standards, such a linear array would not be conscious.  Yet the jump from a line to a two-dimensional grid is enough to light the spark of Mind.

Personally, I give Giulio enormous credit for having the intellectual courage to follow his theory wherever it leads.  When the critics point out, “if your theory were true, then the Moon would be made of peanut butter,” he doesn’t try to wiggle out of the prediction, but proudly replies, “yes, chunky peanut butter—and you forgot to add that the Earth is made of Nutella!”

Yet even as we admire Giulio’s honesty and consistency, his stance might also prompt us, gently, to take another look at this peanut-butter-moon theory, and at what grounds we had for believing it in the first place.  In his response essay, Giulio offers four arguments (by my count) for accepting IIT despite, or even because of, its conscious-grid prediction: one “negative” argument and three “positive” ones.  Alas, while your Φ-lage may vary, I didn’t find any of the four arguments persuasive.  In the rest of this post, I’ll go through them one by one and explain why.

I. The Copernicus-of-Consciousness Argument

Like many commenters on my last post, Giulio heavily criticizes my appeal to “common sense” in rejecting IIT.  Sure, he says, I might find it “obvious” that a huge Vandermonde matrix, or its physical instantiation, isn’t conscious.  But didn’t people also find it “obvious” for millennia that the Sun orbits the Earth?  Isn’t the entire point of science to challenge common sense?  Clearly, then, the test of a theory of consciousness is not how well it upholds “common sense,” but how well it fits the facts.

The above position sounds pretty convincing: who could dispute that observable facts trump personal intuitions?  The trouble is, what are the observable facts when it comes to consciousness?  The anti-common-sense view gets all its force by pretending that we’re in a relatively late stage of research—namely, the stage of taking an agreed-upon scientific definition of consciousness, and applying it to test our intuitions—rather than in an extremely early stage, of agreeing on what the word “consciousness” is even supposed to mean.

Since I think this point is extremely important—and of general interest, beyond just IIT—I’ll expand on it with some analogies.

Suppose I told you that, in my opinion, the ε-δ definition of continuous functions—the one you learn in calculus class—failed to capture the true meaning of continuity.  Suppose I told you that I had a new, better definition of continuity—and amazingly, when I tried out my definition on some examples, it turned out that ⌊x⌋ (the floor function) was continuous, whereas x2  had discontinuities, though only at 17.5 and 42.

You would probably ask what I was smoking, and whether you could have some.  But why?  Why shouldn’t the study of continuity produce counterintuitive results?  After all, even the standard definition of continuity leads to some famously weird results, like that x sin(1/x) is a continuous function, even though sin(1/x) is discontinuous.  And it’s not as if the standard definition is God-given: people had been using words like “continuous” for centuries before Bolzano, Weierstrass, et al. formalized the ε-δ definition, a definition that millions of calculus students still find far from intuitive.  So why shouldn’t there be a different, better definition of “continuous,” and why shouldn’t it reveal that a step function is continuous while a parabola is not?

In my view, the way out of this conceptual jungle is to realize that, before any formal definitions, any ε’s and δ’s, we start with an intuition for we’re trying to capture by the word “continuous.”  And if we press hard enough on what that intuition involves, we’ll find that it largely consists of various “paradigm-cases.”  A continuous function, we’d say, is a function like 3x, or x2, or sin(x), while a discontinuity is the kind of thing that the function 1/x has at x=0, or that ⌊x⌋ has at every integer point.  Crucially, we use the paradigm-cases to guide our choice of a formal definition—not vice versa!  It’s true that, once we have a formal definition, we can then apply it to “exotic” cases like x sin(1/x), and we might be surprised by the results.  But the paradigm-cases are different.  If, for example, our definition told us that x2 was discontinuous, that wouldn’t be a “surprise”; it would just be evidence that we’d picked a bad definition.  The definition failed at the only task for which it could have succeeded: namely, that of capturing what we meant.

Some people might say that this is all well and good in pure math, but empirical science has no need for squishy intuitions and paradigm-cases.  Nothing could be further from the truth.  Suppose, again, that I told you that physicists since Kelvin had gotten the definition of temperature all wrong, and that I had a new, better definition.  And, when I built a Scott-thermometer that measures true temperatures, it delivered the shocking result that boiling water is actually colder than ice.  You’d probably tell me where to shove my Scott-thermometer.  But wait: how do you know that I’m not the Copernicus of heat, and that future generations won’t celebrate my breakthrough while scoffing at your small-mindedness?

I’d say there’s an excellent answer: because what we mean by heat is “whatever it is that boiling water has more of than ice” (along with dozens of other paradigm-cases).  And because, if you use a thermometer to check whether boiling water is hotter than ice, then the term for what you’re doing is calibrating your thermometer.  When the clock strikes 13, it’s time to fix the clock, and when the thermometer says boiling water’s colder than ice, it’s time to replace the thermometer—or if needed, even the entire theory on which the thermometer is based.

Ah, you say, but doesn’t modern physics define heat in a completely different, non-intuitive way, in terms of molecular motion?  Yes, and that turned out to be a superb definition—not only because it was precise, explanatory, and applicable to cases far beyond our everyday experience, but crucially, because it matched common sense on the paradigm-cases.  If it hadn’t given sensible results for boiling water and ice, then the only possible conclusion would be that, whatever new quantity physicists had defined, they shouldn’t call it “temperature,” or claim that their quantity measured the amount of “heat.”  They should call their new thing something else.

The implications for the consciousness debate are obvious.  When we consider whether to accept IIT’s equation of integrated information with consciousness, we don’t start with any agreed-upon, independent notion of consciousness against which the new notion can be compared.  The main things we start with, in my view, are certain paradigm-cases that gesture toward what we mean:

  • You are conscious (though not when anesthetized).
  • (Most) other people appear to be conscious, judging from their behavior.
  • Many animals appear to be conscious, though probably to a lesser degree than humans (and the degree of consciousness in each particular species is far from obvious).
  • A rock is not conscious.  A wall is not conscious.  A Reed-Solomon code is not conscious.  Microsoft Word is not conscious (though a Word macro that passed the Turing test conceivably would be).

Fetuses, coma patients, fish, and hypothetical AIs are the x sin(1/x)’s of consciousness: they’re the tougher cases, the ones where we might actually need a formal definition to adjudicate the truth.

Now, given a proposed formal definition for an intuitive concept, how can we check whether the definition is talking about same thing we were trying to get at before?  Well, we can check whether the definition at least agrees that parabolas are continuous while step functions are not, that boiling water is hot while ice is cold, and that we’re conscious while Reed-Solomon decoders are not.  If so, then the definition might be picking out the same thing that we meant, or were trying to mean, pre-theoretically (though we still can’t be certain).  If not, then the definition is certainly talking about something else.

What else can we do?

II. The Axiom Argument

According to Giulio, there is something else we can do, besides relying on paradigm-cases.  That something else, in his words, is to lay down “postulates about how the physical world should be organized to support the essential properties of experience,” then use those postulates to derive a consciousness-measuring quantity.

OK, so what are IIT’s postulates?  Here’s how Giulio states the five postulates leading to Φ in his response essay (he “derives” these from earlier “phenomenological axioms,” which you can find in the essay):

  1. A system of mechanisms exists intrinsically if it can make a difference to itself, by affecting the probability of its past and future states, i.e. it has causal power (existence).
  2. It is composed of submechanisms each with their own causal power (composition).
  3. It generates a conceptual structure that is the specific way it is, as specified by each mechanism’s concept — this is how each mechanism affects the probability of the system’s past and future states (information).
  4. The conceptual structure is unified — it cannot be decomposed into independent components (integration).
  5. The conceptual structure is singular — there can be no superposition of multiple conceptual structures over the same mechanisms and intervals of time.

From my standpoint, these postulates have three problems.  First, I don’t really understand them.  Second, insofar as I do understand them, I don’t necessarily accept their truth.  And third, insofar as I do accept their truth, I don’t see how they lead to Φ.

To elaborate a bit:

I don’t really understand the postulates.  I realize that the postulates are explicated further in the many papers on IIT.  Unfortunately, while it’s possible that I missed something, in all of the papers that I read, the definitions never seemed to “bottom out” in mathematical notions that I understood, like functions mapping finite sets to other finite sets.  What, for example, is a “mechanism”?  What’s a “system of mechanisms”?  What’s “causal power”?  What’s a “conceptual structure,” and what does it mean for it to be “unified”?  Alas, it doesn’t help to define these notions in terms of other notions that I also don’t understand.  And yes, I agree that all these notions can be given fully rigorous definitions, but there could be many different ways to do so, and the devil could lie in the details.  In any case, because (as I said) it’s entirely possible that the failure is mine, I place much less weight on this point than I do on the two points to follow.

I don’t necessarily accept the postulates’ truth.  Is consciousness a “unified conceptual structure”?  Is it “singular”?  Maybe.  I don’t know.  It sounds plausible.  But at any rate, I’m far less confident about any these postulates—whatever one means by them!—than I am about my own “postulate,” which is that you and I are conscious while my toaster is not.  Note that my postulate, though not phenomenological, does have the merit of constraining candidate theories of consciousness in an unambiguous way.

I don’t see how the postulates lead to Φ.  Even if one accepts the postulates, how does one deduce that the “amount of consciousness” should be measured by Φ, rather than by some other quantity?  None of the papers I read—including the ones Giulio linked to in his response essay—contained anything that looked to me like a derivation of Φ.  Instead, there was general discussion of the postulates, and then Φ just sort of appeared at some point.  Furthermore, given the many idiosyncrasies of Φ—the minimization over all bipartite (why just bipartite? why not tripartite?) decompositions of the system, the need for normalization (or something else in version 3.0) to deal with highly-unbalanced partitions—it would be quite a surprise were it possible to derive its specific form from postulates of such generality.

I was going to argue for that conclusion in more detail, when I realized that Giulio had kindly done the work for me already.  Recall that Giulio chided me for not using the “latest, 2014, version 3.0″ edition of Φ in my previous post.  Well, if the postulates uniquely determined the form of Φ, then what’s with all these upgrades?  Or has Φ’s definition been changing from year to year because the postulates themselves have been changing?  If the latter, then maybe one should wait for the situation to stabilize before trying to form an opinion of the postulates’ meaningfulness, truth, and completeness?

III. The Ironic Empirical Argument

Or maybe not.  Despite all the problems noted above with the IIT postulates, Giulio argues in his essay that there’s a good a reason to accept them: namely, they explain various empirical facts from neuroscience, and lead to confirmed predictions.  In his words:

[A] theory’s postulates must be able to explain, in a principled and parsimonious way, at least those many facts about consciousness and the brain that are reasonably established and non-controversial.  For example, we know that our own consciousness depends on certain brain structures (the cortex) and not others (the cerebellum), that it vanishes during certain periods of sleep (dreamless sleep) and reappears during others (dreams), that it vanishes during certain epileptic seizures, and so on.  Clearly, a theory of consciousness must be able to provide an adequate account for such seemingly disparate but largely uncontroversial facts.  Such empirical facts, and not intuitions, should be its primary test…

[I]n some cases we already have some suggestive evidence [of the truth of the IIT postulates' predictions].  One example is the cerebellum, which has 69 billion neurons or so — more than four times the 16 billion neurons of the cerebral cortex — and is as complicated a piece of biological machinery as any.  Though we do not understand exactly how it works (perhaps even less than we understand the cerebral cortex), its connectivity definitely suggests that the cerebellum is ill suited to information integration, since it lacks lateral connections among its basic modules.  And indeed, though the cerebellum is heavily connected to the cerebral cortex, removing it hardly affects our consciousness, whereas removing the cortex eliminates it.

I hope I’m not alone in noticing the irony of this move.  But just in case, let me spell it out: Giulio has stated, as “largely uncontroversial facts,” that certain brain regions (the cerebellum) and certain states (dreamless sleep) are not associated with our consciousness.  He then views it as a victory for IIT, if those regions and states turn out to have lower information integration than the regions and states that he does take to be associated with our consciousness.

But how does Giulio know that the cerebellum isn’t conscious?  Even if it doesn’t produce “our” consciousness, maybe the cerebellum has its own consciousness, just as rich as the cortex’s but separate from it.  Maybe removing the cerebellum destroys that other consciousness, unbeknownst to “us.”  Likewise, maybe “dreamless” sleep brings about its own form of consciousness, one that (unlike dreams) we never, ever remember in the morning.

Giulio might take the implausibility of those ideas as obvious, or at least as “largely uncontroversial” among neuroscientists.  But here’s the problem with that: he just told us that a 2D square grid is conscious!  He told us that we must not rely on “commonsense intuition,” or on any popular consensus, to say that if a square mesh of wires is just sitting there XORing some input bits, doing nothing at all that we’d want to call intelligent, then it’s probably safe to conclude that the mesh isn’t conscious.  So then why shouldn’t he say the same for the cerebellum, or for the brain in dreamless sleep?  By Giulio’s own rules (the ones he used for the mesh), we have no a-priori clue whether those systems are conscious or not—so even if IIT predicts that they’re not conscious, that can’t be counted as any sort of success for IIT.

For me, the point is even stronger: I, personally, would be a million times more inclined to ascribe consciousness to the human cerebellum, or to dreamless sleep, than I would to the mesh of XOR gates.  For it’s not hard to imagine neuroscientists of the future discovering “hidden forms of intelligence” in the cerebellum, and all but impossible to imagine them doing the same for the mesh.  But even if you put those examples on the same footing, still the take-home message seems clear: you can’t count it as a “success” for IIT if it predicts that the cerebellum in unconscious, while at the same time denying that it’s a “failure” for IIT if it predicts that a square mesh of XOR gates is conscious.  If the unconsciousness of the cerebellum can be considered an “empirical fact,” safe enough for theories of consciousness to be judged against it, then surely the unconsciousness of the mesh can also be considered such a fact.

IV. The Phenomenology Argument

I now come to, for me, the strangest and most surprising part of Giulio’s response.  Despite his earlier claim that IIT need not dovetail with “commonsense intuition” about which systems are conscious—that it can defy intuition—at some point, Giulio valiantly tries to reprogram our intuition, to make us feel why a 2D grid could be conscious.  As best I can understand, the argument seems to be that, when we stare at a blank 2D screen, we form a rich experience in our heads, and that richness must be mirrored by a corresponding “intrinsic” richness in 2D space itself:

[I]f one thinks a bit about it, the experience of empty 2D visual space is not at all empty, but contains a remarkable amount of structure.  In fact, when we stare at the blank screen, quite a lot is immediately available to us without any effort whatsoever.  Thus, we are aware of all the possible locations in space (“points”): the various locations are right “there”, in front of us.  We are aware of their relative positions: a point may be left or right of another, above or below, and so on, for every position, without us having to order them.  And we are aware of the relative distances among points: quite clearly, two points may be close or far, and this is the case for every position.  Because we are aware of all of this immediately, without any need to calculate anything, and quite regularly, since 2D space pervades most of our experiences, we tend to take for granted the vast set of relationship[s] that make up 2D space.

And yet, says IIT, given that our experience of the blank screen definitely exists, and it is precisely the way it is — it is 2D visual space, with all its relational properties — there must be physical mechanisms that specify such phenomenological relationships through their causal power … One may also see that the causal relationships that make up 2D space obtain whether the elements are on or off.  And finally, one may see that such a 2D grid is necessary not so much to represent space from the extrinsic perspective of an observer, but to create it, from its own intrinsic perspective.

Now, it would be child’s-play to criticize the above line of argument for conflating our consciousness of the screen with the alleged consciousness of the screen itself.  To wit:  Just because it feels like something to see a wall, doesn’t mean it feels like something to be a wall.  You can smell a rose, and the rose can smell good, but that doesn’t mean the rose can smell you.

However, I actually prefer a different tack in criticizing Giulio’s “wall argument.”  Suppose I accepted that my mental image of the relationships between certain entities was relevant to assessing whether those entities had their own mental life, independent of me or any other observer.  For example, suppose I believed that, if my experience of 2D space is rich and structured, then that’s evidence that 2D space is rich and structured enough to be conscious.

Then my question is this: why shouldn’t the same be true of 1D space?  After all, my experience of staring at a rope is also rich and structured, no less than my experience of staring at a wall.  I perceive some points on the rope as being toward the left, others as being toward the right, and some points as being between two other points.  In fact, the rope even has a structure—namely, a natural total ordering on its points—that the wall lacks.  So why does IIT cruelly deny subjective experience to a row of logic gates strung along a rope, reserving it only for a mesh of logic gates pasted to a wall?

And yes, I know the answer: because the logic gates on the rope aren’t “integrated” enough.  But who’s to say that the gates in the 2D mesh are integrated enough?  As I mentioned before, their Φ-value grows only as the square root of the number of gates, so that the ratio of integrated information to total information tends to 0 as the number of gates increases.  And besides, aren’t what Giulio calls “the facts of phenomenology” the real arbiters here, and isn’t my perception of the rope’s structure a phenomenological fact?  When you cut a rope, does it not split?  When you prick it, does it not fray?

Conclusion

At this point, I fear we’re at a philosophical impasse.  Having learned that, according to IIT,

  1. a square grid of XOR gates is conscious, and your experience of staring at a blank wall provides evidence for that,
  2. by contrast, a linear array of XOR gates is not conscious, your experience of staring at a rope notwithstanding,
  3. the human cerebellum is also not conscious (even though a grid of XOR gates is), and
  4. unlike with the XOR gates, we don’t need a theory to tell us the cerebellum is unconscious, but can simply accept it as “reasonably established” and “largely uncontroversial,”

I personally feel completely safe in saying that this is not the theory of consciousness for me.  But I’ve also learned that other people, even after understanding the above, still don’t reject IIT.  And you know what?  Bully for them.  On reflection, I firmly believe that a two-state solution is possible, in which we simply adopt different words for the different things that we mean by “consciousness”—like, say, consciousnessReal for my kind and consciousnessWTF for the IIT kind.  OK, OK, just kidding!  How about “paradigm-case consciousness” for the one and “IIT consciousness” for the other.


Completely unrelated announcement: Some of you might enjoy this Nature News piece by Amanda Gefter, about black holes and computational complexity.

Why I Am Not An Integrated Information Theorist (or, The Unconscious Expander)

Wednesday, May 21st, 2014

Happy birthday to me!

Recently, lots of people have been asking me what I think about IIT—no, not the Indian Institutes of Technology, but Integrated Information Theory, a widely-discussed “mathematical theory of consciousness” developed over the past decade by the neuroscientist Giulio Tononi.  One of the askers was Max Tegmark, who’s enthusiastically adopted IIT as a plank in his radical mathematizing platform (see his paper “Consciousness as a State of Matter”).  When, in the comment thread about Max’s Mathematical Universe Hypothesis, I expressed doubts about IIT, Max challenged me to back up my doubts with a quantitative calculation.

So, this is the post that I promised to Max and all the others, about why I don’t believe IIT.  And yes, it will contain that quantitative calculation.

But first, what is IIT?  The central ideas of IIT, as I understand them, are:

(1) to propose a quantitative measure, called Φ, of the amount of “integrated information” in a physical system (i.e. information that can’t be localized in the system’s individual parts), and then

(2) to hypothesize that a physical system is “conscious” if and only if it has a large value of Φ—and indeed, that a system is more conscious the larger its Φ value.

I’ll return later to the precise definition of Φ—but basically, it’s obtained by minimizing, over all subdivisions of your physical system into two parts A and B, some measure of the mutual information between A’s outputs and B’s inputs and vice versa.  Now, one immediate consequence of any definition like this is that all sorts of simple physical systems (a thermostat, a photodiode, etc.) will turn out to have small but nonzero Φ values.  To his credit, Tononi cheerfully accepts the panpsychist implication: yes, he says, it really does mean that thermostats and photodiodes have small but nonzero levels of consciousness.  On the other hand, for the theory to work, it had better be the case that Φ is small for “intuitively unconscious” systems, and only large for “intuitively conscious” systems.  As I’ll explain later, this strikes me as a crucial point on which IIT fails.

The literature on IIT is too big to do it justice in a blog post.  Strikingly, in addition to the “primary” literature, there’s now even a “secondary” literature, which treats IIT as a sort of established base on which to build further speculations about consciousness.  Besides the Tegmark paper linked to above, see for example this paper by Maguire et al., and associated popular article.  (Ironically, Maguire et al. use IIT to argue for the Penrose-like view that consciousness might have uncomputable aspects—a use diametrically opposed to Tegmark’s.)

Anyway, if you want to read a popular article about IIT, there are loads of them: see here for the New York Times’s, here for Scientific American‘s, here for IEEE Spectrum‘s, and here for the New Yorker‘s.  Unfortunately, none of those articles will tell you the meat (i.e., the definition of integrated information); for that you need technical papers, like this or this by Tononi, or this by Seth et al.  IIT is also described in Christof Koch’s memoir Consciousness: Confessions of a Romantic Reductionist, which I read and enjoyed; as well as Tononi’s Phi: A Voyage from the Brain to the Soul, which I haven’t yet read.  (Koch, one of the world’s best-known thinkers and writers about consciousness, has also become an evangelist for IIT.)

So, I want to explain why I don’t think IIT solves even the problem that it “plausibly could have” solved.  But before I can do that, I need to do some philosophical ground-clearing.  Broadly speaking, what is it that a “mathematical theory of consciousness” is supposed to do?  What questions should it answer, and how should we judge whether it’s succeeded?

The most obvious thing a consciousness theory could do is to explain why consciousness exists: that is, to solve what David Chalmers calls the “Hard Problem,” by telling us how a clump of neurons is able to give rise to the taste of strawberries, the redness of red … you know, all that ineffable first-persony stuff.  Alas, there’s a strong argument—one that I, personally, find completely convincing—why that’s too much to ask of any scientific theory.  Namely, no matter what the third-person facts were, one could always imagine a universe consistent with those facts in which no one “really” experienced anything.  So for example, if someone claims that integrated information “explains” why consciousness exists—nope, sorry!  I’ve just conjured into my imagination beings whose Φ-values are a thousand, nay a trillion times larger than humans’, yet who are also philosophical zombies: entities that there’s nothing that it’s like to be.  Granted, maybe such zombies can’t exist in the actual world: maybe, if you tried to create one, God would notice its large Φ-value and generously bequeath it a soul.  But if so, then that’s a further fact about our world, a fact that manifestly couldn’t be deduced from the properties of Φ alone.  Notice that the details of Φ are completely irrelevant to the argument.

Faced with this point, many scientifically-minded people start yelling and throwing things.  They say that “zombies” and so forth are empty metaphysics, and that our only hope of learning about consciousness is to engage with actual facts about the brain.  And that’s a perfectly reasonable position!  As far as I’m concerned, you absolutely have the option of dismissing Chalmers’ Hard Problem as a navel-gazing distraction from the real work of neuroscience.  The one thing you can’t do is have it both ways: that is, you can’t say both that the Hard Problem is meaningless, and that progress in neuroscience will soon solve the problem if it hasn’t already.  You can’t maintain simultaneously that

(a) once you account for someone’s observed behavior and the details of their brain organization, there’s nothing further about consciousness to be explained, and

(b) remarkably, the XYZ theory of consciousness can explain the “nothing further” (e.g., by reducing it to integrated information processing), or might be on the verge of doing so.

As obvious as this sounds, it seems to me that large swaths of consciousness-theorizing can just be summarily rejected for trying to have their brain and eat it in precisely the above way.

Fortunately, I think IIT survives the above observations.  For we can easily interpret IIT as trying to do something more “modest” than solve the Hard Problem, although still staggeringly audacious.  Namely, we can say that IIT “merely” aims to tell us which physical systems are associated with consciousness and which aren’t, purely in terms of the systems’ physical organization.  The test of such a theory is whether it can produce results agreeing with “commonsense intuition”: for example, whether it can affirm, from first principles, that (most) humans are conscious; that dogs and horses are also conscious but less so; that rocks, livers, bacteria colonies, and existing digital computers are not conscious (or are hardly conscious); and that a room full of people has no “mega-consciousness” over and above the consciousnesses of the individuals.

The reason it’s so important that the theory uphold “common sense” on these test cases is that, given the experimental inaccessibility of consciousness, this is basically the only test available to us.  If the theory gets the test cases “wrong” (i.e., gives results diverging from common sense), it’s not clear that there’s anything else for the theory to get “right.”  Of course, supposing we had a theory that got the test cases right, we could then have a field day with the less-obvious cases, programming our computers to tell us exactly how much consciousness is present in octopi, fetuses, brain-damaged patients, and hypothetical AI bots.

In my opinion, how to construct a theory that tells us which physical systems are conscious and which aren’t—giving answers that agree with “common sense” whenever the latter renders a verdict—is one of the deepest, most fascinating problems in all of science.  Since I don’t know a standard name for the problem, I hereby call it the Pretty-Hard Problem of Consciousness.  Unlike with the Hard Hard Problem, I don’t know of any philosophical reason why the Pretty-Hard Problem should be inherently unsolvable; but on the other hand, humans seem nowhere close to solving it (if we had solved it, then we could reduce the abortion, animal rights, and strong AI debates to “gentlemen, let us calculate!”).

Now, I regard IIT as a serious, honorable attempt to grapple with the Pretty-Hard Problem of Consciousness: something concrete enough to move the discussion forward.  But I also regard IIT as a failed attempt on the problem.  And I wish people would recognize its failure, learn from it, and move on.

In my view, IIT fails to solve the Pretty-Hard Problem because it unavoidably predicts vast amounts of consciousness in physical systems that no sane person would regard as particularly “conscious” at all: indeed, systems that do nothing but apply a low-density parity-check code, or other simple transformations of their input data.  Moreover, IIT predicts not merely that these systems are “slightly” conscious (which would be fine), but that they can be unboundedly more conscious than humans are.

To justify that claim, I first need to define Φ.  Strikingly, despite the large literature about Φ, I had a hard time finding a clear mathematical definition of it—one that not only listed formulas but fully defined the structures that the formulas were talking about.  Complicating matters further, there are several competing definitions of Φ in the literature, including ΦDM (discrete memoryless), ΦE (empirical), and ΦAR (autoregressive), which apply in different contexts (e.g., some take time evolution into account and others don’t).  Nevertheless, I think I can define Φ in a way that will make sense to theoretical computer scientists.  And crucially, the broad point I want to make about Φ won’t depend much on the details of its formalization anyway.

We consider a discrete system in a state x=(x1,…,xn)∈Sn, where S is a finite alphabet (the simplest case is S={0,1}).  We imagine that the system evolves via an “updating function” f:Sn→Sn. Then the question that interests us is whether the xi‘s can be partitioned into two sets A and B, of roughly comparable size, such that the updates to the variables in A don’t depend very much on the variables in B and vice versa.  If such a partition exists, then we say that the computation of f does not involve “global integration of information,” which on Tononi’s theory is a defining aspect of consciousness.

More formally, given a partition (A,B) of {1,…,n}, let us write an input y=(y1,…,yn)∈Sn to f in the form (yA,yB), where yA consists of the y variables in A and yB consists of the y variables in B.  Then we can think of f as mapping an input pair (yA,yB) to an output pair (zA,zB).  Now, we define the “effective information” EI(A→B) as H(zB | A random, yB=xB).  Or in words, EI(A→B) is the Shannon entropy of the output variables in B, if the input variables in A are drawn uniformly at random, while the input variables in B are fixed to their values in x.  It’s a measure of the dependence of B on A in the computation of f(x).  Similarly, we define

EI(B→A) := H(zA | B random, yA=xA).

We then consider the sum

Φ(A,B) := EI(A→B) + EI(B→A).

Intuitively, we’d like the integrated information Φ=Φ(f,x) be the minimum of Φ(A,B), over all 2n-2 possible partitions of {1,…,n} into nonempty sets A and B.  The idea is that Φ should be large, if and only if it’s not possible to partition the variables into two sets A and B, in such a way that not much information flows from A to B or vice versa when f(x) is computed.

However, no sooner do we propose this than we notice a technical problem.  What if A is much larger than B, or vice versa?  As an extreme case, what if A={1,…,n-1} and B={n}?  In that case, we’ll have Φ(A,B)≤2log2|S|, but only for the boring reason that there’s hardly any entropy in B as a whole, to either influence A or be influenced by it.  For this reason, Tononi proposes a fix where we normalize each Φ(A,B) by dividing it by min{|A|,|B|}.  He then defines the integrated information Φ to be Φ(A,B), for whichever partition (A,B) minimizes the ratio Φ(A,B) / min{|A|,|B|}.  (Unless I missed it, Tononi never specifies what we should do if there are multiple (A,B)’s that all achieve the same minimum of Φ(A,B) / min{|A|,|B|}.  I’ll return to that point later, along with other idiosyncrasies of the normalization procedure.)

Tononi gives some simple examples of the computation of Φ, showing that it is indeed larger for systems that are more “richly interconnected” in an intuitive sense.  He speculates, plausibly, that Φ is quite large for (some reasonable model of) the interconnection network of the human brain—and probably larger for the brain than for typical electronic devices (which tend to be highly modular in design, thereby decreasing their Φ), or, let’s say, than for other organs like the pancreas.  Ambitiously, he even speculates at length about how a large value of Φ might be connected to the phenomenology of consciousness.

To be sure, empirical work in integrated information theory has been hampered by three difficulties.  The first difficulty is that we don’t know the detailed interconnection network of the human brain.  The second difficulty is that it’s not even clear what we should define that network to be: for example, as a crude first attempt, should we assign a Boolean variable to each neuron, which equals 1 if the neuron is currently firing and 0 if it’s not firing, and let f be the function that updates those variables over a timescale of, say, a millisecond?  What other variables do we need—firing rates, internal states of the neurons, neurotransmitter levels?  Is choosing many of these variables uniformly at random (for the purpose of calculating Φ) really a reasonable way to “randomize” the variables, and if not, what other prescription should we use?

The third and final difficulty is that, even if we knew exactly what we meant by “the f and x corresponding to the human brain,” and even if we had complete knowledge of that f and x, computing Φ(f,x) could still be computationally intractable.  For recall that the definition of Φ involved minimizing a quantity over all the exponentially-many possible bipartitions of {1,…,n}.  While it’s not directly relevant to my arguments in this post, I leave it as a challenge for interested readers to pin down the computational complexity of approximating Φ to some reasonable precision, assuming that f is specified by a polynomial-size Boolean circuit, or alternatively, by an NC0 function (i.e., a function each of whose outputs depends on only a constant number of the inputs).  (Presumably Φ will be #P-hard to calculate exactly, but only because calculating entropy exactly is a #P-hard problem—that’s not interesting.)

I conjecture that approximating Φ is an NP-hard problem, even for restricted families of f’s like NC0 circuits—which invites the amusing thought that God, or Nature, would need to solve an NP-hard problem just to decide whether or not to imbue a given physical system with consciousness!  (Alas, if you wanted to exploit this as a practical approach for solving NP-complete problems such as 3SAT, you’d need to do a rather drastic experiment on your own brain—an experiment whose result would be to render you unconscious if your 3SAT instance was satisfiable, or conscious if it was unsatisfiable!  In neither case would you be able to communicate the outcome of the experiment to anyone else, nor would you have any recollection of the outcome after the experiment was finished.)  In the other direction, it would also be interesting to upper-bound the complexity of approximating Φ.  Because of the need to estimate the entropies of distributions (even given a bipartition (A,B)), I don’t know that this problem is in NP—the best I can observe is that it’s in AM.

In any case, my own reason for rejecting IIT has nothing to do with any of the “merely practical” issues above: neither the difficulty of defining f and x, nor the difficulty of learning them, nor the difficulty of calculating Φ(f,x).  My reason is much more basic, striking directly at the hypothesized link between “integrated information” and consciousness.  Specifically, I claim the following:

Yes, it might be a decent rule of thumb that, if you want to know which brain regions (for example) are associated with consciousness, you should start by looking for regions with lots of information integration.  And yes, it’s even possible, for all I know, that having a large Φ-value is one necessary condition among many for a physical system to be conscious.  However, having a large Φ-value is certainly not a sufficient condition for consciousness, or even for the appearance of consciousness.  As a consequence, Φ can’t possibly capture the essence of what makes a physical system conscious, or even of what makes a system look conscious to external observers.

The demonstration of this claim is embarrassingly simple.  Let S=Fp, where p is some prime sufficiently larger than n, and let V be an n×n Vandermonde matrix over Fp—that is, a matrix whose (i,j) entry equals ij-1 (mod p).  Then let f:Sn→Sn be the update function defined by f(x)=Vx.  Now, for p large enough, the Vandermonde matrix is well-known to have the property that every submatrix is full-rank (i.e., “every submatrix preserves all the information that it’s possible to preserve about the part of x that it acts on”).  And this implies that, regardless of which bipartition (A,B) of {1,…,n} we choose, we’ll get

EI(A→B) = EI(B→A) = min{|A|,|B|} log2p,

and hence

Φ(A,B) = EI(A→B) + EI(B→A) = 2 min{|A|,|B|} log2p,

or after normalizing,

Φ(A,B) / min{|A|,|B|} = 2 log2p.

Or in words: the normalized information integration has the same value—namely, the maximum value!—for every possible bipartition.  Now, I’d like to proceed from here to a determination of Φ itself, but I’m prevented from doing so by the ambiguity in the definition of Φ that I noted earlier.  Namely, since every bipartition (A,B) minimizes the normalized value Φ(A,B) / min{|A|,|B|}, in theory I ought to be able to pick any of them for the purpose of calculating Φ.  But the unnormalized value Φ(A,B), which gives the final Φ, can vary greatly, across bipartitions: from 2 log2p (if min{|A|,|B|}=1) all the way up to n log2p (if min{|A|,|B|}=n/2).  So at this point, Φ is simply undefined.

On the other hand, I can solve this problem, and make Φ well-defined, by an ironic little hack.  The hack is to replace the Vandermonde matrix V by an n×n matrix W, which consists of the first n/2 rows of the Vandermonde matrix each repeated twice (assume for simplicity that n is a multiple of 4).  As before, we let f(x)=Wx.  Then if we set A={1,…,n/2} and B={n/2+1,…,n}, we can achieve

EI(A→B) = EI(B→A) = (n/4) log2p,

Φ(A,B) = EI(A→B) + EI(B→A) = (n/2) log2p,

and hence

Φ(A,B) / min{|A|,|B|} = log2p.

In this case, I claim that the above is the unique bipartition that minimizes the normalized integrated information Φ(A,B) / min{|A|,|B|}, up to trivial reorderings of the rows.  To prove this claim: if |A|=|B|=n/2, then clearly we minimize Φ(A,B) by maximizing the number of repeated rows in A and the number of repeated rows in B, exactly as we did above.  Thus, assume |A|≤|B| (the case |B|≤|A| is analogous).  Then clearly

EI(B→A) ≥ |A|/2,

while

EI(A→B) ≥ min{|A|, |B|/2}.

So if we let |A|=cn and |B|=(1-c)n for some c∈(0,1/2], then

Φ(A,B) ≥ [c/2 + min{c, (1-c)/2}] n,

and

Φ(A,B) / min{|A|,|B|} = Φ(A,B) / |A| = 1/2 + min{1, 1/(2c) – 1/2}.

But the above expression is uniquely minimized when c=1/2.  Hence the normalized integrated information is minimized essentially uniquely by setting A={1,…,n/2} and B={n/2+1,…,n}, and we get

Φ = Φ(A,B) = (n/2) log2p,

which is quite a large value (only a factor of 2 less than the trivial upper bound of n log2p).

Now, why did I call the switch from V to W an “ironic little hack”?  Because, in order to ensure a large value of Φ, I decreased—by a factor of 2, in fact—the amount of “information integration” that was intuitively happening in my system!  I did that in order to decrease the normalized value Φ(A,B) / min{|A|,|B|} for the particular bipartition (A,B) that I cared about, thereby ensuring that that (A,B) would be chosen over all the other bipartitions, thereby increasing the final, unnormalized value Φ(A,B) that Tononi’s prescription tells me to return.  I hope I’m not alone in fearing that this illustrates a disturbing non-robustness in the definition of Φ.

But let’s leave that issue aside; maybe it can be ameliorated by fiddling with the definition.  The broader point is this: I’ve shown that my system—the system that simply applies the matrix W to an input vector x—has an enormous amount of integrated information Φ.  Indeed, this system’s Φ equals half of its entire information content.  So for example, if n were 1014 or so—something that wouldn’t be hard to arrange with existing computers—then this system’s Φ would exceed any plausible upper bound on the integrated information content of the human brain.

And yet this Vandermonde system doesn’t even come close to doing anything that we’d want to call intelligent, let alone conscious!  When you apply the Vandermonde matrix to a vector, all you’re really doing is mapping the list of coefficients of a degree-(n-1) polynomial over Fp, to the values of the polynomial on the n points 0,1,…,n-1.  Now, evaluating a polynomial on a set of points turns out to be an excellent way to achieve “integrated information,” with every subset of outputs as correlated with every subset of inputs as it could possibly be.  In fact, that’s precisely why polynomials are used so heavily in error-correcting codes, such as the Reed-Solomon code, employed (among many other places) in CD’s and DVD’s.  But that doesn’t imply that every time you start up your DVD player you’re lighting the fire of consciousness.  It doesn’t even hint at such a thing.  All it tells us is that you can have integrated information without consciousness (or even intelligence)—just like you can have computation without consciousness, and unpredictability without consciousness, and electricity without consciousness.

It might be objected that, in defining my “Vandermonde system,” I was too abstract and mathematical.  I said that the system maps the input vector x to the output vector Wx, but I didn’t say anything about how it did so.  To perform a computation—even a computation as simple as a matrix-vector multiply—won’t we need a physical network of wires, logic gates, and so forth?  And in any realistic such network, won’t each logic gate be directly connected to at most a few other gates, rather than to billions of them?  And if we define the integrated information Φ, not directly in terms of the inputs and outputs of the function f(x)=Wx, but in terms of all the actual logic gates involved in computing f, isn’t it possible or even likely that Φ will go back down?

This is a good objection, but I don’t think it can rescue IIT.  For we can achieve the same qualitative effect that I illustrated with the Vandermonde matrix—the same “global information integration,” in which every large set of outputs depends heavily on every large set of inputs—even using much “sparser” computations, ones where each individual output depends on only a few of the inputs.  This is precisely the idea behind low-density parity check (LDPC) codes, which have had a major impact on coding theory over the past two decades.  Of course, one would need to muck around a bit to construct a physical system based on LDPC codes whose integrated information Φ was provably large, and for which there were no wildly-unbalanced bipartitions that achieved lower Φ(A,B)/min{|A|,|B|} values than the balanced bipartitions one cared about.  But I feel safe in asserting that this could be done, similarly to how I did it with the Vandermonde matrix.

More generally, we can achieve pretty good information integration by hooking together logic gates according to any bipartite expander graph: that is, any graph with n vertices on each side, such that every k vertices on the left side are connected to at least min{(1+ε)k,n} vertices on the right side, for some constant ε>0.  And it’s well-known how to create expander graphs whose degree (i.e., the number of edges incident to each vertex, or the number of wires coming out of each logic gate) is a constant, such as 3.  One can do so either by plunking down edges at random, or (less trivially) by explicit constructions from algebra or combinatorics.  And as indicated in the title of this post, I feel 100% confident in saying that the so-constructed expander graphs are not conscious!  The brain might be an expander, but not every expander is a brain.

Before winding down this post, I can’t resist telling you that the concept of integrated information (though it wasn’t called that) played an interesting role in computational complexity in the 1970s.  As I understand the history, Leslie Valiant conjectured that Boolean functions f:{0,1}n→{0,1}n with a high degree of “information integration” (such as discrete analogues of the Fourier transform) might be good candidates for proving circuit lower bounds, which in turn might be baby steps toward P≠NP.  More strongly, Valiant conjectured that the property of information integration, all by itself, implied that such functions had to be at least somewhat computationally complex—i.e., that they couldn’t be computed by circuits of size O(n), or even required circuits of size Ω(n log n).  Alas, that hope was refuted by Valiant’s later discovery of linear-size superconcentrators.  Just as information integration doesn’t suffice for intelligence or consciousness, so Valiant learned that information integration doesn’t suffice for circuit lower bounds either.

As humans, we seem to have the intuition that global integration of information is such a powerful property that no “simple” or “mundane” computational process could possibly achieve it.  But our intuition is wrong.  If it were right, then we wouldn’t have linear-size superconcentrators or LDPC codes.

I should mention that I had the privilege of briefly speaking with Giulio Tononi (as well as his collaborator, Christof Koch) this winter at an FQXi conference in Puerto Rico.  At that time, I challenged Tononi with a much cruder, handwavier version of some of the same points that I made above.  Tononi’s response, as best as I can reconstruct it, was that it’s wrong to approach IIT like a mathematician; instead one needs to start “from the inside,” with the phenomenology of consciousness, and only then try to build general theories that can be tested against counterexamples.  This response perplexed me: of course you can start from phenomenology, or from anything else you like, when constructing your theory of consciousness.  However, once your theory has been constructed, surely it’s then fair game for others to try to refute it with counterexamples?  And surely the theory should be judged, like anything else in science or philosophy, by how well it withstands such attacks?

But let me end on a positive note.  In my opinion, the fact that Integrated Information Theory is wrong—demonstrably wrong, for reasons that go to its core—puts it in something like the top 2% of all mathematical theories of consciousness ever proposed.  Almost all competing theories of consciousness, it seems to me, have been so vague, fluffy, and malleable that they can only aspire to wrongness.

[Endnote: See also this related post, by the philosopher Eric Schwetzgebel: Why Tononi Should Think That the United States Is Conscious.  While the discussion is much more informal, and the proposed counterexample more debatable, the basic objection to IIT is the same.]


Update (5/22): Here are a few clarifications of this post that might be helpful.

(1) The stuff about zombies and the Hard Problem was simply meant as motivation and background for what I called the “Pretty-Hard Problem of Consciousness”—the problem that I take IIT to be addressing.  You can disagree with the zombie stuff without it having any effect on my arguments about IIT.

(2) I wasn’t arguing in this post that dualism is true, or that consciousness is irreducibly mysterious, or that there could never be any convincing theory that told us how much consciousness was present in a physical system.  All I was arguing was that, at any rate, IIT is not such a theory.

(3) Yes, it’s true that my demonstration of IIT’s falsehood assumes—as an axiom, if you like—that while we might not know exactly what we mean by “consciousness,” at any rate we’re talking about something that humans have to a greater extent than DVD players.  If you reject that axiom, then I’d simply want to define a new word for a certain quality that non-anesthetized humans seem to have and that DVD players seem not to, and clarify that that other quality is the one I’m interested in.

(4) For my counterexample, the reason I chose the Vandermonde matrix is not merely that it’s invertible, but that all of its submatrices are full-rank.  This is the property that’s relevant for producing a large value of the integrated information Φ; by contrast, note that the identity matrix is invertible, but produces a system with Φ=0.  (As another note, if we work over a large enough field, then a random matrix will have this same property with high probability—but I wanted an explicit example, and while the Vandermonde is far from the only one, it’s one of the simplest.)

(5) The n×n Vandermonde matrix only does what I want if we work over (say) a prime field Fp with p>>n elements.  Thus, it’s natural to wonder whether similar examples exist where the basic system variables are bits, rather than elements of Fp.  The answer is yes. One way to get such examples is using the low-density parity check codes that I mention in the post.  Another common way to get Boolean examples, and which is also used in practice in error-correcting codes, is to start with the Vandermonde matrix (a.k.a. the Reed-Solomon code), and then combine it with an additional component that encodes the elements of Fp as strings of bits in some way.  Of course, you then need to check that doing this doesn’t harm the properties of the original Vandermonde matrix that you cared about (e.g., the “information integration”) too much, which causes some additional complication.

(6) Finally, it might be objected that my counterexamples ignored the issue of dynamics and “feedback loops”: they all consisted of unidirectional processes, which map inputs to outputs and then halt.  However, this can be fixed by the simple expedient of iterating the process over and over!  I.e., first map x to Wx, then map Wx to W2x, and so on.  The integrated information should then be the same as in the unidirectional case.


Update (5/24): See a very interesting comment by David Chalmers.

Retiring falsifiability? A storm in Russell’s teacup

Friday, January 17th, 2014

My good friend Sean Carroll took a lot of flak recently for answering this year’s Edge question, “What scientific idea is ready for retirement?,” with “Falsifiability”, and for using string theory and the multiverse as examples of why science needs to break out of its narrow Popperian cage.  For more, see this blog post of Sean’s, where one commenter after another piles on the beleaguered dude for his abandonment of science and reason themselves.

My take, for whatever it’s worth, is that Sean and his critics are both right.

Sean is right that “falsifiability” is a crude slogan that fails to capture what science really aims at.  As a doofus example, the theory that zebras exist is presumably both “true” and “scientific,” but it’s not “falsifiable”: if zebras didn’t exist, there would be no experiment that proved their nonexistence.  (And that’s to say nothing of empirical claims involving multiple nested quantifiers: e.g., “for every physical device that tries to solve the Traveling Salesman Problem in polynomial time, there exists an input on which the device fails.”)  Less doofusly, a huge fraction of all scientific progress really consists of mathematical or computational derivations from previously-accepted theories—and, as such, has no “falsifiable content” apart from the theories themselves.  So, do workings-out of mathematical consequences count as “science”?  In practice, the Nobel committee says sure they do, but only if the final results of the derivations are “directly” confirmed by experiment.  Far better, it seems to me, to say that science is a search for explanations that do essential and nontrivial work, within the network of abstract ideas whose ultimate purpose to account for our observations.  (On this particular question, I endorse everything David Deutsch has to say in The Beginning of Infinity, which you should read if you haven’t.)

On the other side, I think Sean’s critics are right that falsifiability shouldn’t be “retired.”  Instead, falsifiability’s portfolio should be expanded, with full-time assistants (like explanatory power) hired to lighten falsifiability’s load.

I also, to be honest, don’t see that modern philosophy of science has advanced much beyond Popper in its understanding of these issues.  Last year, I did something weird and impulsive: I read Karl Popper.  Given all the smack people talk about him these days, I was pleasantly surprised by the amount of nuance, reasonableness, and just general getting-it that I found.  Indeed, I found a lot more of those things in Popper than I found in his latter-day overthrowers Kuhn and Feyerabend.  For Popper (if not for some of his later admirers), falsifiability was not a crude bludgeon.  Rather, it was the centerpiece of a richly-articulated worldview holding that millennia of human philosophical reflection had gotten it backwards: the question isn’t how to arrive at the Truth, but rather how to eliminate error.  Which sounds kind of obvious, until I meet yet another person who rails to me about how empirical positivism can’t provide its own ultimate justification, and should therefore be replaced by the person’s favorite brand of cringe-inducing ugh.

Oh, I also think Sean might have made a tactical error in choosing string theory and the multiverse as his examples for why falsifiability needs to be retired.  For it seems overwhelmingly likely to me that the following two propositions are both true:

1. Falsifiability is too crude of a concept to describe how science works.
2. In the specific cases of string theory and the multiverse, a dearth of novel falsifiable predictions really is a big problem.

As usual, the best bet is to use explanatory power as our criterion—in which case, I’d say string theory emerges as a complex and evolving story.  On one end, there are insights like holography and AdS/CFT, which seem clearly to do explanatory work, and which I’d guess will stand as permanent contributions to human knowledge, even if the whole foundations on which they currently rest get superseded by something else.  On the other end, there’s the idea, championed by a minority of string theorists and widely repeated in the press, that the anthropic principle applied to different patches of multiverse can be invoked as a sort of get-out-of-jail-free card, to rescue a favored theory from earlier hopes of successful empirical predictions that then failed to pan out.  I wouldn’t know how to answer a layperson who asked why that wasn’t exactly the sort of thing Sir Karl was worried about, and for good reason.

Finally, not that Edge asked me, but I’d say the whole notions of “determinism” and “indeterminism” in physics are past ready for retirement.  I can’t think of any work they do, that isn’t better done by predictability and unpredictability.

Luke Muehlhauser interviews me about philosophical progress

Saturday, December 14th, 2013

I’m shipping out today to sunny Rio de Janeiro, where I’ll be giving a weeklong course about BosonSampling, at the invitation of Ernesto Galvão.  Then it’s on to Pennsylvania (where I’ll celebrate Christmas Eve with old family friends), Israel (where I’ll drop off Dana and Lily with Dana’s family in Tel Aviv, then lecture at the Jerusalem Winter School in Theoretical Physics), Puerto Rico (where I’ll speak at the FQXi conference on Physics of Information), back to Israel, and then New York before returning to Boston at the beginning of February.  Given this travel schedule, it’s possible that blogging will be even lighter than usual for the next month and a half (or not—we’ll see).

In the meantime, however, I’ve got the equivalent of at least five new blog posts to tide over Shtetl-Optimized fans.  Luke Muehlhauser, the Executive Director of the Machine Intelligence Research Institute (formerly the Singularity Institute for Artificial Intelligence), did an in-depth interview with me about “philosophical progress,” in which he prodded me to expand on certain comments in Why Philosophers Should Care About Computational Complexity and The Ghost in the Quantum Turing Machine.  Here are (abridged versions of) Luke’s five questions:

1. Why are you so interested in philosophy? And what is the social value of philosophy, from your perspective?

2. What are some of your favorite examples of illuminating Q-primes [i.e., scientifically-addressable pieces of big philosophical questions] that were solved within your own field, theoretical computer science?

3. Do you wish philosophy-the-field would be reformed in certain ways? Would you like to see more crosstalk between disciplines about philosophical issues? Do you think that, as Clark Glymour suggested, philosophy departments should be defunded unless they produce work that is directly useful to other fields … ?

4. Suppose a mathematically and analytically skilled student wanted to make progress, in roughly the way you describe, on the Big Questions of philosophy. What would you recommend they study? What should they read to be inspired? What skills should they develop? Where should they go to study?

5. Which object-level thinking tactics … do you use in your own theoretical (especially philosophical) research?  Are there tactics you suspect might be helpful, which you haven’t yet used much yourself?

For the answers—or at least my answers—click here!

PS. In case you missed it before, Quantum Computing Since Democritus was chosen by Scientific American blogger Jennifer Ouellette (via the “Time Lord,” Sean Carroll) as the top physics book of 2013.  Woohoo!!

The Ghost in the Quantum Turing Machine

Saturday, June 15th, 2013

I’ve been traveling this past week (in Israel and the French Riviera), heavily distracted by real life from my blogging career.  But by popular request, let me now provide a link to my very first post-tenure publication: The Ghost in the Quantum Turing Machine.

Here’s the 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 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.

See here (and also here) for interesting discussions over on Less Wrong.  I welcome further discussion in the comments section of this post, and will jump in myself after a few days to address questions (update: eh, already have).  There are three reasons for the self-imposed delay: first, general busyness.  Second, inspired by the McGeoch affair, I’m trying out a new experiment, in which I strive not to be on such an emotional hair-trigger about the comments people leave on my blog.  And third, based on past experience, I anticipate comments like the following:

“Hey Scott, I didn’t have time to read this 85-page essay that you labored over for two years.  So, can you please just summarize your argument in the space of a blog comment?  Also, based on the other comments here, I have an objection that I’m sure never occurred to you.  Oh, wait, just now scanning the table of contents…”

So, I decided to leave some time for people to RTFM (Read The Free-Will Manuscript) before I entered the fray.

For now, just one remark: some people might wonder whether this essay marks a new “research direction” for me.  While it’s difficult to predict the future (even probabilistically :-) ), I can say that my own motivations were exactly the opposite: I wanted to set out my thoughts about various mammoth philosophical issues once and for all, so that then I could get back to complexity, quantum computing, and just general complaining about the state of the world.

“Quantum Information and the Brain”

Thursday, January 24th, 2013

A month and a half ago, I gave a 45-minute lecture / attempted standup act with the intentionally-nutty title above, for my invited talk at the wonderful NIPS (Neural Information Processing Systems) conference at Lake Tahoe.  Video of the talk is now available at VideoLectures net.  That site also did a short written interview with me, where they asked about the “message” of my talk (which is unfortunately hard to summarize, though I tried!), as well as the Aaron Swartz case and various other things.  If you just want the PowerPoint slides from my talk, you can get those here.

Now, I could’ve just given my usual talk on quantum computing and complexity.  But besides increasing boredom with that talk, one reason for my unusual topic was that, when I sent in the abstract, I was under the mistaken impression that NIPS was at least half a “neuroscience” conference.  So, I felt a responsibility to address how quantum information science might intersect the study of the brain, even if the intersection ultimately turned out to be the empty set!  (As I say in the talk, the fact that people have speculated about connections between the two, and have sometimes been wrong but for interesting reasons, could easily give me 45 minutes’ worth of material.)

Anyway, it turned out that, while NIPS was founded by people interested in modeling the brain, these days it’s more of a straight machine learning conference.  Still, I hope the audience there at least found my talk an amusing appetizer to their hearty meal of kernels, sparsity, and Bayesian nonparametric regression.  I certainly learned a lot from them; while this was my first machine learning conference, I’ll try to make sure it isn’t my last.

(Incidentally, the full set of NIPS videos is here; it includes great talks by Terry Sejnowski, Stanislas Dehaene, Geoffrey Hinton, and many others.  It was a weird honor to be in such distinguished company — I wouldn’t have invited myself!)

A causality post, for no particular reason

Friday, November 2nd, 2012

The following question emerged from a conversation with the machine learning theorist Pedro Domingos a month ago.

Consider a hypothetical race of intelligent beings, the Armchairians, who never take any actions: never intervene in the world, never do controlled experiments, never try to build anything and see if it works.  The sole goal of the Armchairians is to observe the world around them and, crucially, to make accurate predictions about what’s going to happen next.  Would the Armchairians ever develop the notion of cause and effect?  Or would they be satisfied with the notion of statistical correlation?  Or is the question kind of silly, the answer depending entirely on what we mean by “developing the notion of cause and effect”?  Feel free to opine away in the comments section.

Why Many-Worlds is not like Copernicanism

Saturday, August 18th, 2012

[Update (8/26): Inspired by the great responses to my last Physics StackExchange question, I just asked a new one---also about the possibilities for gravitational decoherence, but now focused on Gambini et al.'s "Montevideo interpretation" of quantum mechanics.

Also, on a completely unrelated topic, my friend Jonah Sinick has created a memorial YouTube video for the great mathematician Bill Thurston, who sadly passed away last week.  Maybe I should cave in and set up a Twitter feed for this sort of thing...]

[Update (8/26): I've now posted what I see as one of the main physics questions in this discussion on Physics StackExchange: "Reversing gravitational decoherence."  Check it out, and help answer if you can!]

[Update (8/23): If you like this blog, and haven't yet read the comments on this post, you should probably do so!  To those who've complained about not enough meaty quantum debates on this blog lately, the comment section of this post is my answer.]

[Update: Argh!  For some bizarre reason, comments were turned off for this post.  They're on now.  Sorry about that.]

I’m in Anaheim, CA for a great conference celebrating the 80th birthday of the physicist Yakir Aharonov.  I’ll be happy to discuss the conference in the comments if people are interested.

In the meantime, though, since my flight here was delayed 4 hours, I decided to (1) pass the time, (2) distract myself from the inanities blaring on CNN at the airport gate, (3) honor Yakir’s half-century of work on the foundations of quantum mechanics, and (4) honor the commenters who wanted me to stop ranting and get back to quantum stuff, by sharing some thoughts about a topic that, unlike gun control or the Olympics, is completely uncontroversial: the Many-Worlds Interpretation of quantum mechanics.

Proponents of MWI, such as David Deutsch, often argue that MWI is a lot like Copernican astronomy: an exhilarating expansion in our picture of the universe, which follows straightforwardly from Occam’s Razor applied to certain observed facts (the motions of the planets in one case, the double-slit experiment in the other).  Yes, many holdouts stubbornly refuse to accept the new picture, but their skepticism says more about sociology than science.  If you want, you can describe all the quantum-mechanical experiments anyone has ever done, or will do for the foreseeable future, by treating “measurement” as an unanalyzed primitive and never invoking parallel universes.  But you can also describe all astronomical observations using a reference frame that places the earth is the center of the universe.  In both cases, say the MWIers, the problem with your choice is its unmotivated perversity: you mangle the theory’s mathematical simplicity, for no better reason than a narrow parochial urge to place yourself and your own experiences at the center of creation.  The observed motions of the planets clearly want a sun-centered model.  In the same way, Schrödinger’s equation clearly wants measurement to be just another special case of unitary evolution—one that happens to cause your own brain and measuring apparatus to get entangled with the system you’re measuring, thereby “splitting” the world into decoherent branches that will never again meet.  History has never been kind to people who put what they want over what the equations want, and it won’t be kind to the MWI-deniers either.

This is an important argument, which demands a response by anyone who isn’t 100% on-board with MWI.  Unlike some people, I happily accept this argument’s framing of the issue: no, MWI is not some crazy speculative idea that runs afoul of Occam’s razor.  On the contrary, MWI really is just the “obvious, straightforward” reading of quantum mechanics itself, if you take quantum mechanics literally as a description of the whole universe, and assume nothing new will ever be discovered that changes the picture.

Nevertheless, I claim that the analogy between MWI and Copernican astronomy fails in two major respects.

The first is simply that the inference, from interference experiments to the reality of many-worlds, strikes me as much more “brittle” than the inference from astronomical observations to the Copernican system, and in particular, too brittle to bear the weight that the MWIers place on it.  Once you know anything about the dynamics of the solar system, it’s hard to imagine what could possibly be discovered in the future, that would ever again make it reasonable to put the earth at the “center.”  By contrast, we do more-or-less know what could be discovered that would make it reasonable to privilege “our” world over the other MWI branches.  Namely, any kind of “dynamical collapse” process, any source of fundamentally-irreversible decoherence between the microscopic realm and that of experience, any physical account of the origin of the Born rule, would do the trick.

Admittedly, like most quantum folks, I used to dismiss the notion of “dynamical collapse” as so contrived and ugly as not to be worth bothering with.  But while I remain unimpressed by the specific models on the table (like the GRW theory), I’m now agnostic about the possibility itself.  Yes, the linearity of quantum mechanics does indeed seem incredibly hard to tinker with.  But as Roger Penrose never tires of pointing out, there’s at least one phenomenon—gravity—that we understand how to combine with quantum-mechanical linearity only in various special cases (like 2+1 dimensions, or supersymmetric anti-deSitter space), and whose reconciliation with quantum mechanics seems to raise fundamental problems (i.e., what does it even mean to have a superposition over different causal structures, with different Hilbert spaces potentially associated to them?).

To make the discussion more concrete, consider the proposed experiment of Bouwmeester et al., which seeks to test (loosely) whether one can have a coherent superposition over two states of the gravitational field that differ by a single Planck length or more.  This experiment hasn’t been done yet, but some people think it will become feasible within a decade or two.  Most likely it will just confirm quantum mechanics, like every previous attempt to test the theory for the last century.  But it’s not a given that it will; quantum mechanics has really, truly never been tested in this regime.  So suppose the interference pattern isn’t seen.  Then poof!  The whole vast ensemble of parallel universes spoken about by the MWI folks would have disappeared with a single experiment.  In the case of Copernicanism, I can’t think of any analogous hypothetical discovery with even a shred of plausibility: maybe a vector field that pervades the universe but whose unique source was the earth?  So, this is what I mean in saying that the inference from existing QM experiments to parallel worlds seems too “brittle.”

As you might remember, I wagered $100,000 that scalable quantum computing will indeed turn out to be compatible with the laws of physics.  Some people considered that foolhardy, and they might be right—but I think the evidence seems pretty compelling that quantum mechanics can be extrapolated at least that far.  (We can already make condensed-matter states involving entanglement among millions of particles; for that to be possible but not quantum computing would seem to require a nasty conspiracy.)  On the other hand, when it comes to extending quantum-mechanical linearity all the way up to the scale of everyday life, or to the gravitational metric of the entire universe—as is needed for MWI—even my nerve falters.  Maybe quantum mechanics does go that far up; or maybe, as has happened several times in physics when exploring a new scale, we have something profoundly new to learn.  I wouldn’t give much more informative odds than 50/50.

The second way I’d say the MWI/Copernicus analogy breaks down arises from a closer examination of one of the MWIers’ favorite notions: that of “parochial-ness.”  Why, exactly, do people say that putting the earth at the center of creation is “parochial”—given that relativity assures us that we can put it there, if we want, with perfect mathematical consistency?  I think the answer is: because once you understand the Copernican system, it’s obvious that the only thing that could possibly make it natural to place the earth at the center, is the accident of happening to live on the earth.  If you could fly a spaceship far above the plane of the solar system, and watch the tiny earth circling the sun alongside Mercury, Venus, and the sun’s other tiny satellites, the geocentric theory would seem as arbitrary to you as holding Cheez-Its to be the sole aim and purpose of human civilization.  Now, as a practical matter, you’ll probably never fly that spaceship beyond the solar system.  But that’s irrelevant: firstly, because you can very easily imagine flying the spaceship, and secondly, because there’s no in-principle obstacle to your descendants doing it for real.

Now let’s compare to the situation with MWI.  Consider the belief that “our” universe is more real than all the other MWI branches.  If you want to describe that belief as “parochial,” then from which standpoint is it parochial?  The standpoint of some hypothetical godlike being who sees the entire wavefunction of the universe?  The problem is that, unlike with my solar system story, it’s not at all obvious that such an observer can even exist, or that the concept of such an observer makes sense.  You can’t “look in on the multiverse from the outside” in the same way you can look in on the solar system from the outside, without violating the quantum-mechanical linearity on which the multiverse picture depends in the first place.

The closest you could come, probably, is to perform a Wigner’s friend experiment, wherein you’d verify via an interference experiment that some other person was placed into a superposition of two different brain states.  But I’m not willing to say with confidence that the Wigner’s friend experiment can even be done, in principle, on a conscious being: what if irreversible decoherence is somehow a necessary condition for consciousness?  (We know that increase in entropy, of which decoherence is one example, seems intertwined with and possibly responsible for our subjective sense of the passage of time.)  In any case, it seems clear that we can’t talk about Wigner’s-friend-type experiments without also talking, at least implicitly, about consciousness and the mind/body problemand that that fact ought to make us exceedingly reluctant to declare that the right answer is obvious and that anyone who doesn’t see it is an idiot.  In the case of Copernicanism, the “flying outside the solar system” thought experiment isn’t similarly entangled with any of the mysteries of personal identity.

There’s a reason why Nobel Prizes are regularly awarded for confirmations of effects that were predicted decades earlier by theorists, and that therefore surprised almost no one when they were finally found.  Were we smart enough, it’s possible that we could deduce almost everything interesting about the world a priori.  Alas, history has shown that we’re usually not smart enough: that even in theoretical physics, our tendencies to introduce hidden premises and to handwave across gaps in argument are so overwhelming that we rarely get far without constant sanity checks from nature.

I can’t think of any better summary of the empirical attitude than the famous comment by Donald Knuth: “Beware of bugs in the above code.  I’ve only proved it correct; I haven’t tried it.”  In the same way, I hereby declare myself ready to support MWI, but only with the following disclaimer: “Beware of bugs in my argument for parallel copies of myself.  I’ve only proved that they exist; I haven’t heard a thing from them.”

Why Philosophers Should Care About Computational Complexity

Monday, August 8th, 2011

Update (August 11, 2011): Thanks to everyone who offered useful feedback!  I uploaded a slightly-revised version, adding a “note of humility” to the introduction, correcting the footnote about Cramer’s Conjecture, incorporating Gil Kalai’s point that an efficient program to pass the Turing Test could exist but be computationally intractable to find, adding some more references, and starting the statement of Valiant’s sample-size theorem with the word “Consider…” instead of “Fix…”


I just posted a 53-page essay of that name to ECCC; it’s what I was writing pretty much nonstop for the last two months.  The essay will appear in a volume entitled “Computability: Gödel, Turing, Church, and beyond,” which MIT Press will be publishing next year (to coincide with Alan T.’s hundredth birthday).

Note that, to explain why philosophers should care about computational complexity, I also had to touch on the related questions of why anyone should care about computational complexity, and why computational complexity theorists should care about philosophy.  Anyway, here’s the abstract:

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance.  In this essay, I offer a detailed case that one would be wrong.  In particular, I argue that computational complexity theory—the field that studies the resources (such as time, space, and randomness) needed to solve computational problems—leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume’s problem of induction and Goodman’s grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest.  I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.

Weighing in with 70 footnotes and 126 references, the essay is basically a huge, sprawling mess; I hope that at least some of you will enjoy getting lost in it.  I’d like to thank my editor, Oron Shagrir, for kicking me for more than a year until I finally wrote this thing.