Consciousness as a Gödel sentence in the language of science

Why the Hard Problem (might be) so Hard

Art for The Intrinsic Perspective is by Alexander Naughton

Let's say you lived in a universe where you really were some sort of incarnated soul in a corporeal body. Or some sort of agent from a vaster reality embedded in a simulation (depending on definitions, the two scenarios might not be that different). What would the science in such a dualistic universe look like?

It would probably look like this: in an incarnated-soul universe, the laws of physics make sense, and are nice and rule-following, except for one notable absence, which is that no one can explain how exactly corporeal bodies (or simulated ones) have a you. The incarnated/simulated scientists would open up the incarnating/simulating brains and say, “Hmm, it looks like it's following all the physical rules, but we can’t explain how you fit into this thing!” There would be, in other words, an explanatory gap between being you and the incarnation of you. Great scientists and philosophers would write books saying things like:

No explanation given wholly in physical terms can ever account for [it].

I regard [it] as fundamental. I regard matter as derivative from [it].

I believe that the present laws of physics are at least incomplete without a translation into terms of [it].

Which is exactly the kind of thing great thinkers write in our universe (here, David Chalmers, Max Planck, Eugene Wigner, respectively).

What would the state of the rest of science be there? For one, their version of neuroscience would lag significantly behind. Which again, looks like our universe, where there is no accepted lawful way to relate brain states to conscious experiences—what it is like to be you. It is an odd position to be in, given the success of science in so many other domains!

What else? Well, it’s likely that in the incarnated-soul universe, their version of AI would suspiciously have nothing to do with how the brain works. It would be as if they had discovered this totally orthogonal form of intelligence, one based more on all the actual sensible physical rules, and not based off of mysterious confusing soul stuff. An orthogonality which, uh—and things are getting uncomfortable here—is arguably also the case in our universe.

One could go whole hog and say that this all points to proof of religion. But which one? What details? A gap of inexplicability doesn't recommend anything.

And it feels a bit too easy, right? For there is another explanation, one rarely explored, which is that there is no way to set up a universe without this sort of confusion. Which ends up being the same as asking: Are all scientific facts knowable? Or is science fundamentally incomplete?

While I can’t claim certainty, science being fundamentally incomplete is at least conceivable to me. It would mean that there are scientific facts that at first look tantalizingly discoverable, but then their true answers remain closed to us for non-trivial reasons. (“Non-trivial” is important here—there are clearly many facts trivially closed for us, like counting every atom in the universe.) Non-trivial scientific incompleteness would be a different beast. It’d be more like some statements within science end in paradox, the scientific equivalent of “This sentence is a lie.”

Scientists usually shy away from meta-scientific questions like that of scientific incompleteness. I understand why! It seems too much like philosophy, which is dangerous (something I was told during my PhD repeatedly). Dark and deep waters. But of course, a very similar question was famously asked in mathematics, and it has a very famous answer in Gödel's theorems (essentially that yes, mathematics is necessarily incomplete). While the “genre” of incompleteness proofs doesn’t affect most working mathematicians, it also isn’t irrelevant—it crops up like an ominous weed.

Here we must initially be wary: gesturing to incompleteness in mathematics is merely an analogy. It is up for debate (or investigation) to what degree the analogy holds. The various incompleteness results in mathematics (some are more famous than others) conceptualize mathematics as a formal system, an abstract machine. This machine requires defining things like symbols, a grammar, a set of axioms, and inference rules—then you set the machine to run, and see if you can find within its working paradoxes.

What to do such paradoxes look like? A brief example is necessary to capture their gist. Might as well do the most famous.

As with many proofs in the “incompleteness genre” Gödel's theorems are based on self-reference. Basically (and warning: one must radically simplify here) Gödel uses a form of encoding (an interpretation of what the system is doing) to read a formal system S as talking about itself. He then constructs what’s known as a “Gödel sentence.” A Gödel sentence is basically a more complex version of the classic “This sentence is a lie.” It instead claims that “This sentence is not provable in S.” Now just think about it: If that sentence were proved to be true, then it would render S inconsistent, because it’s proving true something it says cannot be proven (inconsistency). On the other hand, if S is consistent, it must not be able to prove the sentence, meaning incompleteness—certain things are true (because the sentence is indeed not provable), but you can’t prove them from within.

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Here's a pause (with smooth jazz) to work through that yourself if you like.

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Similarly, it’s worth asking: (a) can science be thought of as a formal system? And also (b) Is there anything in science that, if you squint, looks like a Gödel sentence?

As to the first ask: obviously science is an incredibly messy process, the details of which change depending on exactly what natural phenomena you seek to explain, how practices vary between fields, heck, even departmental politics. Yet the messiness of science in an applied sense doesn’t outright preclude it being modellable as a formal system. Crack any basic high school textbook and you will often find some (obviously superficial) formal description of science.

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We could imagine empiricism as a very complicated formal system, replete with epicycles, with various inference rules to arrive at scientific truths, and so on. We can even see kernels of formalizing science in ideas like Karl Popper’s notion of science as being primarily about falsification (although Kuhn’s antithesis of science as a series of paradigm shifts seems more difficult to capture formally). Again, it seems at least conceivable that science could be viewed as an abstract machine, perhaps one whose sole goal is even rather simple—like an abstract machine for rejecting hypotheses.

While there has been a smidgeon of modern work on conceptualizing science as a formal system, there’s been not nearly as much as the question deserves, nor much exploration of what it mean for different scientific fields (no asking: Where would incompleteness crop up?).

In fact, incompleteness might necessarily creep into science due to its reliance on mathematics itself. This is what Stephen Hawking thought. While he’s known for his triumphal pursuit of a Theory of Everything, toward the end of his life Hawking concluded that such a theory was impossible. His reasoning? That science owed too much to mathematics, and therefore undecidability crept in—science inherited paradoxes from mathematics.

All this stuff has a long intellectual heritage. The question of whether physics could be grounded in a set of axioms was one of Hilbert’s 23 problems for mathematics that he posed in 1900. He wrote:

The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part.

Good enough for Hilbert, good enough for me. In fact, Hilbert would be interested to learn that we already know that physics appears to include undecidable properties. There are a number of these, again, all remarkably little discussed. E.g., in 2015 a Nature paper literally titled “Undecidability of the spectral gap” showed that an important physical quantity—the difference in energy between ground state and the initial excited state of a material—is formally undecidable. It is also triggered by recursion (basically, they encode questions about spectral gaps into spectral gaps).

So in a sense, we can already say that science is incomplete! At least, we know there are physical properties that cannot be actually discovered (in a rigorous or systematic manner). So it's very much not a foregone conclusion that scientific incompleteness is true. And if it were true, it would be a very good explanation for why the Hard Problem of consciousness is so damn Hard.

I'm certainly not the first person to point out that there are oddities around consciousness and self-reference. But I actually think a lot of the work doesn't directly touch on scientific incompleteness—it's the sort of thing that people think exists fleshed out in detail somewhere, but then struggle to put to a clear source.

E.g., consider Roger Penrose’s The Emperor’s New Mind, which argues that humans can understand undecidable statements in ways that computers can’t, and therefore the human mind is not a computer. Yet, whether or not the mind is a computer doesn’t really tell us if science is complete, nor if the difficulties in understanding consciousness comes from scientific incompleteness.

In Colin McGinn’s The Mysterious Flame he argues for a “mysterian” position wherein consciousness is— you guessed it from the name—necessarily a mystery. However, McGinn thinks that consciousness is solvable scientifically in principle, it’s just that humans themselves lack the intelligence (just as a dog lacks the intelligence to ever understand General Relativity). For, say, a superintelligent AI, consciousness would be no problem.

A third example would be the extremely classic Gödel, Escher, Bach by Douglas Hofstadter—a wonderfully playful book on recursion and the mind. In fact, I’ve found that people often think that Gödel, Escher, Bach’s central hypothesis is about scientific incompleteness! But it’s really not, at least explicitly. If you read the later I am a Strange Loop it becomes clear that what Hofstadter was trying to say is that consciousness just is symbolic recursion in the brain. Personally, I think this is quite a low bar theory, since presumably even a Skyrim NPC might possess rudimentary symbolic recursion, and we don’t think they’re consciousness (an arrow to the knee doesn’t really hurt them). Regardless, the idea of scientific incompleteness is orthogonal to the idea that symbolic recursion is the root of consciousness—here, the claim is not that consciousness comes about via recursion, but rather that within science a theory of consciousness would itself be recursive.

One of the earliest proto-statements of the idea of scientific incompleteness I’ve found is actually in Nobel-Prize-winning economist Friedrich Hayek’s mostly forgotten book on psychology, The Sensory Order. And honestly, his argument is not very good. Basically, he just thinks you can’t understand a natural phenomenon without possessing significantly greater complexity than it, and we don’t possess enough neural complexity to understand our own brains (this seems immediately undermined by counterexamples, like indicating that we could never actually understand how weather works, etc).

But none of these make direct arguments about scientific incompleteness, which is why I spent a chunk of my book The World Behind the World: Consciousness, Free Will, and the Limits of Science, developing it. Trying to make it as direct as possible, there I defined scientific incompleteness as the idea that:

A theory of consciousness is like a Gödel sentence written in the language of science.

I'll openly admit there is no direct evidence of this. But there is circumstantial evidence, and some viable paths toward a philosophical proof.

E.g., no matter which way we broadly turn on what is traditionally called the “mind-body problem” we end in paradox. On one side, once we encode ourselves in the scientific worldview we realize that something is missing (our consciousness). Thomas Nagel, in The View from Nowhere, gives a great metaphor for this: imagine that the scientific worldview is literally a map (of the universe, presumably). It might be complete and coherent, as if axiomatically consistent. But there is a piece of information obviously missing from the map: the YOU ARE HERE sign. And this extra information that relates you, the observer, to the map itself, is not captured anywhere on the map itself.

Edward Steed

Can we then go too far to the other side and believe that consciousness does not exist? This also seems paradoxical. After all, we only know the world through our own consciousness. Every tidbit of science that you know is known only through your consciousness. In fact, it’s arguable that science is just a set of corroborated first-person experiences! Imagining science without consciousness asks us to throw away the entire foundation of our knowledge, the one thing we are truly sure of, and what mediates all our other information—paradoxically, including our certainty of the facts of science itself.

Allow me to get even more head-scratchingly gnarly for a moment with my own example of how paradoxes crop up in our thinking about consciousness in exactly the way you would expect if scientific incompleteness were true.

In the popular “zombie argument” by David Chalmers, the conclusion is that materialism is false because a possible world populated by “zombies” is conceivable (wherein a zombie is someone much like ourselves—but while neurons are firing, and atoms are moving, there is a total lack of subjective experience). If such a world is conceivable, then this means that the normal laws of physics don't fix consciousness in place. It seems like you need some sort of extra laws that go beyond physics. So pure materialism can't be true.

But let's say that a zombie makes the zombie argument. They imagine a possible world filled with zombies and find it conceivable and therefore conclude that materialism is false. But wait! Materialism is true in the zombie world! So therefore, the zombie argument would be wrong. Its validity relies on a hidden premise that the person making the argument is themselves not a zombie.

Fine. That's a premise, but so what? Well, if the zombie argument were to be valid, then no utterance or behavior or experiment could tell the difference between a zombie and a non-zombie. Nor any amount of self-reflection or introspection. There could even be some zombie version of Descartes (let’s call him Zescartes) who concludes he is indubitably conscious, then makes the zombie argument. But for Zescartes, the zombie argument returns the wrong conclusion: materialism is false, despite it actually being true in his world. And there's no way for Descartes or Zescartes to tell themselves apart.

So the zombie argument itself ends in paradox. If it were valid without the premise of already being conscious, a materialist zombie could use it to argue against materialism, indicating inconsistency. On the other hand, if it is valid only with the premise of consciousness, then any conscious being can always make the zombie argument, and then notice that whether or not they are a zombie cannot be resolved by introspection or empirical evidence, indicating incompleteness. The existence of consciousness appears to be the kind of true fact that cannot be proven from the inside.

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Here's a pause (with frenetic jazz) to work through that yourself if you like.

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More evidence for scientific incompleteness comes from an examining how science has historically developed, with consciousness lagging behind.

As philosopher Philip Goff notes in Galileo’s Error: Galileo made specifically sure to carve out qualitative properties (in modern terms, essentially facts about consciousness) from science to begin with. And it was really this removal (in his book The Assayer) that finalized the scientific method—it told scientists what kinds of things focus on, and what kinds of things to ignore. As Goff points out, if Galileo were returned to life in the modern era, he would immediately say that of course science is incomplete, since he, a religious man, never designed science to talk about souls anyways!

Galileo’s restriction of the purview of science to the material and mechanistic looks to me much like Bertrand Russell’s and Alfred North Whitehead’s attempts to remove self-reference (and its associated paradoxes) from mathematics in their infamous Principa Mathematica. Their attempt was based on creating a tier of types of sets (the mathematical entities that they tried to ground all other math in). Their strategy disallowed tiers to “talk about” (contain) members of a similar tier. It only kind of worked, merely delaying paradoxes from creeping in.

This bracketing looks to me a lot like what Galileo did to science, removing consciousness and allowing it to proceed. In fact, even now in consciousness science, people talk about different “strengths” of theories of consciousness. E.g. for a philosopher like Chalmers, the distinction between the Hard versus Easy problems of consciousness, or for a scientist like Anil Seth, the distinction between weak versus strong theories of consciousness. All these are basically admittances that some bracketing is already happening.

In summary, we know this: (a) It is conceivable that science could be some sort of formal system, (b) Some properties of nature have already been proven to be undecidable, (c) Arguments in philosophy of mind often end in paradoxes indicating inconsistency or incompleteness, and (d) The historical development of science, and even the contemporary science of consciousness, resembles a series of bracketings very similar to how mathematicians tried to remove self-reference and paradoxes.

All of this fits perfectly with a world that is scientifically incomplete.

Obviously, we must indulge a final instance of self-reference. If scientific incompleteness is true, how do you—yes, you—exist in a world that is paradoxical? How do you relate to it, I mean? If it is indeed a fact that there are questions about the universe that are true but cannot be proved, this unfortunately recommends no particular religion, points you to no higher mysteries. It is merely a blankness, a clouded-over wall that extends infinitely in all directions.

But still, upon finding such an inaccessible wall, we humans cannot help our nature. We cannot help but wonder: what’s behind the wall?

What do you do when something just