Bobby Azarian is a cognitive neuroscientist and a science journalist.
Before Kurt Gödel, logicians and mathematicians believed that all statements about numbers — and reality more generally — were either true or false, and that there must be a rule-based way of determining which category a specific statement belonged to. According to this logic, mathematical proof is the true source of knowledge.
The Pythagorean theorem, for example, is a mathematical conjecture that is true: It has been proved formally, and in more ways than one. With many theorems, it may be extremely difficult to find proof, but if it is true, it must have a proof — and if it is false, then it should be impossible to prove with the fundamental axioms and the rules of inference of the formal mathematical system.
At least, that was the assumption made by leading mathematicians of the early 20th century like David Hilbert, and later Bertrand Russell and Alfred North Whitehead, who attempted to design an ultimate formal system that could, in theory, prove or disprove any conceivable mathematical theorem. Meanwhile, scientists and philosophers at that time were trying to demystify the mind by showing that human reasoning was the product of purely algorithmic processes. If we could somehow access the exact steps that brains were following to ascertain something, they argued, we would find that they were using strict rules of logic.
A brain, then, was nothing more than a squishy Turing machine — a simple device operating on reasonably simple rules that could compute the solution to any problem solvable with computation, given enough time and memory. This would mean that all the mystery and magic associated with conscious thought could be boiled down to logical operations, or rule-based symbol manipulation. The mind would be no more mysterious than a computer — everything it did would be determinable, definable and understood mathematically. It was a pretty sensible stance at the time.
But Gödel, an eccentric Austrian logician, disproved that view even before Alan Turing invented his abstract machine, in a quite roundabout and loopy way. In 1931, Gödel published his famous incompleteness theorem, as it became known, which called into question the power of mathematics to explain all of reality — along with the hypothesis that the mind works like a formal system, or a mathematical machine.
With a clever use of paradox, Gödel would destroy the idea that truth is equivalent to mathematical proof. Taking inspiration from an old Greek logic statement involving self-reference called the “liar’s paradox,” he constructed a proposition about number theory using a ridiculously complex coding scheme that has become known as Gödel numbering. Although the theorem is virtually impossible to understand for anyone without an advanced degree in mathematics, we can comprehend it by translating it into similar statements in common language.
To really grasp how the liar’s paradox inspired a new kind of theorem — one that would threaten the very foundations of mathematics — you have to explore the logic for yourself. Consider the following sentence, which is a more straightforward variation of the liar’s paradox, and try to determine whether it is true or false:
“This statement is false.”
This exercise works best if you say the statement out loud. Notice that trying to prove the sentence true or false sends you around a loop that does neither. If the statement is true, it would mean the statement is false, because it says that it is. So it can’t be true. But if the statement, “This statement is false,” is false, then that would mean that the statement is true, because it states that it is false. Thus, it can’t be false without being true.
Either pursuit leads to a contradiction, and it is impossible to see the logic of why unless you go around the loop. We are left with a proposition that can neither be proven true nor false only because the statement has this strange and somewhat absurd property of referring to itself. While this discovery might appear to be trivial on the surface, it caused quite a stir among mathematicians and logicians because it demonstrated that no formal system can be considered consistent and complete if it produces what are known as “undecidable” conjectures.
But the true brilliance of Gödel’s theorem was not that it constructed a mathematical statement that could not be proven true or false. Gödel bumped the loopiness up a level by creating a conjecture that was true, but unprovable. Notice that the following self-referential statement is not just about itself, but also its own provability:
“This statement has no proof.”
It was no easy feat, but Gödel created a mathematical statement that was the numerical equivalent of that sentence. The interesting thing about this particular proposition is that it is, in fact, true — it has no proof. We don’t even have to check, because if the proof did exist, it would mean that the statement is true. But it says that it has no proof, so once again, proving the statement would only disprove it.
Even though it cannot be proven with the axioms of the system and the rules of inference, mathematicians can clearly see that its truth is self-evident by focusing on what the symbols mean. Gödel’s true but unprovable proposition proves that there are truths that exist outside the realm of what can be deduced using symbolic logic or computation.
Because mathematicians could see the truth of an undecidable conjecture, the great theoretical physicist Roger Penrose later argued that the mind must be doing something that goes beyond raw computation. In other words, the brain must be more than a symbol-shuffling machine. As he wrote in a 1994 paper:
The inescapable conclusion [of Gödel’s theorem] seems to be: Mathematicians are not using a knowably sound calculation procedure in order to ascertain mathematical truth. We deduce that mathematical understanding — the means whereby mathematicians arrive at their conclusions with respect to mathematical truth — cannot be reduced to blind calculation!
While Penrose can be credited for popularizing this insight, which was proposed by the British philosopher John Lucas nearly three decades earlier, it seems that Gödel himself was aware of that implication of his theorem, as can be seen by this famous quote of his: “Either mathematics is too big for the human mind or the human mind is more than a machine.”
What exactly is the difference between mind and machine? Machines compute, minds understand. They allow us to see truths that a purely algorithmic intelligence would be blind to. What is it that allows this curious ability that we call understanding? Conscious experience, presumably, which enables us to not just reason, but to reflect on reasoning itself.
While Penrose was justified in arguing that the mind is not a Turing machine, he made what many consider an unjustified leap when he proposed that the brain must then be some kind of quantum computer. Although this theory should not be dismissed on the grounds that it invokes a quantum explanation, the truth is that right now it is not taken seriously by most scientists working on the problem of consciousness. The most well-known criticism, supported by physicists like Max Tegmark, says that the brain is too warm, wet and noisy to sustain the kind of coherent quantum state that Penrose believes is responsible for conscious processing.
However, it is worth pointing out that researchers now think a growing number of biological processes exploit quantum mechanics — like bird navigation, which uses quantum entanglement, and photosynthesis, which involves quantum tunneling. If quantum biology is real and takes place inside “warm and wet” systems, who’s to say that quantum neurobiology is impossible? If there’s some computational advantage to a mechanical process that exists in nature, natural selection will typically find a way to leverage it.
While Gödel’s incompleteness theorem made consciousness more mysterious to Penrose, it provided the solution to the puzzle for Douglas Hofstadter, the philosopher who wrote the Pulitzer Prize-winning book “Gödel, Escher, Bach: An Eternal Golden Braid,” which was published in 1979, a decade before Penrose’s book. To Hofstadter, the mystery of subjectivity can only be explained with the concept of self-reference, the same property that allowed Gödel’s statements to transcend formal proof. By referring to themselves, symbols suddenly became meaningful, and semantics sprouted from syntax.
“Something very strange thus emerges from the Gödelian loop: the revelation of the causal power of meaning in a rule-bound but meaning-free universe,” Hofstadter wrote. According to him, he said the self that we associate with subjective experience emerges from the same kind of self-reference “via a kind of vortex whereby patterns in a brain mirror the brain’s mirroring of the world, and eventually mirror themselves, whereupon the vortex of ‘I’ becomes a real, causal entity.”
More specifically, self-reference in the form of self-modeling produces an observer with causal power. Just as Gödel showed that math can reference itself — call it “metamathematics” — minds can do the same by looking back at the model of the world that evolution and adaptive learning have built up in brains. As Hofstadter wrote: “When and only when such a loop arises in a brain or in any other substrate, is a person — a unique new ‘I’ — brought into being. Moreover, the more self-referentially rich such a loop is, the more conscious is the self to which it gives rise.”
The lovably loopy idea that consciousness emerges from self-modeling is supported by some intellectual heavyweights, like Judea Pearl, whose causal calculus forms the backbone of one of today’s most respected consciousness theories, integrated information theory. In a 2019 interview with MIT podcast host Lex Fridman, Pearl was clearly echoing Hofstadter’s big idea:
That’s consciousness. You have a model of yourself. Where do you get this model? You look at yourself as if you are a part of the environment. … I have a blueprint of myself, so at that level of a blueprint I can modify things. I can look at myself in the mirror and say, ‘Hmm, if I tweak this model I’m going to perform differently.’ That is what we mean by free will. … For me, consciousness is having a blueprint of your software.
So how does this idea line up with modern neuroscience? Most neuroscientists believe that consciousness arises when harmonized global activity emerges from the coordinated interactions of billions of neurons. This is because the synchronized firing of brain cells integrates information from multiple processing streams into a unified field of experience. This global activity is made possible by loops in the form of feedback. When feedback is present in a system, it means there is some form of self-reference at work, and in nervous systems, it can be a sign of self-modeling. Feedback loops running from one brain region to another integrate information and bind features into a cohesive perceptual landscape.
When does the light of subjective experience go out? When the feedback loops cease, because it is these loops that harmonize neural activity and bring about the global integration of information. When feedback is disrupted, the brain still keeps on ticking, functioning physiologically and controlling involuntary functions, but consciousness dissolves. The mental model is still embedded in the brain’s architecture, but the observer fades as the self-referential process of real-time self-modeling ceases to produce a “self.”
According to integrated information theory — invented by the neuroscientist Giulio Tononi — a system that has no feedback loops to integrate information can in theory display conscious behavior without having the corresponding experience that a system integrating information would have. Such systems are called “feed-forward systems” because the flow of information only travels one way. An example of a feed-forward system is the cerebellum, which contains more neurons than any other brain region, yet it does not appear to produce an observer. The neuroscientist Christof Koch, one of integrated information theory’s most high-profile supporters, explains the reason no self sprouts from the cerebellum in a 2018 Nature article titled “What Is Consciousness?”:
“The cerebellum is almost exclusively a feed-forward circuit: One set of neurons feeds the next, which in turn influences a third set. There are no complex feedback loops that reverberate with electrical activity passing back and forth.”
An equally famous consciousness theory invented by the neuroscientist Bernard Baars, known as global workspace theory, describes how a stream of conscious experience emerges when multiple sensory streams fuse to form a unified perceptual landscape. In this computational model, consciousness is referred to as a “global workspace” because its contents can be manipulated by the mind and broadcast globally to many regions in the brain at once.
The mental workspace is thought to be produced by feedback loops running from the frontal lobes to the parietal lobes and back again — so-called “fronto-parietal loops” — which integrate information over space and time. When these feedback loops cease, the global workspace ceases to be. Conscious processing is disrupted, and information is no longer made globally available.
What can we conclude from these neuroscience theories of consciousness? That Douglas Hofstadter was right — it is self-reference in the form of self-modeling that conjures up an observer, and it does so through feedback loops that entrain neural activity and integrate information. If it wasn’t for Gödel and his loopy incompleteness theorem, the significance of self-reference might not have been discovered, and Hofstadter probably would have never connected it to self-modeling in minds.
While there is still much for scientists to learn about how the brain generates conscious experience, it is clear that Gödel was correct in his assessment that the mind is more than just a machine. It is a generator of conscious experience that allows beings with brains to reflect on reasoning and to understand the meaning encoded in true but unprovable statements.
Could self-reference be the missing puzzle piece that allows for truly intelligent AIs, and maybe even someday sentient machines? Only time will tell, but Simon DeDeo, a complexity scientist at Carnegie Mellon University and the Santa Fe Institute, seems to think so: “Great progress in physics came from taking relativity seriously. We ought to expect something similar here: Success in the project of general artificial intelligence may require we take seriously the relativity implied by self-reference.”