Gödel’s Ghost and the Left Hemisphere Trap

Gödel, McGilchrist, and why human meaning requires more than logic

Hi everyone,

My sister is visiting from Washington State, so I’ve been explaining fire ants, the odd phenomenon of everyone waving at you (in contrast to the “Seattle Freeze”), and other quirks of life in Texas. I’ve also been planning to write about self-reference, paradox, and Gödel’s Theorem, so I’m sticking to that plan—when I finish, we’re repotting my monstera plant, Fred.

What follows is a reflection on paradox, perception, and the split-brain reality we increasingly inhabit—a strange but illuminating path from Gödel’s Incompleteness Theorem to fire ants, delusions, and digital life.

When Language Loops: Russell, Epimenides, and the Nature of Paradox

Iain McGilchrist devotes a chapter to logical paradox near the end of Volume I of his two-volume magnum opus, The Matter With Things. He doesn’t quite take up Gödel’s theorem, though he comes close—landing on Bertrand Russell’s famous “Barber Paradox” instead:

A barber exists in a town who shaves every man who does not shave himself.
The question: Does the barber shave himself?

Any answer leads to contradiction. If the barber shaves himself, then by definition he must not. If he doesn’t, then—again, by definition—he must. He shaves himself and he doesn’t. The system folds in on itself.

A similar ancient version appears in the Liar’s Paradox, attributed to the Cretan Epimenides: “All Cretans are liars.” Since Epimenides was himself a Cretan, we’re caught—if he’s telling the truth, he must be lying. If he’s lying, maybe he’s telling the truth. Again, the logic loops endlessly.

Gödel’s Leap: Truth Beyond Proof

Self-reference might sound like a philosophical parlor trick, but it’s one of the most powerful ideas in human thought. When a system—like language or mathematics—refers to itself, something profound happens. It reveals not only new insights, but also inherent limits. The brilliant logician Kurt Gödel used a form of the Liar’s Paradox to prove, in his First Incompleteness Theorem, that in any formal system rich enough to express basic arithmetic, there are true statements that cannot be proven within that system. In short: the system cannot see outside itself.

More specifically, Gödel showed that any consistent, formal system with the expressive power of first-order logic—including universal and existential quantifiers—and capable of encoding basic arithmetic (like the successor function and a subset of the Peano axioms), must contain true but unprovable statements. “Proof,” in other words, is not the same as “truth.”

What Gödel demonstrated mathematically, McGilchrist tracks neurologically and existentially: when a system—axiomatic, cognitive, or cultural—can no longer transcend its internal logic, it becomes trapped.

It’s interesting that McGilchrist stops short of discussing Gödel’s theorem, because his interest in logical paradox traces back—as we might expect, given his thesis—to the inability of the left hemisphere to see outside constructed reality. In a formal system, the "constructed" reality is the set of rules and axioms that permit conclusions to be drawn mechanically. McGilchrist poses the problem as one of cognitive limitation, and he introduces Bradley’s Paradox of Relations as a case in point:

If A and B are two terms that are connected, they are necessarily connected by some relation C. Equally, for C to act as a connection, it must differ in some respect from both A and B. Yet C can only relate A and B if there is some connection between A and C and between B and C; so there must be new relations D and E to explain these connections—and so on ad infinitum.

McGilchrist sees in Bradley’s regress a demonstration that continuity can be constructed, but not created, without some sort of leap.

Left Hemisphere Logic, Right Hemisphere Insight

Ah, yes, a leap. A leap of the mind.