The Sentence Outside the System

Written By Arthur Knott

Thinker slot — The Penumbra, Week Four.

A sunflower head places each new floret about 137.5 degrees around from the one before. This is the golden angle, and it packs the most seeds into a disc with no gaps and no spokes. Nothing in the flower measures it. At the growing tip, a new floret forms where the hormone auxin has pooled, and as it grows it draws auxin away from its neighbours, so the next floret can only form in the widest gap left over. Repeat that local rule a few hundred times, across a wide range of growth conditions, and the spacing settles close to an irrational number.

David Deutsch has a plain description of what the flower is doing. Computation, in his account, is a physical process in which one object is made to mimic an abstract one, a number or a relation, so that its behaviour tracks the abstract thing's properties. A calculator is such an object. So is a brain. By that description the growing tip of a sunflower qualifies, though crudely: it tracks an angle, and nothing in it holds the angle as an idea.

The question this week is whether you are the same kind of object with more steps. Every other process neuroscience studies in the brain, from steadying an image to steering a hand to recording a route through a city, is modelled as computation, and the models work well enough to build on. The open question is whether consciousness, and understanding in particular, belongs on the same list. The most serious modern argument that it does not belongs to Roger Penrose. Its weakest point should come first.

Penrose's case, which he developed from the Oxford philosopher J. R. Lucas's 1961 paper, runs through Kurt Gödel. Our short this week sets out Gödel's 1931 result: any consistent formal system strong enough for arithmetic contains a sentence it cannot prove, a sentence that in effect says "I cannot be proved here." "Consistent" means the system never proves both a statement and its opposite. The argument needs one further premise, and it is a large one. It needs us to know that our own reasoning is consistent. The logician Solomon Feferman, the philosopher Hilary Putnam in his 1994 review of Penrose's second book, and most logicians since have said that we cannot know this. Mathematicians have believed contradictions before. In 1902 Bertrand Russell wrote to Gottlob Frege to point out that the system Frege had spent years building contained one, and Frege had not seen it. Without the premise, the argument shows only this: if human reasoning is a consistent formal system, we cannot prove that it is. That is what Gödel's second theorem already says about every such system, and it fits the computational view perfectly well. Alan Turing saw the objection coming in 1950. He granted that any particular machine has limits and noted that no one had shown the human mind lacks them.

Now Penrose, heard against all of that. He was born in August 1931, the year Gödel's paper appeared, and he shared the 2020 Nobel Prize in Physics for showing that black holes are a robust prediction of general relativity. In The Emperor's New Mind (1989) he took aim at what was then called strong AI, the claim that running the right program simply is thinking. His argument goes like this. A mathematician who follows Gödel's construction comes to see that the unprovable sentence is true. Penrose calls that seeing understanding. No algorithm can be the whole of it, because any algorithm would have its own Gödel sentence that it could not reach. Therefore something in the brain is non-computable. Report 3 in this issue separates problems that are merely slow from problems no procedure can answer at all; Penrose's claim is about the second kind. He did not stop at logic. In Shadows of the Mind (1994), and with the anaesthesiologist Stuart Hameroff, he proposed that the non-computable ingredient comes from an unknown physics of quantum collapse operating inside structures in brain cells called microtubules. That second claim is separate from the first and far more speculative. The strongest objection to it is physical: the physicist Max Tegmark calculated in 2000 that quantum states in a warm, wet brain decay far too fast to do any work.

Deutsch supplies the sharpest reply to the logical claim, and it rests on something easy to overlook. A proof is itself a computation. The mathematician who comes to see that the Gödel sentence is true is a physical brain going through physical steps, and those steps are only as reliable as the physics running them. Proof theory tells you what follows from what. It says nothing about which truths can be known in the real world; that depends on what physical objects can be made to do. So the mathematician who steps outside a formal system has stepped into a larger one, and the larger one has its own unprovable sentence. The short ends by telling you to change the question and build a larger frame. The computational view predicts that this is exactly what minds do, and that every frame they build is open to the same construction. Penrose needs a last frame with nothing outside it, and no one has shown one.

The computational view has not won everything either. It explains what brains do. It has not explained why any of it is felt, and Penrose's argument, even where it fails, marks precisely where that burden sits. This week's news shows the layering in practice. A machine search produced a Navier–Stokes blowup, another machine checked the proof in Lean, and human mathematicians are still deciding whether the formal statement matches the problem the Clay Institute posed. Each check stands one frame outside the last.

Go back to the sunflower. Its growing tip tracks the golden angle without holding it as an idea, and the mathematician holds it as an idea. Penrose says that difference is one of kind. The computational view says it is a difference in how many frames a system can build and then inspect from outside. The evidence so far favours the second without settling it, and it suggests that what Gödel drew was never a wall around the human mind but the outline every frame has, the human one included. For the full exchange, search the Lucas–Penrose argument; Feferman's reply is the place to start.

source list:
Arthur Knott
Next

The Spaghetti Vortex