That the laws of physics are computable seems to be an article of faith among some. If the laws of physics are computable, then the brain (and consciousness) would be computable unless we are willing to entertain supernatural exemptions from the laws. The computability of reality, however, is actually a conjecture. It certainly can’t be proven because we can never be sure we might come up with physical phenomena that could never be computed.
We have good reason to suspect that the conjecture is false. There is the fermion problem in the Standard Model that I’ve discussed previously. There are also the problem of computation with many-body problems in condensed matter physics. A particular case of the many-body problem called the “spectral gap problem” seems to demonstrate that at least one physical phenomena cannot be computed.
Interestingly, recent work in condensed matter quantum physics indicates that—possibly—quantum many-body systems could infringe the Total thesis. In 2012, Eisert, Müller and Gogolin established the surprising result that
the very natural physical problem of determining whether certain outcome sequences cannot occur in repeated quantum measurements is undecidable, even though the same problem for classical measurements is readily decidable. (Eisert, Müller & Gogolin 2012: 260501.1)
This was a curtain-raiser to a series of dramatic results about the uncomputability of quantum phase transitions, by Cubitt and his group (Cubitt, Perez-Garcia, & Wolf 2015; Bausch, Cubitt, Lucia, & Perez-Garcia 2020; Bausch, Cubitt, & Watson 2021). These results concern the “spectral gap”, an important determinant of the properties of a substance. A quantum many-body system is said to be “gapped” if the system has a well-defined next least energy-level above the system’s ground energy-level, and is said to be “gapless” otherwise (i.e., if the energy spectrum is continuous). The “spectral gap problem” is the problem of determining whether a given many-body system is gapped or gapless.
The uncomputability results of Cubitt et al. stem from their discovery that the halting problem can be encoded in the spectral gap problem. Deciding whether a model system of the type they have studied is gapped or gapless, given a description of the local interactions, is “at least as hard as solving the Halting Problem”
https://plato.stanford.edu/entries/church-turing/
There is a great account of the development of the Cubitt et al proof written by the researchers themselves in a Scientific American article called the “The Unsolvable Problem.”
If the laws of physics are not completely computable, then the question of whether the brain is computable becomes an empirical question.
Is there empirical evidence that suggests activity in the brain that is not Turing computable?
I think the answer is yes.
Cognition seems to be accompanied by synchronous firings of groups of neurons sometime in distant parts of the brain. There is evidence some of this is generated from a form of communication that is not mediated by chemicals or physical connections and goes under the general term of “ephaptic coupling.”
In the present study, we show that slow periodic activity in the longitudinal hippocampal slice is a self-regenerating wave which can propagate with and without chemical or electrical synaptic transmission at the same speeds. We also show that applying local extracellular electric fields can modulate or even block the propagation of this wave in both in silico and in vitro models. Our results support the notion that ephaptic coupling plays a significant role in the propagation of the slow hippocampal periodic activity. Moreover, these results indicate that a neural network can give rise to sustained self-propagating waves by ephaptic coupling, suggesting a novel propagation mechanism for neural activity under normal physiological conditions.
https://pubmed.ncbi.nlm.nih.gov/30295923/
Travelling waves propagate in different directions during separate cognitive processes. In episodic memory, travelling waves tended to propagate in a posterior-to-anterior direction during successful memory encoding and in an anterior-to-posterior direction during recall. Because travelling waves of oscillations correspond to local neuronal spiking, these patterns indicate that rhythmic pulses of activity move across the brain in different directions for separate behaviors.
https://www.nature.com/articles/s41562-024-01838-3#:~:text=Travelling%20waves%20propagate%20in%20different,to%2Dposterior%20direction%20during%20recall.
The traveling waves – think of a stadium wave – have an uncanny resemblance to turbulence. They began to appear in the conscious brain on waking and mostly vanish during sleep and unconsciousness.
Furthermore, we build a whole-brain model with coupled oscillators to demonstrate that the best fit to the data corresponds to a region of maximally developed turbulent-like dynamics, which also corresponds to maximal sensitivity to the processing of external stimulations (information capability). The model shows the economy of anatomy by following the exponential distance rule of anatomical connections as a cost-of-wiring principle. This establishes a firm link between turbulent-like brain activity and optimal brain function.
https://pubmed.ncbi.nlm.nih.gov/33296654/
A complete description of turbulence is one of the unsolved problems in physics.
The brain consists of connections of neurons called the connectome. The number of neurons, of course, varies by species. The C. elegans brain has a few more than a hundred. The human brain has around 85 million in total with about 16 billion in the cortex. Undoubtedly communication of information in either brain is largely through the connectome. The connectome might explain completely the operation of the C. elegans brain. In that sense, its brain might be computable. The human brain on the other hand seems to have supra-connectome properties. Turbulent, wave-like, and vortical activity arise as emergent properties as a function of the complexity, size, and structure of the connectome. This activity has causal force since it produces real neural firings that might not be predictable from the connectome itself.
A supra-connectome might be the evolutionary solution for communicating tightly coupled data across a fragmented and asynchronous brain. By tightly coupled data, I mean data that couldn’t be broken into chunks without losing meaning. For example, this paragraph could be broken into words but, if the words arrive in pieces and at different times, it might be impossible to reassemble the paragraph and understand its meaning. Turbulent activity in the brain may have arisen evolutionarily as a side effect of size and been detrimental. Rather than eliminating it, however, evolution might have learn to control it through a critical balance between excitatory and inhibitory pressures and to use it as an information transmission mechanism over and above the connectome itself.
Broad Speculations 