A really good article. Thanks.
That said, what is computational cognitive neuroscience? By definition, I believe, it asserts the truth being sought is a model that mimics. Those of us who believe (have unsupported faith) that there is a "universal cortical algorithm" believe we might have one in our brains, but deduction is not the goal.
This is not Newell and Simon's General Problem Solver. Darwin's science, if you recall, was just a collection of self-evident truths that hung together with a story. A model. Yes, as you put it perfectly, a theory.
So, I would argue that symmetries which service physics very well, do not necessarily service other sciences. I would argue science is the art of careful observation, and sometimes it begins and ends with observation and proof of a model that simulates the observations (the self-evident truths under study).
Eventually, a deeper mechanical understanding, like DNA discovered after Darwin was long dead, might create a richer model. But it is still a model which is the scientific result and describes the laws governing the self-evident observations -- both the old self-evident truths and the new ones we measure. One more comprehensive model follows another.
I've argued that cognitive neuroscience fails to notice the self-evident universal truths of human natural language, and that these reveal a lot about neocortical computation. Because, they exist.
Compared to physics, this is pre-science science because it simply attempts to mimic and not deduce based on accounting only for the self-evident truths. As Darwin originally did with his Origin and his pre-scientific, but nevertheless scientific, theory of evolution.
Put another way, there are those of us who believe the brain operates with innumerable parameters. We like explanatory spaces of very high dimensionality because we think that is what we observe in brain computation. And to try to reduce a scientific theory of neural cognitive computation needs all those parameters under potentially simple controls.
For a reason. Because that is what we observe directly. And yes, the objective is a model of exceptionally high dimensionality. The details of the computations may come many years after we develop such simplified models of what we can directly observe of the results that inform us. But we believe, based on what we have observed, that even once those details are known, the high dimensionality of the explanatory model will persist. It's not physics. But it is science nevertheless.
Read/audible Terry Sejnowski for a book on this argument by a fellow computational cognitive neuroscientist:
https://www.amazon.com/Deep-Learning-Revolution-audiobook/dp/B07MM8F42R/
But right this very minute. We still don't have a clue. We have some guesses in our models that seem like they move, non-symmetrically, but observationally, forward. The human brain fundamentally does compute in "=" (equal) signs, we think, but in that sense, trillions or more categorically different 'forced by learning and DNA' symmetries. Not a few. No matter what you physicists think we should be doing. We think our models are great science, and subtle deductive inference driving our quest is a long way off right now.
Here's a summary of our deductive math .. such as it actually is ... with trillions of variables...
