Thanks for the comments about the neurosciences! As I said I don’t know that much about those, so it’s always fun to learn I would be interested if you could give any examples of neurological structures laid in a matrix (I did an online search to try to find those, but it failed, I suspect due to a lack of knowledge of the relevant words in the area).

Regarding the neurosciences, I would propose several answers:

If you consider that something must remain fixed so as to be a lookup table, then you’ll hardly find anything in our brains that fits this description.

If what matters is a structuration laid in a n-dimensional matrix, then many neurological structures are known to be more or less structured this way -it’s a general rule, not an exception. The combination of the activities of these different maps is however unlikely trivial, to say the least.

Finally, one may wish to consider that lookup tables are a special case of a deeper principle: save processing time at the expense of memory space. If this idea belongs to the Platonic heaven, then our brain is its incarnation.

In the non brute-force case, we could have a system such that when n = 1, no activity is detected other than a copy of m to the output. Otherwise, the part of the system that holds the answer, holds first m, then the answer to 2*m , then the answer to 3*m, and so on, until it finally holds the answer n*m. One could generalize this to other problems in terms of the system holding during its computation answers to partial subproblems which combine to give the final answer in a nontrivial way (I add the last requirement to try to avoid the case where the system could be seen as having different brute force tables whose answers are easily combined, in the multiplication case this could be different brute force tables for different digits of the answer).

In the brute force case, we could be registering activity in a different spatial region of the system for each possible input, and after that a copy of the state of that region of the system to the part of the system holding the final answer. In the multiplication case, this would be even clearer if the system was laid in a 2 dimensional way, and as we varied the m/n input, the region of the system that is activated would move along the x/y axis.

One possible path toward formalizing these issues would be to require (as part of the Turing Test) a witness extraction procedure that would require of any sentient algorithm the feasible extraction of a witness, with said witness amounting (in essence) to a short explanation if “How I work”.

The intuition being, that thereby any “brute force” Turing Machine would be obligated — as an required ingredient of sentience — to provide (in essence) a short algorithm for emulating itself … whereupon the “brute force” algorithm renders itself nugatory.

In human terms, the required process of witness extraction would be, very broadly speaking, the TM equivalent of a psychoanalysis — which indeed is a quintessentially human activity.

In accord with ordinary human understanding, we would scarcely call an entity that could not concisely rationalize its actions “sentient” in the common sense of the word.