Thanks for your provocative and interesting post.
I believe if you and Penrose consider the nested structural coding on just the genetic and epi-genetic levels, and add in metabolic and protein-folding (expressive) features -- even if you don't dare venture into the stacks and sequences in the underlying 10^20 per second hydrogen bonding packets forming within distributed respiration sites' transactions -- you may notice that the ~system is shifting from one structurally coded resonance point to another, basically, adjusting in terms of energy -- conserving energy, hashing out balance within the nested structured~duality.
Also, the ~system is running these transactions in analog mode, naturally. The ~routines that succeed do so naturally, energetically.
With this backdrop, that is, conceptualizing reality as nested fields within nested fields, then oddities such as Pi and infinity, etc., are a lot more like special system resonance points rather than just special numbers on a single idealized number line. Similarly, primes, etc.
As for sums and strings...
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1. Only humans can do sums.
2. Sums use strings.
3. The genetic change that allowed us to do sums allowed our brains to use strings internally.
In contrast I think the correct conclusion is
3. The genetic change that allowed us to do sums allowed our brains to interface internal non-string computation to external string signalling. A mechanism was acquired that could convert temporal sequences into spatial patterns and out again.
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I wonder whether #1 thru #3a or 3b is correct. Though limited, animals appear to keep track of things and do a balance with lack of food/food. And, it seems the key difference is adding nested level of association which seems to mainly be a reverberating memory type of thing begining with a one-to-one alignment of artifacts of interest/energy and one's fingers and toes, later to marks in sand or on bark or skin and the associated protein-folding expressions. An increase in level of association can also be gained by inhibition or reduction of dissociation -- which again circles around to increased empathy relating perhaps just to an increase in mirror neurons.
I'm not tracking on why you suppose or suggest a "mechanism to convert temporal sequences into spatial patterns and out again". Can you clarify, please? It seems to me that habituating and then falling asleep to the constant heart-beating and in-out breathing (plus orbits and seasons), prompts assuming that the erroneous temporal relation exists. This flaw seems more like a natural initial approximation -- a precursor to discerning that reality is actually nested fields within nested fields.
I suspect all this may become easier to provisionally accept once there are some repeatable instances of multiple-state transportation kicking around. We can't walk or count our way to infinity; still it's just a matter of hitting a resonance point and shifting state.
Nested structural coding... genetics, epi-genetics, metabolism, protein-folding, reproduction, respiration... Think about it.