As a standard disclaimer, I am not a mathematician, and you should take these as observations rather than arguments. (For example, the amount of time I would have to invest to understand even this discussion is large.) But these observations add up to an assertion that, rather than numbers and there being an abstract relationship between two separate domains, numbers are a type of geometrical entity, though they are arbitrary and do not exist separate from our nervous systems, much like pain or non-spectral colors like pink. As in that argument, again observations from neurology are central.
In Gerstmann syndrome, patients who have had a dominant (usually left) parietal lobe injury, often from a stroke, lose the ability both to distinguish left and right, and to do arithmetic. (They also lose the ability to distinguish individual fingers, suggesting how important they are for counting. Children born blind often begin counting on their fingers without being taught.)
Related, I've long been mystified why in OCD, among the more abstract obsessions patients develop, are not just symmetry, but counting, and if you have one, you're more likely to have the other than random chance would predict. In terms of evolutionary psychology, many psychiatric illnesses are easily understood as hyperactive subsystems that in their normal setting and function, would be quite adaptive. In paleolithic Africa, you need anxiety to survive, you should be afraid of heights and snakes - and you should be able to frequently and easily orient yourself. Consequently, symmetry being part of OCD makes sense. But why counting, or especially fixation or avoidance with certain numbers? Some people insist on doing things in multiples of 3 or 5, or must avoid certain numbers (7 is frequent, possibly because it's the highest single digit prime?) to the point that it impacts their life, despite knowing how irrational it is - a smart-aleck roommate calls them to say they turned their TV volume to 7 and this is so uncomfortable to the OCD patient that she has to drive home from work to adjust it even at the risk of getting fired. It's hard to explain why the number 5 (or any number) would be good, or 7 would be bad, in the Pleistocene - unless it's actually the symmetry system that's hyperactive here.
Unsurprisingly, the same brain region (the inferior parietal sulcus, Brodmann area 7) is involved in both symmetry and counting. Therefore, that a lesion to this area would result in deficits in both these abilities is completely predictable; but why would the brain have evolved so the same area was serving both purposes? As before, even proving that numbers are a geometric entity in their representations in our brains does not give us a generalizeable argument about what numbers are, and it's possible that even if this is true for humans, it's only provincially true for the way our nervous systems work - butwhen AI systems develop the same way without being directly instructed, it does increase our confidence that our nervous systems and the AIs are converging in the way they represent something because that's really what they objectively are. On that note, it was discovered that what one AI was actually doing, when it is doing arithmetic, was spontaneously turning a modeled shape and counting vertices. It "evolved" this system on its own without being instructed.
Musings on Reported Cost of Compute (Oct 2025)
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