We Make Numbers From Shapes: Hints from Neurology

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.) It's also notable that almost all blind mathematicians are geometers.

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. It's obviously suggestive that when people with number-avoidance describe why they don't like certain numbers, they often describe them as "spiny" or "lopsided" or in other spatial terms.

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 - but when 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.

Most "Emergent Properties" Are Either Ignorance (or, Once We Understand Them, Merely Shorthand)

Searle's famous example of an emergent property is that a single water molecule is not wet. It's only when there are a lot of them together under certain conditions that they occur in something meaningfully called a liquid.

More generally in this view: emergence occurs when the interaction of multiple entities (often ones outside the direct perceptual abilities of humans, like water molecules), produces a behavior qualitatively different from the single entities, which can be perceived directly (like water.)

This concept has been rightly rejected but it's worth being clear about exactly why it should be rejected, in order to make a general argument against the idea.

Viewed in terms of predicting the behavior at the more complex (usually directly perceived macro) level, from the simpler entities, emergence is only, and always, ignorance. By "ignorance" I mean "an inability to predict that is based on limited knowledge of the observer, rather than a property of the entity that is lacking when the entity exists in isolation, but apparent when the entity interacts with other entities." Water's behavior in aggregate as a liquid is determined by the masses and electrochemical properties of its atoms It is determined by the water molecules, inherent to their properties. If the water molecules are replaced with ammonia or methane, the properties of those atoms and their relationship in ammonia or methane molecules, create a different sort of liquid, with different properties.

Of course, we CAN now predict fairly well from the intermolecular forces of water molecules at various temperatures and pressures (or of multiple molecules) where they will be liquids, and how those liquids will behave. Now that we can predict it, does this mean it's no longer emergent? And for those places (still in the large majority) in chemistry where we cannot yet make the prediction, does that mean reality is fundamentally irrational with no causal association between the simple entity and the aggregate, or we just aren't smart enough to see it yet? Therefore, the superficial countenance of an "emergent" property has nothing to do with the entities themselves, it's just the result of our own provincial limits on cognition that keep us from predicting how they will behave together.

Viewed in reverse (trying to apply the property we think has emerged at the higher complex level to the simple entities), it's clear that "wet" is shorthand for aggregate interactions of myriad small entities. In principle, you could understand (read: predict) the bucket of water at the level of individual molecules, but we use a (in this case, very good) approximation for their behavior which we describe as "wet". This is based entirely on the provincial bounds on human perception, cognition, and the tools we have to observe water molecules. In fact it is the cases where evolution has given us very good cognitive and perceptual approximations that the appearance of emergence is most compelling. But leave the realm of entities and events that our ancestors needed good cognitive/perceptual shorthand for, and the idea starts getting less interesting. I haven't heard anyone saying that quantum entanglement, or Bose-Einstein condensates at near-absolute-zero temperatures are examples of emergent properties, even though they weren't immediately predictable - though they fulfill the conditions for "emergency" seen above, both the simple and complex entities are outside of our meter-second-kilogram realm of experience, and there's no cognitive/perceptual shorthand for them. In the same way, we could talk about a building a Gothic or art deco instead of describing the spatial relationships of every brick (that is to say - a brick is not Gothic or art deco but this is equally "emergent" as wetness from water.)

A charitable interpretation of the traditional idea of emergence then does exist: we directly perceive certain properties like wetness at the macro level in the manifest world. We can actually predict the occurrence of this directly-perceived property based on what we observe at a simpler, smaller level. But the provincial limitations of our eyes and brains do not constitute a dividing line between properties where the universe carves itself at its joints.

This paper by Darley refers to work on cellular automata (of course) and argues that it is not Turing-decidible whether an infinite system would demonstrate emergence, and is "only" NP-hard whether a finite system is. I will go out on a limb and say this is at least in practice a reductio ad absurdum for such a property, and shows at least that ignorance can (formally!) never be ruled out in cases of apparent emergence of this type. What's more, the "emergent" phenomenon (emergent by this definition) is often not really that interesting - for example, the behavior of a three-body system. Does unpredictability really put it on another "level" of understanding? Actually, the opposite - in such a case it is only the individual elements that seem to form a unified entity. If each body is a water molecule, then there is no analogous "wetness" in the holistic full three bodies.

Darley, V. Emergence Phenomena and Complexity.