Pointers as generalizations of talk and pin pricks

The science of development, whether motor development, cognitive development, or morphogenesis, has often sought to know where information is stored. For example, walking behavior has been attributed to a mental walking program that somehow contains the walking behavior within it, and sometimes imposes that behavior on the rest of the body. But we know from personal experience that outcomes can be reliably shaped without information being stored anywhere. For example, if you’re eating a meal with someone and ask them to pass you the salt, you can reliably expect that they will do so even though the words “Can you pass the salt?” are just noises or symbols arranged in an order and carry no inherent meaning about salt being handed from one person to another. The words really function to point to a class of outcomes, relying on the hearer of the words to make meaning of them and arrange the outcome accordingly. We could say that the words function as a pointer that helps the physical system discover the intended pattern or relational web.

Pointers are conceptually related to the idea of a control variable in dynamic systems theory. Sometimes, adjusting a variable leads to dramatic reorganization of a system, such as the emergence of novel behaviors or novel forms, such as how adjusting temperature can lead to boiling or freezing water. It is tempting to attribute the information of the reorganization to the control variable, but this is not so. For example, if you poke an infant with a pin hard enough, they will cry. However, the pin obviously does not carry with it any information about crying. The infant constructs the crying even though the force of the pin prick is the control variable. As with words, the pin prick prompts the system to reach a particular state without conveying anything about that state to the system.

Pointers are an interesting because they help to explain how developing systems get things for free. Developing systems rely on the “free lunches” provided by physics. For example, humans are able to walk by taking advantage of the spring-like, pendulum-like properties of the legs, allowing them to walk without instructions for walking being written down somewhere. Morphogenesis also uses physical properties that are not written down in the genes, for example, the exploitation of gravity and pressure to reach and maintain a form.

Developing systems also get free lunches from mathematics, for example being able to produce geometric arrangements that are algebraically consistent because of the isomorphism between geometry and algebra. Somehow, certain patterns are exploited or instantiated without them ever being written down. The standard definition of the Cartesian product, for example, does not actually define the Cartesian product but simply points to the correct pattern, relying on the mind of the reader of the definition to infer what is intended. The actually pattern of the Cartesian product cannot be pinned down to a specific object or even a specific web of relationships but only up to unique isomorphism between one web of relationships and another.

From a dynamic systems view, such relational templates might serve as attractor states that systems tend to seek out. Such templates are efficient in a clear mathematical sense, but it is not obvious to me why mathematical systems would prefer efficient states to inefficient ones. Unlike physical systems, they don’t need to worry about dying of starvation or old age. Math and economics may be the same thing, in which case it should seem intuitive that math should prefer efficiency, but I don’t understand why or how. For example, you can do a stupid version of the real numbers that includes an extraneous element that does nothing. This would be bad for humans because it means extra work for them, but does math care? Apparently, but I don’t know why. Consider also the case of mathematicians choosing for 1 not to be a prime number because it would be very annoying if it was a prime number. Maybe there are deeper ideas here, I don’t know.

I’ve never seen a mathematical form that isn’t a pointer: equation, graph, and proof are all pointers. Because of the relationship between concepts in dynamic systems theory and the pointers concept, it’s possible that dynamic systems theory does an adequate job of explaining the emergence, maintenance, and transformation of mathematical forms.