Abstraction is for the weak

The idea of abstraction as a formalism largely emerged from computer science, though it has been the driver of human progress for a long time. It is the process of hiding the complexity of a system, often with the goal of building more complex systems on the shoulders of the old ones.

This is a powerful concept, and has enabled the human race to build things of remarkable complexity. One only needs to look at the features of microprocessor, billions of layered bumps carved into silicon with light and chemicals to see evidence of this.


No one mentions, though, the reasons for the “invention” of abstraction. Abstraction would not be necessary if humans could mentally store and manipulate systems of arbitrary complexity in the first place. At some point, thinking about complex emergent behavior in terms of lower level agents (i.e., thinking about a processor in terms of transistors) becomes infeasible as the capacity of the typical human’s working memory is surpassed. Abstraction is, fundamentally, a crutch.

Sometimes, the low-level agents (i.e., transistors) in a particular discipline naturally combine into a small set of convenient abstractions (i.e. logic gates). Other times, there is no natural “next step” in the process of abstraction, such as in modeling the macroeconomy or nonlinear fluid flows. In these cases, the burden of modeling the full complexity of multiple interacting agents falls to the practitioners of the field. This sort of thinking is difficult for the inherently linear, “serial” processes of formal thought. In these cases, a deep understanding of the system, which takes a relatively large amount of time to develop, is called intuition. Intuition is the best we can do as humans to understand a complex system that isn’t conducive to abstraction, especially when its complexity exceeds our working memory and the computing power of rational thought. In these cases, the brain seems to repurpose the non-serial (spatiotemporal, parallel) thought processes at our disposal, jury-rigged to understand the complex system before us.

However, the example primarily used above (digital logic and transistors) does not accurately represent one property of abstraction: it is typically lossy. In this case, we are using “lossy” to convey a loss of information, robustness, or scope when a new layer of abstraction is formed. A perfect example of this is Snell’s Law. Snell’s Law is taught as an absolute scientific fact in primary schools, though in reality is an abstraction invented to make solving a particular problem (finding the refraction angle of light entering a new medium) extremely simple. This “law” only governs the behavior of non-polarized light as it passes from one non-dissipative medium through a perfect planar boundary into another medium. This is an incredibly limited set of agents operating under specific conditions in a particular topology. In fact, Snell’s Law can be derived from Maxwell’s equations, a far more robust generalization of electromagnetic phenomena. Yet this is scarcely obvious Yet it gives us a way to powerfully model light and it’s interactions with materials.

This idea of lossy abstraction gives rise to some interesting and disturbing ideas. What if there are useful or important abstractions that lurk unnoticed, too complex to fit in working memory? To extend the digital logic analogy, imagine that logic gates turned out to be extremely complex circuits requiring dozens of transistors in a very peculiar topology. The revolution in VLSI computing may have never happened. In fact silicon may never have been considered as a substrate for computation.

Why are some problems made easy through abstraction, while others, such as modeling dynamical systems or designing artificially intelligent machines, stay so hard? Maybe we’re just lacking the right abstractions.


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