abstraction

Abstraction

Abstraction turns particulars into categories; ignores distinctions that don’t matter for a specific purpose.

$$f:X\to Y,\text{where}|Y|<|X|$$

The pre-image $f^{-1}(y)$ is what’s treated as the same.

All categories are an imperfect depiction of reality. The truth always lies in the concrete.

Create abstractions from the concrete, to the degree they are useful for the task at hand.

Link to original

Examples & Different terms

Raw observations → categories: abstraction/categorization
Microstates → macrostatess: coarse-graining
Tokens → Type: typing/classifying

Abstract vs general vs category hierarchy vs abstraction layer

Concrete → Abstract: Which? → What kind?
Specific → General: Narrow kind→ Broad kind

Specific → General operates within the space of categories.
Intension … defining properties of a category (what it is)
Extension … the set of things that fall under that category
Shrinking the intension → expanding the extension (and vice versa)

	Specific	General
Abstract	Int	Num
Concrete	42 :: Int a ⇒ a	42 :: Num a ⇒ a

Category hierarchy: A is-a B is-a C … requires only shared properties (e.g.: tabby $\subset$ cat $\subset$ mammal)
Abstraction layer: hides a mechanism, exposes an interface … requires near-decomposability (e.g.: the structure of reality)

that-s-not-an-abstraction-that-s-just-a-layer-of-indirection

Do perfect abstractions exist?

I think there are perfect abstractions when considering a specific problem from a specific perspective. Like those that allow you to actually solve the problem (most) efficiently/robustly/whatever you have in mind. So in that sense abstractions/representations are highly subjective.

And from a philosophical perspective, I don’t think that there are ideal forms or essences of things that are ontologically prior (“treating the map as the territory”); inhabitants of a separate ideal realm. Ideas are historically produced abstractions and a result of material interactions.

That’s why I find it hard to grapple with some aspects of what levin argues.
Specifically that he argues that these patterns “ingress” from some realm of ideas, independently causing the physical. Yes, there are objective, lawlike structures (symmetries, attractors, …) and we can and should map and experimentally verify those. The “necessitation” we feel from mathematics is a feature of how well our abstractions capture the immanent constraints of matter in motion. Like if a system has certain material constraints, then any realization of those constraints will exhibit the same pattern, which we can then abstract. Saying the math causes the physical world has it backwards, it’s just compressing what those constraints entail (that’s also why it’s obviously important to anchor in empirical reality; else you study things not as they are, but as they ought to be … perfectly circular orbits).

Minds, embryos, and algorithms realize and exploit these (and we should too – I think that’s his main point)
Explanations run upward (composition) and downward (constraint) (picture) but all causation is materially implemented.
Like, when levin says abstract properties from maths / geometry are true no matter the laws of physics … well, but we pick the axioms and rules of our mathematical language based on what helps us model the physical world.