Algebraic Cognition?

July 11, 2021

Whenever you solve an equation, you never actually do anything. There is a set of manipulations available (+, *, -, /, etc) and you use them to isolate the variable. To do this, you are "massaging" the equation, continuously cycling representations of the same relation (doing the same thing to both sides).  So an equation doesn’t have an answer, per se. You just eventually find a representation where the quantity you care about is "on top".  The relations  x=4 and (4x-12)/4 = 1 mean the same thing, but the former is more perspicuous. 

However, the quantity of interest (what x equals) is arbitrary - it’s dictated by the mathematician. Why are we satisfied with  x=4, but not (4x-12)/4 = 1? The algebra doesn't really care, they encode the same relation.


As humans, we look for the representation that is most "cognitively composable". We wish to make an inference and  then fold that inference  into a more complicated mental model. In this case, it’s most likely better to incorporate the representation x=4 than (4x-12)/4 = 1 into the next layer of thought. 

x=4 is not always the best representation, x-.25=0 is better in some situations. The choice completely depends on the target domain - the place where the reduced representation is meaningful.

This is important because it lets you easily compose insights from completely different domains.  It lets you abstract the details of the previous relation by destroying the structure from which the insight came from. You most likely don't care how x=4 was decided, you just care that x=4.


The goal of philosophy is the same. We seek to see  the underlying , abstract structure of a relation and feed it forward to more complicated models. In the context of, say, a political debate, a trained mind will do some "algebraic pre-processing" on a question. She will manipulate it in her mind to extract its essence, an equivalent representation that is more tractable (more composable into larger mental construction). She doesn't "filter noise", she folds the noise back into the question, in the same way an algebraist would fold superfluous features back into the equation. The mind never computes new information, it discovers essential structure.  The existence of a situation may hinge on a small set of facts. If you can find  those facts, you are representing a topologically equivalent version which may be useful.


This is a critical to remember, because the "right answer" has just as much to do with your goal. When you do algebra, you a priori commit to what it means for representations to be equivalent. I took for granted that we all agree that x=4 and (4x-12)/4 = 1 are equivalent, but are they? Can you think of other equivalence relations that makes sense? If asked why those two equations are the same, a 6 year old may say  "they both have the number 4 in it". That’s a pretty reasonably inference if you don’t understand the context.

This may sound like an obvious or stupid insight, but it becomes a lot more subtle in more complex algebras. When discussing political solutions, our philosopher may see two representations as equivalent for her. To a white person, for example, America with Robert E. Lee statues may be equivalent to an America without, but that is clearly not the case. 

This is the fundamental danger to algebraic cognition. As mentioned, the goal of algebra is to find a representation where superfluous source structure is destroyed. But, if unquestioned, that representation may not compose as we think it will. If the safety of an airplane is contingent on the fact that x=4 but that fact was calculated under the 6 year old’s notion of equivalence (its correct if it contains 4), will that airplane fly as expected? What if a politician thinks that any solution protecting their donor's interest is the same?

My point is we need to discuss more processes as sorting and filtering operations on complex, abstract structures. They say “garbage in, garbage out”, but that only makes sense when you believe an algorithm is detached from the data it operates on, which is not true. Focus on the substrate of inference, the ontology which you work on, all the answers are already in there.