Abstractions and Generalizations I: Background and Thinking

This is the first part, the motivation and background for my own personal exploration of the abstraction an generalization. Please follow along and correct me when you think I am off my rockers.

In the essay titled: The Vessel and the Filter in The Creative Act: A Way of Being (Strauss 2023) a collection of essays by Rick Rubin on creativity. He noted that:

“Each of us has a container within. It is constantly being filled with data. It holds the sum total of our thoughts, feelings, dreams, and experiences in the world. Let’s call this the vessel.”

“Information does not enter the vessel directly, like rain filling into a barrel. It is filtered in a unique wat for each of us.”

“To navigate our way through this immense world of data, we learn early in life to focus on information that appears essential or of particular interest. And to tune out the rest.”

This quote seek to describe our unique way to filtering information, the boiling down of information to only the essentials or of particular interest.

I first became interested in the idea of using abstraction as a tool to stimulate thinking after I read Sparks of Genius: The 13 Thinking Tools of the World’s Most Creative People (Root-Bernstein 1999), Chapter 5 of the book is devoted to abstraction as a tool for the artist to use for concentrating the essential details so that we can create and innovate. There is an impressive list of people mentioned in the chapter as examples of those who used abstraction to create original works. These works span the broad spectrum of human endeavors, are extremely abstract, and broad in scope: art, physics, poetry, prose writing, and mathematics. The originality of these works are often beyond the understanding of those of us who are not as abstract in our outlooks; yet we are often emotionally moved by the work and we often wonder: how did they get to this level of abstraction?

Thus inspired, I dug into the subject of Abstraction, which is defined by Wikipedia as:

Abstraction is a process wherein general rules and concepts are derived from the usage and classification of specific examples, literal (real or concrete) signifiers, first principles, or other methods.

"An abstraction" is the outcome of this process—a concept that acts as a common noun for all subordinate concepts and connects any related concepts as a group, field, or category.

Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only those aspects which are relevant for a particular purpose. (Wikipedia n.d.)

One can loosely associate abstraction with induction, an inferential method to reach a generalized conclusion from specific experiences. Induction draws an inferential conclusion from the common factors existing in many examples; whereas abstraction collects separate and disparate samples from reality and distilling the common and essential details from the samples to redefine the collection of separate and disparate events into a simple concept.

Abstraction helps us simplify and prune all the unnecessary and inconsequential details not pertinent to the focus of the exercise so that the post abstraction concepts appear to be less encumbered, i.e., the act of abstraction turns detailed and complex reality into general and simple concepts.

Root-Bernstein asserts that abstracting is closely synonymous with intelligence itself because abstraction develops our talent for simplicity. One salient result of that assertion is that abstraction trains us to be more sensitive to reality; reality can be, according to Root-Bernstein, the sum of all the abstractions that we can imagine and that in knowing all the possible abstractions, we will understand the complete reality. This was a new and intriguing thought for me to think about.

Some quotes from Root-Bernstein in which they used to illustrate the concept of abstraction:

· See with your mind, not your eyes. Don’t just look—think! Which on a high level makes perfect sense but difficult to deploy if one had been habituated to be grounded in the material world.

· The simplest abstractions are hardest to perceive or devise yet also yield the most insight. This is an admonition to take the abstraction to the most fundamental level.

· The essence of writing is not putting words on the page but learning to recognize and erase the unnecessary ones.

Root-Bernstein’s descriptions of abstraction are seemingly ambiguous, ethereal, and are far away from reality, which as stated previously, is the exact point of abstraction; but this exercise in the amorphous and ethereal does not help our minds, rooted in reality and the concrete, determine how to use and apply the concept of abstraction in reality.

More specifically, in the Wikipedia definition, the statement in the last paragraph states: “Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only those aspects which are relevant for a particular purpose.” While Root-Bernstein’s definitions conjures up romantic notions of artists, art, and the mysterious process that artists undergo to be inspired by their muses, what is missing is the relevance factor for a particular purpose. This aspect of abstraction is central to my interpretation of how the concept of abstraction can serve my purposes.

While reexamining my interpretation of abstraction for my purposes, I thought about the above paragraph; while I am sure that the artists that Root-Bernstein cited as examples also had specific purposes when they used their abstracting abilities in their vision, I am also not sure that they deliberately made a specific purpose as the focus of their exercise in creativity. I am not an artist, so my interpretation of their use of abstraction is probably erroneous; my conjecture is that the purpose of their abstraction is more ambiguous and amorphous than what I had in mind.

Even though the Wikipedia definition for abstraction includes the phrase: a process wherein general rules and concepts are derived from the usage and classification of specific examples. I think of abstraction and generalization as complementary halves of a process of thinking which allows us to extend our ability to create concepts in our minds that are beyond the known and familiar. Right or wrong, my personal investigation places generalization as a companion idea to abstraction.

Wikipedia defines Generalization as:

A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims.

Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics, and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation.

Generalization can also be used to refer to the process of identifying the parts of a whole as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them.

However, the parts cannot be generalized into a whole — until a common relation is established among all parts. This does not mean that the parts are unrelated, only that no common relation has been established yet for the generalization. (Wikipedia n.d.)

In parallel with abstraction and induction, generalization can be associated with deduction. Deduction applies a general conclusion or truism to be true broadly without advantage of supporting empirical evidence; whereas generalization is the mental process of taking the simplified concept that we had obtained from making abstractions of the complex reality and try to apply it generally to reality, after having stripped away those things that are considered to be extraneous and are distractions, applying the abstracted concept for general purposes.

Root-Bernstein considers generalization to be a deeper level of abstraction, they refer to a Heisenberg quote to explain their thinking about generalization: “The step toward greater generality is always itself a step into abstraction —into the next level of abstraction; to unite the wealth of diverse individual things.”

Abstraction and generalization are mental processes that happen within our thought, processes that we cannot turn on or off, our cognitive abilities automatically abstract and generalize in order to process the information that our senses are continuously collecting. In many ways, humans are abstraction and generalization machines because our minds are having to, by necessity, reconcile the incessant flow of new information and unknown experiences with known information and existing experiences. We take the detailed, contextual, and complexity of each new situation, create analogies and make connections between the unknown and the known through abstraction; we create narratives to coalesce the unknown into the known. Abstraction gives us an internal sense of order which ties our cognitions together with common factors while filtering out the uncommon factors. Generalization resolves the abstractions to create reasonable conclusions so that we can come to “reasonable” predictions in an open-loop manner. This is an iterative process; the cycle of abstractions and generalizations adjusts and iterates as more information and experiences become known to us. Our ability to abstract and generalize is how we have been able to adapt to our circumstances, which allows us to survive for as long as we have.

Indeed, I had imagined a “conversation” amongst a group of our hunter-gatherer forebears abstracting and generalizing about which plants are edible and which are not, taking important note of the vegetations which a smaller group of them had eaten and had dropped dead as an important part of their abstraction to avoid eating those vegetations in the future.

I became interested in abstractions and generalizations because I realized that even though this is an intuitive and natural human process, I personally did not have any conscious idea about HOW I am abstracting nor HOW I am generalizing. It also occurred to me that I don’t know whether my exercises in abstracting and generalizing have any basis in reality nor whether I can trust my generalizations since my abstraction and generalization exercises are completely based on my previous experiences, my intuition, and my perception of reality, which is also besotted with my own logical fallacies and cognitive biases. Referencing the recent tomes on the our faulty and self-deluding cognitive nature: Annie Duke’s Thinking In Bets (Duke, Thinking in Bets 2018), Emre Soyer and Robin Hogarth’s The Myth of Experience (Emre Soyer 2020), David Epstein’s Range (Epstein 2019), and Daniel Kahneman’s defining book on our proclivity to believing in the wrong things: Thinking: Fast and Slow (Kahneman 2013); I have become convinced that our ability to accurately abstract and generalize is severely compromised by our flawed ability to discern facts from fiction due to our very dodgy beliefs, beliefs that we anchor on for our decision making.

We rarely, if ever, work to verify the veracity of the information and experiences that we use as basis for our abstraction and generalization. If the information and experiences that we count on as the basis of our decision making are so adversely affected by our logical fallacies and cognitive biases, then the basis of our ability to abstract and to generalize is similarly compromised by those logical fallacies and cognitive biases. If our experiences exist in such a state of acatalepsy, how is it that we have survived through history using our ability to abstract and generalize? (https://thecuriouspolymath.substack.com/p/ruminations-acatalepsy)

One answer is in how our generalizations are endlessly iterated upon and refined, gaining new experience and iteratively eliminating the unnecessary elements in our abstractions to sharpen our perceptions to better reflect reality, which in turn improves our ability to generalize. Unfortunately, there can never be enough information, knowledge, and experiences to tune the abstraction and generalization cycle as reality is continuously varying.

One of my friends often responds to some people’s broad conclusions with: “you are generalizing!” My smartass response is usually: “of course, that is one of the things that we humans do best!” My kinder self should remind him that it isn’t the generalizing that is the problem, it is the overgeneralizing, which is also normal for humans. The real problem is that our decision making mechanism is operating in open loops, we rarely have guardrail on our generalization machine, we don’t follow any sort of reasoning or logic which acts as scaffolding for our abstraction or generalizing, we just do it. Nothing exists to keep our thoughts from over abstracting or over generalize. We are dependent on our memories of information and experiences to guide us, without knowing whether the information and experiences are applicable in that context and that situation. We subconsciously use Kahneman and Tversky’s System 1, the unconscious and procedural thinking mode to abstract and generalize; we depend on Kahneman and Tversky’s System 2 mechanism, our conscious and conceptual thinking mode to discern whether the information and experiences are applicable in that context and that situation. As Kahneman wrote, System 2 is both conscious and lazy, so the probability of our very human mind consciously verifying the veracity of our memories is quite small.

Our lazy System 2 will more often than not, defer to Occam’s Razor (The Law of Parsimony) to such extremes that we overgeneralize and oversimplify to attain the simplest possible conclusion as a matter of following the mythology.

This philosophical razor advocates that when presented with competing hypotheses about the same prediction and both theories have equal explanatory power one should prefer the hypothesis that requires the fewest assumptions and that this is not meant to be a way of choosing between hypotheses that make different predictions. Similarly, in science, Occam's razor is used as an abductive heuristic in the development of theoretical models rather than as a rigorous arbiter between candidate models. (Wikipedia n.d.)

I started to think about how bad we are at using our natural abilities to abstract and generalize, the predicaments that we force ourselves into because of our inability to control the level of our abstraction and generalization, especially since our lazy means of abstracting and generalizing first automatically assumes that each act of abstracting and generalizing is context agnostic and recycling procedural solutions without thought is the most expedient path forward. This is clearly untrue, and yet even when we are able to consciously slow our thought process down to conceptually consider the situation and context, we are operating open-loop, without a plan or a blueprint or a process to compare against or to guide our decision making to assure that our subconsciously held fallacies and biases are held in check.

I toyed with the idea that what is needed was seemingly an impossibility: an abstraction and generalization process which can be applied consciously to an infinite array of situations while accounting for context and also encompassing the broadest applicable domain without misrepresentation and falling into illogic. I was thinking about an abstraction and generalization process where I can use for teaching, coaching, engineering, and for idea generation/evaluation. I am under no delusion that others have not thought about this previously of course, but I was keen on coming to terms with my own method, something that makes some sense, but I was struggling with the breadth, depth, and constraints to use to make this idea effective and efficient.

I asked my wise friend Erica Lucast Stonestreet (https://substack.com/@ericalucaststonestreet), a philosophy professor and a trained mathematician about the best resources, books, and articles that I can consult to clear up my confusion concerning the ways abstraction and generalization are used. She recommended Eugenia Cheng’s How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics, (Cheng, How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics 2016), it is a unique book devoted to explaining the most theoretical features and concepts in mathematics through using analogies with baking. I was leery about this approach at first, but it became a valuable reference in explaining mathematical concepts to an engineer, abstraction and generalization happened to be two of them. Eugenia Cheng is a mathematician working in the area of Category Theory, probably one of the most abstract areas of mathematics. She also wrote a more advanced book on mathematics and Category Theory six years after the first book, The Joy of Abstraction: An Exploration of Math, Category Theory, and Life (Cheng, The Joy of Abstraction: An Exploration of Math, Category Theory, and Life 2022) It is through my own amateurish attempts to understand her explanations about abstraction and generalization in mathematics that I came to the idea of using mathematical abstractions and generalizations as the blueprint of my own ideas about using abstraction and generalization in my search for a methodology which can be applied to my own thinking process.

First, how Does Mathematics use abstraction and generalization? Eugenia Cheng had written two books on the subject; I am sure others have studied the idea in fine granularity so who am I to expound on the ideas as an amateur? I am just taking the bare essence of the material Cheng wrote about and attempt to abstract and generalize her work for my purposes. 😊

According to Cheng, mathematics is a realm that completely depends on abstractions. This is the theoretical mathematician’s main skill set, and it is what mathematicians do to produce original mathematical works — contrary to the way we teach mathematics — which is mostly through computation and quantitative exercises, i.e. number crunching.

The determiner of whether mathematical abstractions and generalizations are true or false, while also ensuring that the mathematician’s exercise in abstraction does not veer into fanciful yet wrong conclusions is mathematical logic. Mathematicians apply mathematical logic to their exercises in abstraction. It is this mathematical logic that guides their abstraction, generalization, and creating proofs, and it is also this mathematical logic that other mathematicians use to independently check their peer’s works. I believe that the mathematical approach to abstractions is arguably the strictest and most limiting approach to performing abstraction, which assures us that any results from the rigorous practice of mathematical abstraction are assuredly correct and are probably the most difficult and restrictive set of guardrails to apply our thinking. This is why I think applying the mathematical framework for abstraction and generalization as the initial model for application to our everyday abstraction and generalization is a solid place to start, as it is easier to apply the strictest constraints and then relaxing them than it is to apply constraints after the abstracting and generalizing. What we lose is the ability to freely and flexibly abstract and generalize without constraints, as our mind is wont to do; we are sacrificing the freedom that Root and Bernstein and Rubin described so romantically. This also means that we are switching from purely procedural and System 1 exercises to conceptual and System 2 processes; we are consciously disengaging our subconscious and consciously engaging our consciousness to be more innovative. I don’t know how or if this is going to work out. It might be that we are too constrained in our conscious thinking to think innovatively, but I don’t think there is any harm in trying. This is, after all, an exercise in abstraction and generalization; in other words, abstracting and generalizing about abstracting and generalizing.

We shall see.

I am writing about this idea with a plan of splitting this topic into separate articles. This is the initial summary of why I am writing about this topic, the background for my thinking, and my initial thought process; the rest of the articles will hopefully result. But this is the tentative order of the articles.

Abstraction and Generalization II: The Mathematical Abstraction

Abstraction and Generalization III: Ideas from Engineering and Volleyball

Abstractions and Generalization IV: A Shot at a Process

References

Cheng, Eugenia. 2016. How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics. Basic Books.

—. 2022. The Joy of Abstraction: An Exploration of Math, Category Theory, and Life. Cambridge UK: Cambridge University Press.

Duke, Annie. 2022. Quit: The Power of Knowing When to Quit. London: Penguin.

—. 2018. Thinking in Bets. New York: Penguin.

Emre Soyer, Robin M. Hogarth. 2020. The Myth of Experience: Why we Learn the Wrong Lessons, and Ways to Correct Them. New York: Hatchett Book Group.

Epstein, David. 2019. Range, Why Generalists Triumph in a Specialized World. New York : Riverhead Books.

Kahneman, Daniel. 2013. Thinking Fast and Slow. NYC: Farrar, Straus and Giroux.

Root-Bernstein, Robert and Michele. 1999. Sparks of Genius-The 13 Thinking Tools pf the World's Most Creative People. New York: Houghton Mifflin.

Strauss, Rick Rubin and Neil. 2023. The Creative Act: A Way of Being. New York City: Penguin Press.

Wikipedia. n.d. "Abstraction." Wikipedia . Accessed May 5, 2024. https://en.wikipedia.org/wiki/Abstraction.

—. n.d. "Generalization." Wikipedia. Accessed May 5, 2024. https://en.wikipedia.org/wiki/Generalization.

—. n.d. "Occam's Razor." Wikipedia. Accessed May 5, 2024. https://en.wikipedia.org/wiki/Occam's_razor.