Abstract knowledge

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This article by Dr Piotr Wozniak is part of SuperMemo Guru series on memory, learning, creativity, and problem solving.

Abstraction, applicability and usefulness

Abstract knowledge provides for the core of human intelligence. It is knowledge that forms the "big picture" in abstraction from detail. It relies on rules and formulas rather than a mass of facts.

I often say that in learning we need to maximize usability, applicability, and abstractness of knowledge. You can probably sense that all these three words may mean roughly the same thing. I often get asked what I mean when I say we should aim at "abstract knowledge". This is an issue of monumental importance, and the confusion in terminology leads to a great deal of confusion in learning strategies employed by students. In this text I try to provide some simple definitions that will help me explain my articles on learning, creativity, and problem solving.

Confusing abstraction with irrelevance

The terminological problem is made worse by schools that condition bad learning habits in which abstractness is underappreciated. Moreover, abstraction can be actively eradicated as it may also mean disassociation from reality. The same term can then mean two opposite things: (1) the most desirable: applicable knowledge or (2) the most abhorred: disconnected knowledge!

A 16-year-old wrote to me:

I was reading your article on The Roots of Creativity and Genius and I had trouble understanding the part of gaining abstract knowledge. Did you mean gain knowledge in areas one is not familiar with, or just gain knowledge in mathematics such as probability or statistics?

Abstraction vs. Generalization

Abstraction is a universally accepted term used in computer science and mathematics, however, it carries negative vibes such as "kids hate to learn abstract knowledge. They like things associated with real life". This confuses the precious process of generalization with the scourge of learning: incoherent knowledge. Knowledge that lacks in coherence is labelled abstract for it is not well connected with the rest of the body of knowledge. In other words, abstraction can be associated with things that are detached from reality. I prefer the term generalization and will try to limit the use of the adjective abstract without qualifiers. Unfortunately, general knowledge may be understood as knowledge related to many aspects of life (as opposed to a specific domain such as chemistry). In other words, I cannot say generalization leads to general knowledge nor can I be easily understood by saying abstraction leads to abstract knowledge.

Terminology: Abstract knowledge

It seems important to organize the terminology associated with generalization. I will try to consistently use a limited set of terms associated with the power of abstract knowledge. Here is the list of overlapping terms that often lead to confusion:

Abstract knowledge terminology

  • generalization: a process in which detail is ignored to reveal a deeper structure. The term overlaps with abstraction, conceptualization, inductive reasoning, modeling, theorization, categorization, conclusion, unification, colligation, de-concretization, pattern extraction, pattern separation, and more. Example: Trump is a winner is a gross generalization that ignores Donald Trump's failures
  • concept: a generalization of a set of objects/nouns. It overlaps with idea, entity, notion, group, etc. For example, animal is a concept derived from objects such as specific cats, birds, etc. Perhaps this should also include: property, attribute, quality, etc. i.e. the abstraction of object characteristics (e.g. the concept of yellowness)
  • rule: a generalization of an observed regularity. It overlaps with formula, theorem, principle, proposition, law, statement, and more. Example: "no pleasure, no good learning" is a fundamental law of learning. It is an example of a general rule that determines learning strategies
  • model: set of rules that apply to a specific phenomenon. It overlaps with theory, metaphor, opinion, schema, view, (concept) map, (formal) system, and more. Example: jigsaw puzzle metaphor of learning is a model of how knowledge coherence emerges in the process of learning
  • abstractness: universality of a concept or a rule, e.g. 2 apples and 2 apples add up to 4 apples is less abstract (i.e. more concrete or more specific) than 2+2=4
  • applicability: usefulness of a rule or model. It overlaps with usability. Example: 2+2=4 is useful in counting apples, but not-too-helpful in memorizing song lyrics
  • abstract knowledge: well-generalized, highly applicable knowledge that is conceptual/abstract in nature. It overlaps with: "big picture", set of rules/formulas, abstract set, theory, etc. Example: mathematics is the queen of abstract knowledge
  • abstract thinking: conceptual computation on rich and complex abstract knowledge

If mathematics is the queen of abstract knowledge, natural language should follow the lead and strive at mathematical precision. This will be helpful in natural language processing and artificial intelligence. Providing a well-established terminology is an effort in abstraction on its own. This way we can speak of mathematical concepts using natural language, without the need to produce a formal language for the algebra of abstraction.

The above list is an attempt to organize a tiny subset of terminology related to abstract knowledge. Sadly, my effort will certainly be undone by others who will make their own attempts to make order in the chaos of words. Language darwinism is welcome. However, if my own texts become easier to understand, my minimum goal will have been reached.

In the statement knowledge is power we almost certainly put abstract knowledge to the forefront. Wisdom is not about memorizing facts (e.g. thousands of phone numbers), but about coherent abstract knowledge that can be effectively used in creative problem solving.

Definition: Abstract knowledge

The process of generalization is based on compiling facts, deriving concepts and rules, building higher-level models, and accumulating knowledge that is abstract/conceptual in nature. Abstract knowledge is knowledge of high abstractness that leads to high applicability that makes up the core of human intelligence

Example: building abstract knowledge

Abstract knowledge by example: rabbits and foxes
Abstract knowledge can be illustrated with an example. While looking at the world we see rabbits and foxes. These are our objects that we label as animals. Animal is then a concept derived in the process of generalization from objects such as foxes. While watching rabbits, we can establish a rule for rabbit survival in the presence of a fox. Deriving the rule is also based on a generalization (i.e. conclusion drawn from multiple situations). After a while we can build a model of an ecosystem that involves rabbits and foxes. That model will contain a great deal of facts (e.g. rabbit Joe is lazy) and a great deal of rules (e.g. white rabbits are faster). Once we develop a set of good models, we may acquire a good knowledge of an ecosystem. That knowledge will help us survive in that ecosystem. Abstract knowledge based on rules and models will be particularly applicable. For example, we may employ a rule: to make my carrots thrive, I cannot hunt for foxes

Generalization is inherent in neural networks

The power of generalization is a property that comes with neural networks. Abstraction is often attributed to humans and the development of the language, however, even a fish can generalize the concept of the predator, pray or mate. The power of generalization is an attribute of the simplest neural networks in existence.

Validity of generalization

For centuries, philosophers struggled with the concept of inductive reasoning, in which general principles (rules, formulas, etc.) are derived from specific observations (data). Unlike deductive reasoning, inductive reasoning can lead to a wrong conclusion when we trip on the proverbial exception to the rule. In the human brain, inductive reasoning is based on generalization. It is not valid, but it is usually probabilistically valid, which means it can be valid most of the time. This is the precious part of the generalization: ignoring detail, reducing costs, compressing knowledge, increasing applicability, etc. All that value comes at a cost of a possible error in reasoning. On occasion, we can confuse a rabbit with a hare or even with a fox. The evolution provided the proof: it works and it is worth it. All intelligence is based on the generalization power of neural networks. This is why forgetting is precious.

All generalizations throw away detail. They are not a form of unzooming from a picture when details of the picture get blurry, but can be recovered by zooming back in. Generalization loses information, and is only probabilistically valid (e.g. carrying a high probability of making accurate predictions). Probabilistic validity is helpful even if it carries a large margin of error. For a good example see: Brett Kavanaugh guilt estimated at 87%.

Positive and negative shades of abstract knowledge

To resolve the problem of the pejorative use of the term abstract we only need to observe that abstract knowledge that is not derived via generalization may lose its applicability. Without the process of generalization, i.e. learning by example, abstract knowledge may become detached from reality. In simple terms, abstract knowledge that is not backed up by comprehension becomes abstract in a pejorative sense.

In 20 rules of formulating knowledge, I insist on the obvious: Do not memorize if you do not understand. Without comprehension, there is no coherence and no applicability. In incremental reading, a seeming violation occurs when we differentiate actual memorization from introducing a piece of knowledge into the learning process.

In school lingo, abstract knowledge is nearly always bad. In my dictionary, abstract knowledge is the key to intelligence. This is why the use of the term is always putting the writer on a shaky ground.

Schooling leads to the injury of generalization skills

While generalization is natural, schools often lead to the injury of the generalization process. This occurs via bad conditioning, in which factual knowledge, or literal representation of abstract knowledge, are employed in the process of learning. This is most pronounced in cramming for a test, when time is of essence, and comprehension is secondary.

Free creative abstraction, accurate or erroneous, is essential for modeling and building comprehension. Intense focus on factual knowledge at school may inhibit attempts at generalization. If a child's brain produces an abstraction that quarrels with the presented factual knowledge, she may be penalized. Instead, literal and accurate rendition of factual knowledge is rewarded.

As a result, a student may know that 2+2=4, but will not know that two apples and two pears add up to four fruits. He may be fluent in the use of Laplace operators, but never truly aware of what they signify. After many years of bad conditioning, students disengage the power of generalization they used in toddlerhood, and approach knowledge as a system of strings that need to be encoded in memory. This ways schools condition children to give up on their intelligence and become tape recorders. See: On the superiority of a rat over a schooled human.

Further reading

Learn more from Wikipedia:

For more texts on memory, learning, sleep, creativity, and problem solving, see Super Memory Guru