Where do mathematical concepts come from? What are they? How do we create them? What is the role of definition? How do we judge a mathematical statement with concepts to be true or false? When do we really learn math concepts and statements? How do we integrate them into our existing knowledge?
Here are some excerpts from Ayn Rand’s philosophy of Objectivism which has an entirely new theory about knowledge.
It is fuzzy thinking on the questions above that lead to the “new new fuzzy math” collaborations that educators expect from Sixth Graders – this to “construct” heir own math concepts. Teaching concepts involves much more than facilitating a discovery learning event.
“A concept is an intellectual abstraction drawn from two or more percepts. Concepts are built on percepts and represent a new scale of consciousness, a scale that leaps beyond the perceptual limits of animals. Concepts allow humans to generalize, to identify natural laws, to understand what they observe.
Differentiation and Integration as the Means to a Unit-Perspective
A unit is an existent regarded as a separate member of a group of two or more similar members. The ability to regard entities as units is man’s distinctive method of cognition. The processes of differentiation and integration of attributes among observed entities allow a person to make an abstraction of these entities into a single unit, which a person can then store mentally as a word.
Concept-Formation as a Mathematical Process
An attribute of an entity is any characteristic reducible to a unit of measurement, such as shape, length, velocity, weight, color, etc. The Conceptual Common Denominator (CCD) between two or more entities is the commensurable (commonly measurable) attribute between those entities. For example, tables and chairs have the commensurable attribute of shape, while tables and red objects have the incommensurable attributes of shape and color. In turn, the CCD of shape allows a differentiation between chairs and tables and an integration of all tables into a single concept called “table”. The field of pure mathematics offers the deductive method of reasoning, while the process of concept-formation offers the first step in inductive reasoning. Conceptual awareness is the algebra of cognition.
Concepts of Consciousness as Involving Measurement-Omission
A first-level concept is abstracted directly from concrete percepts. A higher-level concept is abstracted from abstractions. Concepts differ not only in their concrete referents, but also in their distance from the perceptual level. Concepts of consciousness, such as “thought” and “love”, are formed by the same mathematical process as concepts of existence, such as “table” or “organism”. For example, two fundamental attributes of every process of consciousness (“thought”) are content and intensity of action. These two attributes of every mental process are measurable relative to each other by introspection. By omitting the measurements of these attributes, the concept of “thought” is abstracted.
Definition as the Final Step in Concept-Formation
The basic function of a definitionis to distinguish a concept from all other concepts and thus to keep its units differentiated from all other existents. A definition identifies a concept’s essential characteristics, which are the genus (CCD) and thedifferentia (differences from other existents that share the same genus). These characteristics must be fundamental, i.e. they must be responsible for all or most of the units’ remaining distinctive characteristics. An excellent metaphor for the term “definition” is that of a file folder with a label. The file folder represents the concept, while the label represents the definition. The contents of the folder can increase as more sensory knowledge of the concept is obtained, but the definition remains the same.
Concepts as Devices to Achieve Unit-Economy
A mind can only retain conscious focus upon a limited number of concrete percepts. A concept allows the conscious mind to cluster related percepts together as a single unit, e.g. perceiving many chairs, observing their similarities and differences, and then forming the concept “chair”. Thus, concepts allow the mind to condense or economize an unlimited amount of information into a finite number of easily processed, abstract units. Concepts empower the mind to process far larger amounts of information than it could on a strictly perceptual level, and thus enhance its ability to survive. Human beings are the only creatures on earth known to possess the ability to form concepts.”
For more see Luke Selzer Book Summary of “Objectivism” by Leonard Peikoff