How do we form and define concepts?

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.

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“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

How do we conceptualize?

It is important to always reduce the most abstract mathematical concepts in a chain to the level of perceptual concretes. Our knowledge – our mathematical knowledge – is hierarchical and contextual.  Our knowledge items must not be held to be arbitrary. 

Ayn Rand has advanced the field of philosophy – epistemology (the study of how we know what we know) – beyond the work of  “The Father of Logic” Aristotle over 2000 years ago. 

In this article we collect some of her thoughts on “conceptualizing” in mathematics:

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“Man’s sense organs function automatically; man’s brain integrates his sense data into percepts automatically; but the process of integrating percepts into concepts—the process of abstraction and of concept-formation—is not automatic.  …

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The process of concept-formation does not consist merely of grasping a few simple abstractions, such as “chair,” “table,” “hot,” “cold,” and of learning to speak. It consists of a method of using one’s consciousness, best designated by the term “conceptualizing.” It is not a passive state of registering random impressions. It is an actively sustained process of identifying one’s impressions in conceptual terms, of integrating every event and every observation into a conceptual context, of grasping relationships, differences, similarities in one’s perceptual material and of abstracting them into new concepts, of drawing inferences, of making deductions, of reaching conclusions, of asking new questions and discovering new answers and expanding one’s knowledge into an ever-growing sum. The faculty that directs this process, the faculty that works by means of concepts, is: reason. The process is thinking.”

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“… Concepts “represent classifications of observed existents according to their relationships to other observed existents.”

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“To form a concept, one mentally isolatesa group of concretes (of distinct perceptual units), on the basis of observed similarities which distinguish them from all other known concretes (similarity is “the relationship between two or more existents which possess the same characteristic(s), but in different measure or degree”); then, by a process of omitting the particular measurements of these concretes, one integrates them into a single new mental unit: the concept, which subsumes all concretes of this kind (a potentially unlimited number).

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The integration is completed and retained by the selection of a perceptual symbol (a word) to designate it.

“A concept is a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted.”

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“Bear firmly in mind that the term “measurements omitted” does not mean, in this context, that measurements are regarded as non-existent; it means that measurements exist, but are not specified. That measurements must exist is an essential part of the process. The principle is: the relevant measurements must exist in some quantity, but may exist in any quantity.”

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“Concepts are not and cannot be formed in a vacuum; they are formed in a context; the process of conceptualization consists of observing the differences and similarities of the existents within the field of one’s awareness (and organizing them into concepts accordingly). From a child’s grasp of the simplest concept integrating a group of perceptually given concretes, to a scientist’s grasp of the most complex abstractions integrating long conceptual chains—all conceptualization is a contextual process; the context is the entire field of a mind’s awareness or knowledge at any level of its cognitive development.

This does not mean that conceptualization is a subjective process or that the content of concepts depends on an individual’s subjective (i.e., arbitrary) choice. The only issue open to an individual’s choice in this matter is how much knowledge he will seek to acquire and, consequently, what conceptual complexity he will be able to reach. But so long as and to the extent that his mind deals with concepts (as distinguished from memorized sounds and floating abstractions), the content of his concepts is determined and dictated by the cognitive content of his mind, i.e., by his grasp of the facts of reality.

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“A commensurable characteristic (such as shape in the case of tables, or hue in the case of colors) is an essential element in the process of concept-formation. I shall designate it as the “Conceptual Common Denominator” and define it as “The characteristic(s) reducible to a unit of measurement, by means of which man differentiates two or more existents from other existents possessing it.”

The distinguishing characteristic(s) of a concept represents a specified category of measurements within the “Conceptual Common Denominator” involved.

New concepts can be formed by integrating earlier-formed concepts into wider categories, or by subdividing them into narrower categories (a process which we shall discuss later). But all concepts are ultimately reducible to their base in perceptual entities, which are the base (the given) of man’s cognitive development.”

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“When concepts are integrated into a wider one, the new concept includes allthe characteristics of its constituent units; but their distinguishing characteristics are regarded as omitted measurements, and one of their common characteristics determines the distinguishing characteristic of the new concept: the one representing their “Conceptual Common Denominator” with the existents from which they are being differentiated.

When a concept is subdivided into narrower ones, its distinguishing characteristic is taken as their “Conceptual Common Denominator”—and is given a narrower range of specified measurements or is combined with an additional characteristic(s), to form the individual distinguishing characteristics of the new concepts.”

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“Let us now examine the process of forming the simplest concept, the concept of a single attribute (chronologically, this is not the first concept that a child would grasp; but it is the simplest one epistemologically)—for instance, the concept “length.” If a child considers a match, a pencil and a stick, he observes that length is the attribute they have in common, but their specific lengths differ. The difference is one of measurement. In order to form the concept “length,” the child’s mind retains the attribute and omits its particular measurements. Or, more precisely, if the process were identified in words, it would consist of the following: “Length must exist in some quantity, but may exist in any quantity. I shall identify as ‘length’ that attribute of any existent possessing it which can be quantitatively related to a unit of length, without specifying the quantity.”

The child does not think in such words (he has, as yet, no knowledge of words), but that is the nature of the process which his mind performs wordlessly. And that is the principle which his mind follows, when, having grasped the concept “length” by observing the three objects, he uses it to identify the attribute of length in a piece of string, a ribbon, a belt, a corridor or a street.

The same principle directs the process of forming concepts of entities—for instance, the concept “table.” The child’s mind isolates two or more tables from other objects, by focusing on their distinctive characteristic: their shape. He observes that their shapes vary, but have one characteristic in common: a flat, level surface and support(s). He forms the concept “table” by retaining that characteristic and omitting all particular measurements, not only the measurements of the shape, but of all the other characteristics of tables (many of which he is not aware of at the time).”

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“Observe the multiple role of measurements in the process of concept-formation, in both of its two essential parts: differentiation and integration. Concepts cannot be formed at random. All concepts are formed by first differentiating two or more existents from other existents. All conceptual differentiations are made in terms of commensurable characteristics (i.e., characteristics possessing a common unit of measurement). No concept could be formed, for instance, by attempting to distinguish long objects from green objects. Incommensurable characteristics cannot be integrated into one unit.

Tables, for instance, are first differentiated from chairs, beds and other objects by means of the characteristic of shape, which is an attribute possessed by all the objects involved. Then, their particular kind of shape is set as the distinguishing characteristic of tables—i.e., a certain category of geometrical measurements of shape is specified. Then, within that category, the particular measurements of individual table-shapes are omitted.

Please note the fact that a given shape represents a certain category or set of geometrical measurements. Shape is an attribute; differences of shape—whether cubes, spheres, cones or any complex combinations—are a matter of differing measurements; any shape can be reduced to or expressed by a set of figures in terms of linear measurement. When, in the process of concept-formation, man observes that shape is a commensurable characteristic of certain objects, he does not have to measure all the shapes involved nor even to know how to measure them; he merely has to observe the element of similarity.

Similarity is grasped perceptually; in observing it, man is not and does not have to be aware of the fact that it involves a matter of measurement. It is the task of philosophy and of science to identify that fact.”

For more see Ayn Rand, Introduction to Objectivist Epistemology

What is the purpose of a mathematical education?

Mathematical education equips students with the knowledge to help them them function in daily life, develop their rational faculty, become able to acquire new knowledge as needed, be responsible citizens and develop lucrative careers.

Ayn Rand writes:  “The only purpose of education is to teach a student how to live his life—by developing his mind and equipping him to deal with reality.

The training he needs is theoretical, i.e., conceptual. He has to be taught to think, to understand, to integrate, to prove.

He has to be taught the essentials of the knowledge discovered in the past—and he has to be equipped to acquire further knowledge by his own effort.”

Here is a audio interview with Ayn Rand on education issues.      Ayn Rand on Education.