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