In my study of artificial intelligence, I have come across an important contemporary philosopher and cognitive scientist who should be of particular interest to students of Ayn Rand’s theory of concepts. Although the published work of Ayn Rand predates his published work (and almost all of the voluminous other work that he cites), Peter Gärdenfors seems to have arrived independently at important similar ideas, and he seems to have developed some of these ideas much further mathematically.

In this brief post, just a few of these similarities are noted. For more, I highly recommend the following works from these thinkers.

Rand, Ayn. [1966–1967] 1990. Introduction to Objectivist Epistemology. The Objectivist 5(7)–6(2). Reprinted in Introduction to Objectivist Epistemology, Expanded Second Edition. Edited by Harry Binswanger and Leonard Peikoff. New York: Meridian.

Gärdenfors, Peter. 2000. Conceptual Spaces: The Geometry of Thought. Cambridge, Massachusetts: MIT Press.

Gärdenfors, Peter. 2014. The Geometry of Meaning: Semantics Based on Conceptual Spaces. Cambridge, Massachusetts: MIT Press.

Gärdenfors has a third book on the subject forthcoming in May 2026.

I have just completed a first reading of Gärdenfors 2000 and have just started reading Gärdenfors 2014 with intense interest.

Ideas from Ayn Rand

Note these passages from Ayn Rand ([1966–1967] 1990) on her theory of concepts:

The process of concept formation is, in large part, a mathematical process.” (p. 7.)

A concept is a mental integration of two or more units [particular instances] possessing the same distinguishing characteristic(s), with their particular measurements omitted. (p.13.)

The element of similarity is crucially involved in the formation of every concept; similarity, in this context, is the relationship between two or more existents which possess the same characteristic(s), but in different measure or degree. (p.13.)

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. (p.13.)

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. (p.13.)

Another example of implicit measurement can be seen in the process of forming concepts of colors. Man forms such concepts by observing that the various shades of blue are similar, as against the shades of red, and thus differentiating the range of blue from the range of red, of yellow, etc. [Note from RP: That is, similarity is less difference.] Centuries passed before science discovered the unit by which colors could actually be measured: the wavelengths of light—a discovery that supported, in terms of mathematical proof, the differentiations that men were and are making in terms of visual similarities. (pp. 13–14.)

Ideas  from Gärdenfors

Now consider these passages from Gärdenfors (2000):

Concept learning is closely tied to the notion of similarity … . (p.1.)

We frequently compare the experiences we are currently having to memories of earlier episodes. Sometimes, we experience something entirely new, but most of the time what we see or hear is, more or less, the same as what we have already encountered. This cognitive capacity shows that we can judge, consciously or not, various relations among our experiences. In particular, we can tell how similar a new phenomenon is to an old one.

With the capacity for such judgments of similarity as a background, philosophers have proposed different kinds of theories about how human concepts are structured. For example, Armstrong (1978, 116) presents the following desiderata for an analysis of what unites concepts:

If we consider the class of shapes and the class of colours, then both classes exhibit the following interesting but puzzling characteristics which it should be able to understand:

(a) the members of the two classes all have something in common (they are all shapes, they are all colours)

(b) but while they have something in common, they differ in that very respect (they all differ as shapes, they all differ as colours)

(c) they exhibit a resemblance order based upon their intrinsic nature (triangularity is like circularity, redness is more like orangeness than redness is like blueness), where closeness of resemblance has a limit in identity.

(d) they form a set of incompatibles (the same particular cannot be simultaneously triangular and circular, or red and blue all over). (pp. 4 –5).

[See Armstrong, D M. 1978. A Theory of Universals. Cambridge: Cambridge University Press.]

Judgments of similarity … are central for a large number of cognitive processes. … [S]uch judgments reveal the dimensions of our perceptions and their structures … . (p. 5)

The structure of many quality dimensions of a conceptual space will make it possible to talk about distances along the dimensions. There is a tight connection between distances in a conceptual space and similarity judgments: the smaller the distances is between the representations of two objects, the more similar they are. In this way, the similarity of two objects can be defined via the distance between their representing points in the space. Consequently, conceptual spaces provide us with a natural way of representing similarities. (p.5.)

A phenomenally interesting example of a set of quality dimensions concerns color perception. According to the most common perceptual models, our cognitive representation of colors can be described by three dimensions: hue, chromaticness, and brightness. …

The first dimension … is hue, which is represented by the familiar color circle. The value of this dimension is given by a polar coordinate describing the angle of the color around the circle … .

The second phenomenal dimension of color is chromaticness (saturation), which ranges from grey (zero color intensity) to increasingly greater intensities. This dimension is isomorphic to an interval of the real line. The third dimension is brightness which varies from white to black and is thus a linear dimension with two end points. (pp. 9–10.)

In a conceptual space that is used as a framework for a scientific theory or for construction of an artificial cognitive system, the geometrical or topological structures of the dimensions are chosen by the scientist proposing the theory or the constructor building the system. The structures of the dimensions are tightly connected to the measurement methods employed to determine the values on the dimensions in experimental situations … . (p. 21.)

The notion of a domain is central in this book, and it is used in connection with concept formation in chapter 4, with cognitive semantics in chapter 5, and with induction in chapter 6. Using the concepts of this section, I can now define a domain as a set of integral dimensions that are separable from all other dimensions. The three-color dimensions are a prime example of a domain in this sense since hue, chromaticness, and brightness are integral dimensions that presumably are separable from other quality dimensions. Another example could be the tone domain with the basic dimensions of pitch and loudness. The most fundamental reason for decomposing a cognitive structure into domains is the assumption that an object can be assigned certain properties independently of other properties. An object can be assigned the weight of “one kilo” independently of its temperature or color.

A conceptual space can then be defined as a collection of one or more domains. …

The domains of a conceptual space should not be seen as totally independent entities, but they are correlated in various ways since the properties of the objects modeled in the space covary. For example, ripeness and color domains covary in the space of fruits. These correlations are discussed in connection with the model of concepts in section 4.3 and in connection with induction in section 6.6.33

Conceptual spaces will be the focus of my study of representations on the conceptual level. A point in a space represents a possible object (see section 4.8). The properties of the objects are determined by its location in the space. As will be argued in chapter 3, properties are represented by regions of a domain. (p. 26.)

I take a categorization to be a rule for classifying objects. … The result of applying a categorization rule will be a number of categories. In the model presented here, where (possible) objects are represented as points in conceptual spaces, a categorization will generate a partitioning of the space and a concept will correspond to a region (or a set of regions from separable domains) of the space. {p. 60)

In effect, Gärdenfors is working to implement the program, discussed by Ayn Rand, of mathematizing similarity and concepts.

Of course, there are important differences between the two thinkers. But each thinker has important original ideas to contribute (to say the least). Anyone interested in knowing the state of the art of understanding the formation and use of concepts must read both Gärdenfors and Ayn Rand.