What are “conceptual spaces”?

After having sketched what hides behind the term “Artificial General Intelligence” in my last post, today I would like to give a short introduction to conceptual spaces.

The term “conceptual spaces” describes a framework proposed by Peter Gärdenfors [1] that aims at a geometric representation of concepts. It is the starting point of my PhD research on concept formation.

But first things first: what is a concept?

Well, one can start with the Wikipedia definition:

A concept is an abstract idea representing the fundamental characteristics of what it represents.

That might sound reasonable, but is not too helpful to grasp the concept behind the word “concept”. Jokes aside, here’s a more practical example: The concept of an apple is the abstract idea of what an apple is about. It is your internal representation of a category of things – you have probably observed many apples in your life, but you would put all of them into the category “apple”. The concept of an apple includes information about the typical shape, size, taste, weight, etc. of apples. From another point of view, one could also argue that a concept corresponds to the meaning behind a word (if we ignore synonyms and multiple word meanings for the time being).

Now the conceptual spaces framework basically claims that these concepts can be represented as regions in a high-dimensional space. How does this work?

Well, first of all, we need some dimensions that span our conceptual space. Gärdenfors calls them quality dimensions. A quality dimension provides us with means to judge the similarity of two stimuli. Examples of such quality dimensions include brightness, pitch, temperature, weight, and length.

Each stimulus (or observation or instance – however you would like to call it) can then be represented by a point in this space, that is by a value for each of the dimensions.

Some of the quality dimensions inherently belong together – for instance, the dimensions of hue, saturation, and brightness (aka “lightness” or “value”). These three dimensions form the so-called color domain. A domain is just a set of dimensions that inherently belong together – other domains are the temperature domain (consisting only of the temperature dimension) and the sound domain (consisting of pitch and loudness).

Illustration of the color domain (from Wikipedia).

Now the main idea of the conceptual spaces framework is the following:

Concepts (like “apple” or “dog”) can be represented as convex regions in a conceptual space. Properties (like “red” or “sour”) can be represented as convex regions within a single domain.

This means, that we should be able to find for instance a convex region in the color domain that represents the property of being “red”. Similarly, we should be able to find a convex region in the overall conceptual space that represents the concept “apple”. Remember that all observations are represented as points in our space. We can thus say something like: “If the point representing a specific observation is located in the region describing a given concept, then this observation is an example of this concept”.  So if our concept would be the color “red”, we could say: “If this observation falls in the red-region, we call it red.” If we consider the concept “apple”, we could say: “”If this observation falls in the apple-region, we classify it as an apple.”

One important point is that we are not talking about arbitrary regions here, but about convex regions. A region R is called convex if the following holds true:

If we have two points x and z that both belong to this region R, then any point y that lies somewhere between x and z must also belong to R.

Applied to our red-example, this basically means: If we know of two observations that they are red (i.e., they belong to the red-region), then any observation between those two (i.e., any “morph” between the two original observations) must also be called red. Again, the same also holds for apples: Given two apples, any morph between them must also be an apple.

All right, that’s already it – the basic idea of conceptual spaces. There is of course a lot more going on with respect to the details, but for now I think this brief introduction is sufficent.

So conceptual spaces are just spaces spanned by quality dimensions. These dimensions can be grouped into domains. Instances are represented as points in this space and concepts are represented by convex regions.

I will elaborate on the structure and mechanics of conceptual spaces in future blog posts.


[1] Gärdenfors, Peter. Conceptual spaces: The geometry of thought. MIT press, 2004.