In an earlier post, I hinted that I might take up the question of conceptual schemes, and the question of whether conceptual relativity should be seen as a problem. Donald Davidson has argued that the idea of a conceptual scheme is incoherent. I disagree, and will explain why.

## A brief detour

I’ll start with a detour into my mathematical background. I learned linear algebra primarily from Halmos: “Finite Dimensional Vector Spaces.” It presents an elegant, but abstract, account of vector spaces (linear algebra). Briefly, a vector space consists of a set of things (vectors) on which certain operations are permitted (addition, scalar multiplication), and some axioms or rules that apply when those operations are used.

I had, previously, been introduced to vectors in calculus classes. There, a 3-dimensional vector was a triple *(x,y,z)* of numbers. One might perhaps describe the approach in the calculus class as based on coordinates (as in coordinate geometry), while Halmos presented an essentially coordinate free account of vector spaces.

The coordinate based treatment of vectors and the coordinate free treatment complemented on another. What became clear, is that the choice of coordinate system is arbitrary, with the coordinate vectors selected for pragmatic reasons. That is to say, the choice of coordinate vectors is dictated by practical consideration in how we wish to apply linear algebra, and is not in any way dictated by the nature of the vector space itself. Mathematicians describe this by saying that there is no canonical choice of coordinate vectors. That’s perhaps about the same as saying that the coordinate vectors are not metaphysical.

## Back to conceptual schemes

In my earlier post, “A semantic conception of mind,” I gave what could be considered a coordinate free account of knowledge. I treated knowledge as a repertoire of perceptual functions that we can use to access properties of the world. I mentioned particular functions such as temperature and distance only to illustrate the ideas. The overall account that I gave was coordinate free. However, practicalities require that if we are to use these property functions, then we must single out some and give them names and rules of usage. I see concepts as the particular functions that we have singled out for explicit use, and I see a conceptual scheme as a collection of such functions that we have singled out.

In the linear algebra case, the choice of coordinates is arbitrary, selected on a pragmatic basis. In much the same way, the choice of which property functions we single out can be seen as arbitrary, and selected on a pragmatic basis. In other words, the choice of conceptual scheme is not canonical (in the mathematical sense), so is not metaphysical. The conceptual scheme that we use is partly a result of our cultural history. We can see this with the concept of mass, which is a fundamental concept of physics. But it was introduced by Galileo and Newton, so probably unknown before that time.

With a finite dimensional vector space, any vector has a finite representation in terms of whatever coordinate vectors we choose. It will be a bit more complex for concepts. If we have an adequate set of concepts, then any property will be representable as a limit of what can be expressed finitely in terms of our selected concepts. But there is no guarantee that properties will have finite expression. So two different conceptual schemes might be able to describe the same world, yet there might fail to be exact finite translatibility between those descriptions. So meaning incommensurability can exist, but it is not nearly as important as Kuhn took it to be.