## On conceptual schemes

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.

### 3 Comments to “On conceptual schemes”

1. “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.”

I think that this is arguable. If the incommensurability is a result of different property functions being singled out within those vector spaces, or even one property function intuitively or otherwise negating others in some way, then I can see this incommensurability as not only existing but also having significant importance as Kuhn did.

Whether it’s trying to reconcile the quantum realm with that of the macroscopic, there are certain ways of looking at the world (and any resultant property functions) that are incompatible with others (e.g. Copenhagen interpretation vs. the MWI for quantum mechanics). I do like the idea of “property functions being singled out for pragmatic reasons”, but I also realize that they may only look like functions when in reality there are either other hidden variables which cause them to not reconcile with other property functions, or they are just coincidental close approximations. It’s hard to say.

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• I only used finite dimensional vector spaces as a simple illustration. Once we start talking about properties, we are talking about an infinite dimensional function space.

Take the unit interval (the real numbers between 0 and 1, inclusive). Every continuous function on the unit interval is a limit of polynomial functions (that’s the Weierstrass approximation theorem). Likewise, any continuous function is a limit of sums of trigonometric functions (that’s Fourier series). But a trig function is not finitely expressible in terms of polynomial functions, and polynomial functions are not finitely expressible in terms of trig functions.

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• Yes, and I think that the same thing applies with property functions in our world. They may not be translatable and this obviously leads to the incommensurability that Kuhn described.

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