Saturday, October 16, 2004

reductionism in geometry

Here is, some, background.

Analysis is an investigative technique where you break the subject down into parts. It is a very successful technique and leads to some important simplifications in our view of the universe. For example, we analyze human bodies into parts. If someone is sick, it is often because one of the parts is not functioning correctly. Identify the faulty part, and you are halfway to a solution.

Analysis within a single domain is an extremely useful, even vital form of reasoning. A domain (as I defined it in a previous post) is a class of things. Concepts are in a different domain from objects. Physical objects are in a different domain from thoughts. You can tell if two objects are from different domains because they have different properties; not just different values of the same properties, but entirely different properties.

For example, physical objects have a size, a mass, a temperature. Every physical object has those properties to a greater or lesser extent. Thoughts have none of those properties at all. Thoughts have subjects and emotional content and other non-physical properties. Physical objects have none of those properties. That is why thoughts are in a different domain from physical objects.

It makes sense to analyze an object into other objects that share the same domain. You analyze a physical object into other physical objects that have smaller size and mass. It makes a lot less sense to analyze an object into other objects from a different domain. When you do this, you have to show not only how the original object is made up of the other objects, but also how its properties are made up from aspects of the other domain. This sort of analysis is called reduction because you reduce one domain to another.

One success story for reductionism can be taken from geometry. Ancient geometry had two distinct concepts: points and lines. Some modern geometry reduces the domain of a lines to the domain of points. This mathematical form of reductionism offers some insights that can be used to talk about other kinds of reductionism.

As in any other instance of reduction, the point reductionist is faced with a serious problem from the very beginning: lines clearly aren't points. Lines have direction, points don't. Lines have length, points don't. Lines can intersect with other, different lines, points do not intersect with other, different points.

In geometry, we get around this problem by defining a line as a set of points. A set of points is not a point: for example, a set of points can intersect with another, different set of points, a single point can't do that. A set of points has a cardinality, a single point doesn't.

In mathematical logic we say that a set of points has a different type than a point. This difference in type is obviously similar to a difference in domains and the point reductionists exploit it to get their reduction. If a set of points were the same type as a point, we would have to explain how something that seems so different from a point (namely a line) can be reduced to something that is the same as a point. But now we can say even though lines are reduced ultimately to points, they are things of a different type and that is what gave us the illusion that they are of a different domain.

So, does this example from geometry prove that reduction is sometimes possible?

I'll give you a moment to think about it. You might want to compare and contrast the point/line dichotomy with the physical object/thought dichotomy.

Back? OK, here's my take: the problem with this example from geometry is that points and lines are only from partially different domains. Even though lines have some properties that points don't have, there are other kinds of properties that they both have in common. They both have positions, for example. In fact, it was always understood that lines contain points or even that points are "parts" of lines in some sense. The only thing the point reductionist really had to do was explain how lines could be dispensed with entirely in favor of points.

By contrast, it is highly controversial whether physical objects can be "parts" of thoughts in any sense or thoughts parts of physical objects. Yet there is some overlap of properties. Both a thought and physical object, for example, exist at some particular time.

It turns out that the concept of domain is not perfectly precise. Some kinds of properties are shared by many different kinds of things. We will have to examine each individual case of reductionism to see if it is plausible or not.

To begin, lets examine the reduction of line to point. Have modern mathematicians proven that a line is nothing more than a set of points? Many people think they have. But couldn't we just as well reduce points to sets of lines? Define a point as the set of all lines that pass though it.

So, if a line can be reduced to a set of points and a point can be reduced to a set of lines, what possible justification can you have for claiming that one of these reductions is somehow more fundamental? Isn't it more reasonable to just observe that point and lines are distinct concepts and that there are interesting homomorphisms between individuals of one and sets of the other?

It turns out that even this uncontroversial example of reduction isn't especially convincing. It's only a mathematical convenience. It turns out to be convenient to use sets of points rather than lines for some purposes. Our innate talent for physical analysis misleads us into thinking that we have found the parts of lines when all we have found is an interesting homomorphism.

I just realized that I never got around to explaining what a homomorphism is when I was talking about abstract algebras. I'll try to rectify that in the near future. For now, think of a homomorphism as a mapping from one domain to another in a special way. For example, you can map lines into sets of points in lots of different ways, but there is one "right" way to do it: map each line into the set of points that are in the line.

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