Reduction

Standard way of eliminating objects you don't like.

Used by science: Molecules (chemistry) reduce to atoms reduce to subatomic particles.

The idea in science is often thought to be that the laws of each level follow from the laws of the level below, and so all there "really" is is collections of elementary particles governed by their laws.

Emergentists think that high-level laws are not reducible to low-level laws—there are new laws at higher levels.

Example: Economics—Gresham's law: bad money drives out good.

There is a further question of whether high-level things are "new": is there, in addition to chemistry a "life force," not found at lower levels that makes living things alive? Such theories are alive and well, even if that one isn't: consciousness.

No one thinks that there is "monetary value," an irreducible addition to money. Every piece of money is a physical object, nothing more. Money might nonetheless obey new laws.

In addition, money supervenes on physics: every individual bit of behavior of money is describable in terms of physical laws. The slogan: there is no monetary difference without a physical difference.

There are many simple attempts at reduction in every area of metaphysics. We have already seen, for example, attempts to reduce universals to sets or properties or similarity.

It is therefore worthwhile to get clear on what is required for a successful reduction. To do that, it is helpful to have clear examples. Because mathematical objects are bound by precise rules that we know, the easiest case in which to get reductions that are clearly successful is in mathematics, which is therefore important to look at to understand reduction.

Two examples of successful reductions:

Here are some reductions of intermediate success. All are attempts to reduce natural numbers to sets or classes.

Any of the proposals will serve as the numbers, but, for that very reason, there is a problem: for any them to be "the numbers" we would have to have a reason to prefer it to the others. Thus, to reduce one class of things to another, it is not enough to find things in the reducing class that have the right structural properties: if there are many such, then it turns out that we have replacements for our original things, instead of a stipulation of what they are.

In the case of mixed numbers, we eliminated them in favor of a single candidate, but in the case of natural numbers, we instead got lots of candidates, which are therefore substitutes but not simple replacements.

Most who haven't liked some metaphysical entities or other have been content with replacements. That is a good enough form of reduction.

It has been common to analyze such examples by saying that all that is required is a way to map claims about the things to be eliminated onto claims about other things, preserving truth value. As Quine points out, that won't do: the number of sentences about any subject is at most countable—we can match them up with the natural numbers. Thus, if the analysis were correct, all we would need are the numbers and suitable properties on them: replace each sentence by a number and let the property be the one that holds only of the numbers that correspond to true sentences.

Quine says that what we need is to reduce, not the truths, but the objects: there needs to be a "proxy function" that, if the objects to be reduced had existed, would map each one to an actual object that stands in its place.

Examples:

-- ShaughanLavine - 03 Mar 2008