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Monday, June 2, 2014

Exploring Operational Definitions: Part II - Distance

So in Part I, I began discussing operational definitions. In particular, I've been focusing on the concept of distance.

I proposed that we might consider our intuitions regarding answering the question - what is distance? - by means of early experience of operational definitions of distance. The idea being that what we consider distance is going to be closely related to ways we were taught to measure distance.

These procedures for measuring distance are all fundamentally normative; there are correct and incorrect ways of proceeding.  There's always the practical question of whether or not we're measuring correctly. For example, if we're attempting to see if one child is taller than the other by having them stand back to back, we need them both to stand on a level surface.

But what is it that determines correct procedure? And how do we know that a particular measure of distance is really measuring "distance" and not something else?

Measuring Distance

Here are three examples:

1) Meter Stick (or Yard Stick)

Using a meter stick is fairly common so I won't go into details on how to use it. To actually spell out explicitly in detail all of the various considerations on how to correctly use a meter stick could very well take an entire blog in itself (and may not be possible.) I believe this is important for understanding normative practices in general (and also important for the philosopher Ludwig Wittgenstein's later works such as his Philosophical Investigations.)

There are two considerations for any measuring instrument: (1) over what range of values does it give an accurate measurement and (2) how precise is that measurement.

With a meter stick, the precision will depend on a number of factors. While there are markings for one millimeter units, it's unlikely that everyone competent in using a meter stick will be able to agree on the the precise value. What we get with that are a range of values over which we agree (e.g. we agree that this item is between 0.563m and 0.566m.)

With a meter stick, the range is between zero and one meter. However, it may be possible to use the meter stick in a way that allows one to measure more than one meter. One can stand the meter stick from where it previously ended and therefore measure between one meter and two meters. This process, in theory, can be repeated indefinitely.

Using it in this manor, will introduce more uncertainty as there will be some uncertainty in each successive placement of the meter stick.

2) Calipers

Another way to measure distance is using calipers. Interestingly, calipers actually have several techniques for measuring distance.

For example, looking at this Vernier caliper (image courtesy of wikipedia):

we can actually see three techniques for measuring distance (I think there may be a fourth too but it's not coming to me at the moment.)

The outside large jaws (marked item 1 in the picture) can measure outside of a component. The inside small jaws (marked as 2) can measure inside distance. The depth probe (marked as 3) can measure depth.

Like the meter stick, calipers have both a range of accuracy and a precision associated with them. The range for most calipers is zero to six inches (some are larger, however). This limits its use compared with the meter stick. (Granted, I suppose you use the approach with the meter stick of successively applying it from end to end but I doubt that's terribly useful.)

As far as precision goes, these are far more precise than a meter stick which can potentially be as precise as 0.001 inches (which comes out to 0.0254mm or 0.0000254m). Practically speaking, there is some play in calipers so competent users may only agree to, say, within about 0.002 inches which is still far more precise than the meter stick.

3) Parallax

What's the distance between the earth and some star out in the sky? Clearly we can't use a caliper or a meter stick for the task (for example, as you approach the star, the heat will expand the device and burn/melt/etc the instrument.) So how do we measure such things?

The simplest way to show it is by placing your finger in front of your eyes. Now close your left eye. Now open your left eye while closing your right eye. You'll see your finger "move" so to speak. Now move your finger closer or further away from your eyes and repeat the process. This is the basic idea behind parallax.

The geometry looks something like this:

The distance to the star, $D$, is then given by:
$$D = d \tan (90^{\circ} - \theta)$$

For a more detailed description, see here.

The precision of this method will depend upon how precise we can measure the angle $\theta$ and the distance $d$. The uncertainty of those two elements will carry through to the uncertainty of the distance to the star ($D$). 

These are all methods for measuring distance. There are of course many more. Consider the number of different ways one can measure the height of a building with a barometer.

What is Distance?

One question we can ask is this: to what extent are all of these operational definitions actually measuring the same "property"?

As I mentioned in Part I, I made mention of Percy Bridgman's operationalism (described in the Logic of Modern Physics). In his view, each operation describes a different property. Bridgman puts it this way:
In principle the operations by which length is measured should be uniquely specified. If we have more than one set of operations, we have more than one concept, and strictly there should be a separate name to correspond to each different set of operations.
So his contention is that if we measure distance with a meter stick or a caliper or barometer or via parallax, what we are measuring (abstract "distance") is not the same; they're entirely different and they ought to be called something else.

I doubt too many physicists or mathematicians will share this view. But there is an interesting ontological question here. What's the ontological status of "distance"? In what sense do all of these techniques, these operational definitions, measure this concept "distance"?

One possible answer to this would be to suggest that all of these techniques are "equivalent". The idea here is that no matter which method we choose to use, we'll get "the same" result.

But what does "the same" result mean in this context? As noted, there is a range of applicability as well as an uncertainty. If I measure the thickness of a table to be 1" +/- .1" with a yard stick and .998 +/- .002 with a calipers, in what sense are these "the same" result? So we'd need to get into some detail in what we mean by "equivalent".

But there's also the latter question: if sets of operations are equivalent, do they measure the same concept? I'll leave these ontological questions aside. I think it's safe to say that operational definitions play an important part of the sciences. They enable us to take what are essentially abstract mathematical models and conduct experiments with them to see how well they perform.

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