Exploring Operational Definitions: Part I

An operational definition is a way to define a concept by the set of operations or procedures which are used to make a judgement with that concept. This idea, along with a philosophical thesis on meaning known as operationalism, was popularized by Percey Bridgman (see his The Logic of Modern Physics).

One example he liked to use is the concept of distance.

What is distance?

Suppose you were to ask this question and your interest in this question is an ontological one. One might try to place the concept of distance within the concept of space. Now I'm not sure what the ontological status of space is but I have to wonder if this would provide any insight.

Mathematically speaking, each space has distinct metrics which ultimately measure distance. For example, Euclidean space is typically measured with a Pythagorean metric:
$$ds^2 = dx^2 + dy^2 + dz^2$$
But why should I measure it that way and not this way?
$$ds^2 = dx^2 + dy^2 + dz^2 - cdt^2$$
This is what distance looks like in Minkowski spacetime per special relativity (no acceleration, please!)

Regardless of the choice, that still doesn't tell me what it is that I'm measuring or how to measure the three components.

We might answer this with appeals to one's intuition regarding distance, but what is the basis for our intuition on distance? 

I would like to suggest that the bulk of our intuitions about distance stem from early experience with measuring distance. This takes us back to Bridgman's operational definitions.

So what is distance, operationally speaking?

Perhaps one of my earliest recollections of measuring distance as a child stems from measuring height. One question children often posed to one another is this: who is taller? A dispute may take place and there needs to be some way to resolve this dispute.

The generally accepted way to do so is as follows:

1) Have the two children stand on a flat, level surface.
2) Both children stand next to each other, back to back.
3) Then proceed to draw a line that is tangent to the top of one child's head across to where the other child's head is. This can be achieved using straight objects such as a ruler.

If the line is above the other child's head then the first child is taller; otherwise the other child is taller.

Now there's a lot that can go wrong in this sort of procedure. Is the surface the children are standing really flat and level? How can we know whether the line being drawn is even a straight line or is really tangent to the top of one child's head?

Ignoring the methodological difficulties in applying this (children probably have less difficulty in doing so than a philosopher), we can see that we've utilized an operational definition to determine which child is taller. These operations did not spell out how one might actually measure distance in the sense that we can associate the distance with a precise number.

We have, however, set up a set of normative procedures which we agree upon and accept as determining whether or not one child is taller.

Furthermore, these procedures can be generalized to determine whether or not any two objects are "taller" or "longer" than the other.

I posit, therefore, that operational procedures such as this example form the very basis for our "intuitions" about distance. The question then we might ask is this: Do we really need any other concept of distance except for these operational definitions?

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