On Counting (Exploring Operational Definitions Part III)

For the first two parts in this series see:

Exploring Operational Definitions: Part I
Exploring Operational Definitions: Part II - Distance

Perhaps the "simplest" procedure that most folks have learned is the technique(s) of counting. What I would like to explore is that there are a variety of techniques that we call counting. In some cases they build on one another. In other cases, they are techniques which give "approximate" solutions.

Of course not all societies count things (see here). Nonetheless, I suspect that many of our "intuitions" about mathematics ultimately stem from our earlier experience with counting. Our attachment to such intuitions will somewhat determine how willing we are able to accept alternative definitions and techniques for counting. Today I'll explore a few of these definitions.

What are numbers?

Before we can learn one basic technique for counting we need numbers. Or more specifically, we need some technique by which we can generate numbers.

One procedure is something often referred to as a "successor function". Peano's axioms take this approach of course that raises the question of what a function is. . . In any event, the procedure amounts to starting with an initial number (call it "zero") and then generating subsequent numbers as "successors". Here are Peano's axioms (adapted from Wolfram):

1) 0 is a number.
2) If $a$ is a number then $s(a)$ is a number (where $s(x)$ is the successor function.)
3) If $a$ is a number then $0\ne s(a)$. 
4) If $s(a) = s(b)$ then $a=b$.
5) If $S$ is a set and $0 \in S$ and for every number $x \in S$, $s(x) \in S$ then $S = \mathbb{N}$.

There is a "formal" elegance to this but it's certainly not "intuitive" and certainly doesn't match the way numbers were taught. The number 8, for example, would look like $s(s(s(s(s(s(s(s(0))))))))$ which is a bit clumsy to use and work with.

Now we could use Roman numerals but that system is also not convenient, especially as we start developing higher ordered mathematics like addition and multiplication.

Most learn numbers by some process of memorizing particular names and symbols. Eventually a general technique (using a base 10 convention) is established for writing numbers and their successors. The former would restrict ourselves to some finite amount (e.g., if our number system only contained 100 elements, then we wouldn't be able to count past 100.). The latter would allow us to potentially go to infinity (a subject that may come up later.)

What is counting?

Going with the theme here, our goal is not to establish some abstract "essence" for what counting IS but rather to establish an operational definition that will enable us to count.

To begin with, let's suppose we have a collection of objects. We're going to assign each object a number with the following stipulations:

1. Every object is to be assigned a number.
2. No two objects are to be assigned the same number. 
3. No object is to be assigned two numbers.
4. The numbers are to be assigned in order. (e.g. if 4 is assigned, then 3, 2 and 1 must be already assigned.)

In this explicit formulation, I'm already presupposing what it means to say there is "one" or "two" of something. This need to be some foundationalist problem; such stipulations would be implicitly learned through a process of training (e.g when a child counts an object twice or forgets to count an object, the child will be corrected.)

The "number" of items will be the last number assigned.

As an example, here we have paired up each penny in a collection of pennies with a number (in this case, we'll be using the standard natural numbers: 1, 2, 3, etc.)

 We have paired up each penny with a distinct natural number (in order) and then say that there are "7" pennies since we "paired up" each penny from 1 to 7.

Counting Uncertainty

Using the above technique is not problematic when we only have, say, 7 items. There is a problem when attempting to count a large number of items. It's possible for a counter to count a large number of items and get a different result each time she counts. This is a problem of uncertainty.

Let's suppose I have 100 different individuals and I ask them to count a large collection of objects. While these 100 individuals may be entirely competent on the counting procedure, they may still come up with different results. The distribution of results will be associated with the uncertainty of the counting procedure. The uncertainty may be measured by some calculation such as the range or standard deviation.

For example, one individual might count 671 items. Another will count 673 and yet another will count 668 items. The "truth" of the matter is likely somewhere in there but there's an uncertainty or lack of precision in our counting procedure.

How to Address Uncertainty

Thus far, I have insisted that counting be given an operational definition whereby I assign each object one unique number. As we deal with larger collections of items, this introduces more uncertainty into the mix.

To address this, other techniques are used instead of counting (as described by the above procedure).

Addition as a Form of Counting

An alternative procedure would be to group items into multiple smaller groups.

For example, if I have 22 items I might group them into a collection of 10 and a collection of 12 items. We then count one group and it totals to 10 and the other totals to 12. We then add the two collections together and arrive at:
$$10+12=22$$
It should be noted that this procedure is different from the previous procedure. If I were an operationalist of the  Percey Bridgman sort, I would contend that these are two distinct concepts.

What I would like to argue is that we may construct what I'll call equivalence classes of procedures. I am here using a more technical concept (equivalence class) in a more metaphorical sense.

Now I contend that this procedure will produce the same result as counting one collection in total. What does that mean?

Roughly what this means is that if I count two groups of items and add them together then I will get the same result as counting the entire group.

But there's a caveat here. Recall the issue of uncertainty. If we have a large enough collection of items then using the counting procedure will get a wide variety of results. So obviously the precise result will not be the same. But the range of results will overlap.

For the time being we will leave this concept of "equivalence classes" of operational definitions aside.

While this procedure of adding multiple groups of items has its appeal, we can still go much further.

Multiplication as a Form of Counting

So another technique to reduce the uncertainty of counting large collections of objects would be to create multiple groups of equal numbers of items. For example, consider this collection of pennies:

With the exception of the stack on the bottom, all of the stacks of pennies can be counted to stacks of 10 pennies in them. The stack at the bottom has only 5.

Since we have 10 stacks of 10 pennies and 1 stack of 5 pennies, we can say that there are 105 pennies:
$$10 \times 10 + 5 = 105$$
or alternatively, we have 2 rows of 5 stacks of 10 pennies plus the 5:
$$2 \times 5 \times 10 + 5 =105$$
In both cases we get the same result as if we were to count them individually.

Now it should also be noted here that I only counted 2 of the stacks of pennies (to 10). The rest I stacked them up until they were the "same height" as the other stacks. So I slightly modified the procedure a bit. I'll say more on that (and other techniques) at a later time.

Quick Summary

I have suggested several distinct methods for counting. The first method consists in "pairing up" each item with a unique number and then saying that the "number" of the collection of items is the last number that was "paired up".

But as I noted, this technique has a decent amount of uncertainty when dealing with large quantities. As a result, two other procedures were suggested: adding and multiplication. I have suggested that these two procedures are forms of counting because they give the "same" results.

While I have not gone into any detail, I have suggested that the range of values that each method provides should overlap with all the others.

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