An analysis of counting and countability

This article by Dan Polansky investigates the applicability of the concept of counting. It focuses on counting the entities of the empirical world; it mentions counting of mathematical objects in passing.

Counting of abstract mathematical objects (which are immutable) is generally straightfoward as long as there are finitely many of them. Counting infinitely many mathematical objects is a problem solved with the use of set theoretical cardinal numbers. Thus, we can ask how many integers there are, how many prime numbers, how many prime numbers less than 1,000,000, how many nodes in a graph, how many strongly connected components in a graph, but also how many real numbers, how many subsets of a particular infinite set, how many functions from N to N, how many functions from R to R, etc.

It must have been Wiener in his Cybernetics who mentioned that it is much easier to count stars than it is to count clouds. Sometimes, clouds do have an appearance of discrete objects that can be counted; often, not so much.

Counting well defined physical objects not undergoing multiplication seem straightforward enough. Thus, we can count plates in a box, lentils in a package, or planets in the Solar system (but beware of the dwarf planet Pluto).

Counting of cells in a biological body presents a problem of counting of entities that are in the process of being split and thereby multiplied. A cell that is in the middle of the ongoing split can arbitrarily be counted as either one or two cells. An objector to the concept of counting could thus charge there is no such thing as the number of cells in the body at a given point in time. But that is untenable. At least, we can obtain a low estimate and a high estimate of the cell count by either erring on the lower side for all cells being split or on the upper side. We could also assign a non-integral number to each cell that is being split depending on how far it is in the process of splitting, add up these numbers, and take the floor value of the sum. Thus, we are forced to admit that the concept of the cell count is not as clearly defined as it is for counts of mathematical objects, but we also point out that it still makes sense to talk of cell count for various analytical purposes.

Counting of events can have very material consequences. It must have been 9/11 attacks that lead to a dispute concerning how many insurance events this produced. Thus, counting something as two insurance events rather than one leads to receipt of the double of a large amount of money; that seems worth philosophising about the concept of counting.

Counting abilities have been found in animals.

Counting entities of the empirical world makes some basic things clear: 1) there are sufficiently well delineated/well defined and sufficiently stable individual instances in the world (e.g. the present speaker, this cat right here, that tree over there, the main river in the city where the speaker lives, etc.); 2) there are classes and characteristics that as if "reside" in these instances and enable selections of instances into sets for the purpose of counting. Thus, starting with the subject of counting, one may branch off to an arguably rather off-topic tangent of ontology.

Counting units of measure seems to be a special case. Thus, if one wants to count the centimeters that make up the length of a desk, one has to accept that the centimeters are not visually or haptically delineated in any way. Rather, one perhaps asks the following question: if I had same-length objects called centimeters and I would place one next to the other along the length line of the desk, how many of these objects would I need to place to cover the length? (The result would not necessarily be integral, but that is a separate issue.)

As something of an aside, it is the positional numeral system (contrasted to e.g. Roman numerals) that makes counting very practical. But Roman numerals also do. For counting small counts, one may even use an unary system, which is typographically exeplified by stating, that the number of seven is the number of "s" in "sssssss".

As something of another aside, let us have a closer cursory look at cardinal numbers in mathematics. A cardinal number is something that captures one class of cardinality, that is, one answer to the question "how many?". Two sets belong to the same class of cardinality (informally, have the same number of elements) if there is a one-to-one mapping between the two sets. Thus, there is a mapping (let's call it "friendship") of one set to another such that each element in any of the two sets has exactly one friend: no element is left alone and no element has two or more friends. A key result using this definition is that a power set has strictly more elements than the base set, where the power set of X is the set of all subsets of X. The proof uses Cantor's diagonalization.

Further reading

• Counting, wikipedia.org
• Cardinal number, wikipedia.org
• Positional notation, wikipedia.org