This article nicely articulates why counting might be problematic if counts are not representative of what they’re proposing to measure.
Another way in which it’s very hard to get counting right is when the phenomenon being observed cannot be broken into discrete & fungible pieces — the essence of counting is that there is a meaningful fungible “unit” worth talking about.
One of my favorite illustrations of this is a beautiful excerpt [1] in an RK Narayan novel (Swami and friends) through the eyes of a child who can’t fathom the weird problems in his math homework — because he has a much richer sense of reality than his textbook can ever hope to convey.
Another example context (slightly more “high-brow”) in which fungibility clearly breaks down is fat-tailed distributions. In a thin-tailed distribution (like a bell curve) the fluctuations from “typical” can often be assumed small enough that any measurement/sample could be fungibly replaced by the typical (Eg: designing furniture for the typical person). Counting, averages, etc are only guaranteed to make sense in this situation. In a fat-tailed distribution, OTOH, it’s cumulative properties might be dominated by outliers — so trying to characterize the system based on counts might often be quite meaningless! (Eg: designing buildings for the typical earthquake or the typical tsunami)
(There’s a lot of nuance, but I’ll leave it at that)
> Rama has ten mangoes with which he wants to earn fifteen annas. Krishna wants only four mangoes. How much will Krishna have to pay?
If Rama has a transactional perspective on business and has the local monopoly on mangoes, and Krishna really needs mangoes right now and will accept no substitute, Krishna might have to pay fifteen annas for four mangoes. Or maybe one hundred annas per mango. Or one thousand. Is there any limit?
If there are competing mango vendors who are offering five mangoes for five annas, Rama may have difficulty convincing Krishna to pay more than four annas for four mangoes.
If Rama expects that competing mango vendors will run out of stock early, but some customers are likely to still come toward the end of the day, Rama's price per mango may depend upon the time of day - it may remain quite high throughout the day compared to other vendors and perhaps only drop if Rama fears ending the day without selling all the stock.
I remember the first story my (baby) economics professor told me:
A local gas station changed gas prices to above a dollar and got a visit from the Sheriff's office. The demand was simple: you can either continue to charge ninety nine cents or you can close the pump. However, due to a chance of public unrest, you may not charge a dollar or more at this time.
What the professor didn't have to say is that in real life local authorities have incredible latitude and we must comply at least while we "work the courts". I think the morale was that free enterprise doesn't exist anywhere in real life and probably for good reason.
Very good, and I'll add the "denominator problem."
Whenever you take an average, you divide by some count. What you choose to count matters, and people can disagree about what belongs in this set. It's often a judgement call.
For example, just recently there was a study of vaccine effectiveness in Israel and it appears they made a mistake about the denominator. (Or did they?)
I feel like this article uses the term counting, when really they are more generally talking of the process of measuring and/or sampling.
Every measurement is a process of some imprecision - in experiment in itself lending itself to needing some sort of calibrating process or set of experiments.
I think using the terms counting and even systemization instead of the term measurement or sample would cause the readers to miss tapping into a large body of existing scientific and technical work that is relevant to what is being discussed in the article. Eg of you look for measurement theory a reader would find a lot of relevant info.
Maybe though to give credit to the author, they’re looking for a focus on a place where one is just starting to form different concept of a particular estimate.
Another way in which it’s very hard to get counting right is when the phenomenon being observed cannot be broken into discrete & fungible pieces — the essence of counting is that there is a meaningful fungible “unit” worth talking about.
One of my favorite illustrations of this is a beautiful excerpt [1] in an RK Narayan novel (Swami and friends) through the eyes of a child who can’t fathom the weird problems in his math homework — because he has a much richer sense of reality than his textbook can ever hope to convey.
Another example context (slightly more “high-brow”) in which fungibility clearly breaks down is fat-tailed distributions. In a thin-tailed distribution (like a bell curve) the fluctuations from “typical” can often be assumed small enough that any measurement/sample could be fungibly replaced by the typical (Eg: designing furniture for the typical person). Counting, averages, etc are only guaranteed to make sense in this situation. In a fat-tailed distribution, OTOH, it’s cumulative properties might be dominated by outliers — so trying to characterize the system based on counts might often be quite meaningless! (Eg: designing buildings for the typical earthquake or the typical tsunami)
(There’s a lot of nuance, but I’ll leave it at that)
[1] https://potterfiend.blogspot.com/2006/03/swami-and-friends-m...