I was working on a paper, writing the introduction to a new section that deals with an extension of the basic model. It’s a relevant extension because it fits many real-world applications. So naturally I started to list the many real-world applications.

“This applies to X, Y, and….” hmmm… what’s the Z? Nothing coming to mind.

But I can’t just stop with X and Y. Two examples are not enough. If I only list two examples then the reader will know that I could only think of two examples and my pretense that this extension applies to many real-world applications will be dead on arrival.

I really only need one more. Because if I write “This applies to X, Y, Z, etc.” then the Z plus the “etc.” proves that there is in fact a whole blimpload of examples that I could have listed and I just gave the first three that came to mind, then threw in the etc. to save space.

If you have ever written anything at all you know this feeling. Three equals infinity but two is just barely two.

This is largely an equilbrium phenomenon. A convention emerged according to which those who have an abundance of examples are required to prove it simply by listing three. Therefore those who have listed only two examples truly must have only two.

Three isn’t the only threshold that would work as an equilibrium. There are many possibilities such as two, four, five etc. (ha!) Whatever threshold N we settle on, authors will spend the effort to find N examples (if they can) and anything short of that will show that they cannot.

But despite the multiplicity I bet that the threshold of three did not emerge arbitrarily. Here is an experiment that illustrates what I am thinking.

Subjects are given a category and 1 minute, say. You ask them to come up with as many examples from that category they can think of in 1 minute. After the 1 minute is up and you count how many examples they came up with you then give them another 15 minutes to come up with as many as they can.

With these data we would do the following. Plot on the horizontal axis the number x of items they listed in the first minute and on the vertical axis the number E(y|x) equal to the empirical average number y of items they came up with in total conditional on having come up with x items in the first minute.

I predict that you will see an anomalous jump upwards between E(y|2) and E(y|3).

This experiment does not take into account the incentive effects that come from the threshold. The incentives are simply to come up with as many examples as possible. That is intentional. The point is that this raw statistical relation (if it holds up) is the *seed *for the equilibrium selection. That is, when authors are not being strategic, then three-or-more equals many more than two. Given that, the strategic response is to shoot for exactly three. The equilibrium result is that three equals infinity.

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November 2, 2011 at 1:23 am

Will JohnsonAs interesting as your theory is, I think it instead has much more to do with sound / flow / rhythm / etc. (ha!) than with actual truth. Specifically, what you’re describing is just a specific example of the rule of three:

http://en.wikipedia.org/wiki/Rule_of_three_%28writing%29

This rule is a rule not just in academic papers but also in jokes, slogans, stories, etc. (ha!)

November 2, 2011 at 1:44 pm

dan s3 implies E(examples)> 3, but not infinity. readers know not all authors who state 3 have unlimited

November 2, 2011 at 1:56 pm

WernherWhy three? – Well: if you have a hypothesis to prove false or right, then a population of three is the minimum you need for a majority vote in any empirical approach. That’s why I favour three validation cases over two. Since there is a cost associated with validation (e.g. of large software solutions as developed in European research projects), you try to minimize the cost that way. Nothing to do with infinity – on the contrary: the minimalist meaningful statement you can make about your hypothesis.

yours truly – 50something🙂

November 3, 2011 at 12:43 am

AnonymousIn medicine, if I’ve seen a disease once I can say ” In my experience”. Two cases- “In my series of cases I have seen” but if I have seen something three times , I have seen it again and again.

November 4, 2011 at 8:22 am

Dan HirschmanThis reminds me of Asimov’s “The Gods Themselves” where scientists determine that at least two universes exist, and then argue that “[T]he number two is ridiculous and can’t exist.” See also http://en.wikipedia.org/wiki/Zero_One_Infinity

November 10, 2011 at 3:30 am

anonymousi’m sure rick perry agrees with your view.