December 21, 2006

The magic number 17

A while back I considered Christopher Buckley's picking 17 as the number of Eskimo words for snow:

Buckley has obviously pulled the number 17 out of his, um, hat.  This number is what you're likely to come up with when you're asked to pick a random number: it's the smallest prime number without any special cultural significance.  The numbers 2, 3, 5, 7, and 13 are clearly special; 11 is not quite so special, though it is the number of players on a football team (American or Association), and then you're up to 17.

That elicited a wry comment from John Cowan, plus several queries about whether research had actual been done on the question.  I didn't know of any at the time, and I still don't, but I can report on some famous research on a related question.

Cowan got the paradox in my original posting -- I thought about whether I should point it out and eventually decided to just leave it out there -- and elucidated it poetically:

Thanks for explaining to us
    readers of Language Log the special cultural
        significance of the number 17.

Then came Mike Morrison and Natan Cliffer, asking about research.  Cliffer had a special interest:

I lived in Random Hall (a dorm) during my undergrad years [at MIT], and that assertion is part of the mythology of the culture there...

Well, it wouldn't be terribly hard to do research on the question, though just asking your friends to name a random number wouldn't do; you'd need a carefully designed questionnaire, with suitable distractor items ("Name a bright color", that sort of thing).  Whether such a study would come up with 17 as the favorite random number is an open question, but it's very likely it would show that people treat randomness as a gradable property of numbers: some numbers are more random than others, in the minds of ordinary people, even though such a view doesn't make sense mathematically.

This is the sort of result that Armstrong, Gleitman (of this parish), and Gleitman got when they investigated people's views of exemplars for various categories that are usually thought of as well-defined rather than gradable: odd number, even number, female, and plane geometry figure.  The paper is:

Armstrong, Sharon Lee; Lila R. Gleitman; & Henry Gleitman.  1983.  What some concepts might not be.  Cognition 13.263-308.  [Available on-line here.]

Their study also looked at exemplars of prototype categories that other people had studied: sport, vehicle, fruit, vegetable.  Rather than directly asking people to supply exemplars, they replicated two experiments reported on by Eleanor Rosch in 1973 -- one in which people were ask to give a rating to exemplars on a 7-point scale, the other measuring response times in judging whether sentences like "An orange is a fruit" and "An orange is a vehicle" were true or false (the idea here is that less good exemplars of a category take more time to verify as members of that category than good examples do).   The results for "odd number" etc. were comparable to those for "fruit" etc.  (There was a third set of experiments not directly relevant here.)

AG&G also mention an earlier similar study by Eric Wanner on "prime number".

So if anyone decides to replicate Rosch's experiments with "random number", or just to look at the exemplars people supply for the category "random number", we should expect that they'll find gradability here too.  But the final answer doesn't seem to be in yet on 17.

zwicky at-sign csli period stanford period edu

[More here and here.]

Posted by Arnold Zwicky at December 21, 2006 12:33 PM