December 22, 2006

Another trip down Random Rd.

In my last adventure with the random number 17, I followed it back to Princeton mathematicians in the 1960s and speculated about where I got the idea.  My senior thesis adviser Paul Benacerraf was one of three likely sources.  Now Mark Kalderon has written from the Department of Philosophy at University College London to fix on Paul:

Paul Benacerraf was my advisor as well. Seventeen was indeed his favorite number and used it in many examples. He was desperately disappointed when his, then, young son reported that his favorite number was eight. (I tried to cheer him up by suggesting that it was a coded representation of seventeen.) Anyway, Paul attributed his obsession with seventeen to Hilary Putnam, his advisor, who had a pseudo-proof that seventeen was the most arbitrary number. I once saw a copy of the proof in Hilary's hand in Paul's office, but I cannot now remember how it went.

Hilary and Paul confirm most of this, though Paul's not so sure that the proof was written down.  Paul continues:

But it was surely from him that I got it, although I learned later that it was a well-known fact:  True, it was known hereabouts as the Feller Number.  According to my story, he frequently said: "Take a number, any number, say, 17."  Feller, by the way, lived on Random Road -- a further, if indirect, proof of the randomness of 17.  As for Hilary's proof, the one I remember best is this [a proof by cases]:  A completely random number can't [=should not] be too large. Say, </=20. After that it's a breeze.  Working from below, it clearly can't be 1, nor can it be even, and hence neither 2, 4, 6, 8, 10, 12, 14, 16, 18, nor 20 [proof left to the reader]. 3 is for the Trinity and 5 is too important in base 10 notation; 7 and 11 are lucky, and 9 is a perfect square and hence hardly random.  13 is unlucky; 15 is a multiple of 5; and 19 is too close to 20.  That leaves 17 as the only possible candidate. Q.e.d.

If you have a good memory and have been following things carefully, you'll recognize the Putnam proof as a more detailed version of the argument I gave in favor of 17 back in my first posting on the matter.  Pretty clearly I got it from Paul, who got it from Hilary, and then things disappear in the mists of time past.

Some notes and additions:

We had mathematicians Kelly and Spivak putting 17 together with yellow pigs in the 1960s, and then Lander creating a club devoted to the randomness of 17 in the 1970s.  Now Lance Knodel recalls that the name of Lander's club was, yes, the Yellow Pigs.  Cultural continuity!

Once you start looking for 17, you of course find it all over the place.  Stalag 17.  Band names, including (again) Stalag 17, Heaven 17 (in Burgess's A Clockwork Orange and now in real life), East 17.  Alex Baumans suggests that 17 has a special resonance for bands -- though there's the Mile 21 Band, also Level 42, with other significant numbers in their names.

Meanwhile, other candidates vie with 17 for exemplary random-number status.  Don Porges says:

I've always heard that 37 is the number that most commonly comes to mind when people are asked for a random number (between 1 and 100, maybe?)  I was gratified a few weeks ago when Penn Jillette was on Stephen Colbert, and Colbert asked him "What number am I thinking of?" and Penn quite offhanded answered "37".  See video here, 5:00 minutes in...

(Note:  Colbert continues with " was 4".)

Obviously, it's time for someone to do that research.

You could try various ranges.  Maybe 1-20 instead of 1-100.  Here's Jonathan Ferro writing on the range 1-4:

One of my favorite bar tricks is prepared by writing the numbers "1 2 3 4" with a fat pen on one side of an index card, and the number "3" alone on the reverse.  When I have an excuse to be looking through my wallet, or at some other lull in the conversation, I pull out the card and shove it under the nose of an unsuspecting neighbor with the instructions "Pick a number".  I can flip over the card to reveal that I predicted his answer far more often than the 25% predicted by pure chance.  Out of these four numbers, "3" appears to be "more random" than the others.

Well, 1 and 3 are the odd numbers, and 1 is very special indeed, so that leaves 3.  Simple Putnamian reasoning.

zwicky at-sign csli period stanford period edu

Posted by Arnold Zwicky at December 22, 2006 02:25 PM