From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4024 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Benford's law Date: Tue, 16 Oct 2007 19:51:49 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019672 11375 80.91.229.2 (29 Apr 2009 15:41:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:41:12 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Wed Oct 17 14:15:22 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 17 Oct 2007 14:15:22 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IiCFm-0000k8-Ig for categories-list@mta.ca; Wed, 17 Oct 2007 14:00:18 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 81 Original-Lines: 62 Xref: news.gmane.org gmane.science.mathematics.categories:4024 Archived-At: Mike writes: >I haven't yet looked at Minhyong Kim's work, and I don't know how this >fits in with number theory or categories, but a friend is encouraging me >to go to the following conference on Benford's Law: >http://www.ece.unm.edu/benford . > >Does anybody on this list (including you, John) know of a connection >between Benford's Law and any work in category theory? I would really >like to hear about it if so. I don't know any interesting connection between this law and category theory or number theory. I didn't know it was called "Benford's law", but I knew the idea: if you take a table of widely spread numbers (say the gross national products of nations, or the incomes of Americans), often about log 2 ~ 30% will have 1 as their first digit, about log 3 - log 2 ~ 17% will have 2 as their first digit, and so on. It's easy to derive this law from the assumption that the data is distributed in an approximately scale-invariant way within a certain range. (That is, the percentage of numbers in your table between X and cX is about equal to the percentage between Y and cY, for c not too big, and X and Y within some large but finite range. Or: the logarithms of the numbers are approximately uniformly distributed over some interval.) So, the mystery of Benford's law reduces to the mystery of this fact: in practice, widely spread numbers are often distributed in an approximately scale-invariant way, within some range. (Perhaps some people find Benford's law mysterious because it's impossible for a probability distribution to be *perfectly* scale-invariant. But that's a red herring. It's enough to have approximate scale-invariance within some range, for example a couple powers of 10.) Why is approximate scale-invariance so common? People have written books on this! Here's a nice one: Manfred Schroeder, Chaos, Fractals, Power Laws, W. H. Freeman, 1992. Or, for starters: http://en.wikipedia.org/wiki/Power_law I would rather go to a conference on power laws than a conference on Benford's law, which seems like just a spinoff. Best, jb