This message was stimulated by John Baez's week257 which, though interesting as usual, has one item of special interest to me at this time. 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. Thanks, Mike
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
mjhealy@ece.unm.edu wrote:
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 doubt if there is much of one. I suppose _some_ sort of connection might be made through the concept of "invariance" - Benford's law holds for distributions that are wide enough not to have a "natural scale". If you cannot give an approximate answer to "how big is a (river/piece of string/data file/bank deposit)?" then the distribution of first digits in (say) centimeters should [waving hands hard] be the same as that in furlongs or wavelengths of green light; and from that property Benford's law follows. On the other hand, humans are approximately of a height, to the point that the foot, hand, cubit, fathom, etc. can be used as rough units in their natural form. Thus there is a natural scale for human heights, and we are not surprised that almost all human heights in meters have a first digit 1 and very few do in inches. -Robert
participants (3)
-
John Baez -
mjhealy＠ece.unm.edu -
Robert J. MacG. Dawson