Saturday, May 28, 2016
Recently, I have been rereading chapters in two or my earlier books, from 1996 and 2003 (2nd edn, 2013), with echos from my 1992 (2nd edn 2012) book. I have been trying to understand why various counting procedures, arithmetically adding up individual interactions in a city--lead to macroscopic accounts that exhibit scaling symmetry (eg. fractals)--things look "the same" at various scales. (In the last twenty years this has been a recurrent theme in empirical research, and actually goes back a hundred years at least, as in Bachelier's work on security prices.)
There are similar phenomena in mathematics, so that a way of packaging all the prime numbers into a single function--Riemann's zeta function--was shown by Riemann (1850s) to be related to a function that exhibited scaling, the theta function (earlier in the century, Fourier used theta to describe the flows of heat), where the world looks the same at various scales. [You know this from the central limit theorem of statistics, where a sum of gaussians is a gaussian at a larger scale.] Or, somewhat differently, in a spatial model of a city, where interactions among people or institutions is only with your neighbors, we might have remarkable city-wide phenomena such as homogeneous neighborhoods and heterogeneity among neighborhoods. [That this might have been sourced in zoning, red-lining, discrimination, Tiebout sorting, geomorphology, is surely the case. But even without such, you get this heterogeneity of homogeneity.] Yet, if it becomes a bit more difficult to maintain such neighborhood interactions (much as amazon.com has upset neighborhood commerce), one again sees such scaling behavior--with homogeneous neighborhoods adjacent to differently homogeneous neighborhoods, and regions that are comparatively homogeneous next to differently homogeneous other regions (each region itself exhibiting that heterogeneity of homogeneity).
My point is that one wants to write in such a way that if you look at what you wrote 20+ years ago, you want to feel that you did a good job then, even if you have subsequently discovered improvements or errors. You want to tell yourself something to the effect that you are surprised that you had been able to do that then. "I knew that then?" (Surely, in retrospect there will be passages you are embarrassed about, as well. Even whole articles.)
The secret is to write carefully, to say just what you know to be the case, to separate out speculation, and to make your arguments clear and cogent. For 20 years from now, you will be the audience, the readers of your work (now) who are not you.