Size-Density from Time-Minimization

Philosophy is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.

Galileo Galilei (1564-1642)

The line is pictured in Fig 8-1. We may write its formula as

(1)

FIG. 8-1. WORLD REGRESSION LINE (N = 1,764 primary administrative subdivisions of 98 nations)

Between 1972 and 1975 a number of studies produced size-density slopes which did not depart significantly from this -2/3 value. Each time it happened I wondered Why? I tried to derive the slope theoretically (i.e., mathematically, from initial assumptions), but I failed each time until Spring term, 1975, when I was working with Jay Callen.

Dimensional Analysis: Time

Jay had been doing a lot of work with "central place theory", much of which involves explaining empirical regularities in terms of least distance models.^[1] As it happened, I had been re-reading Zipf's principal work in which empirical regularities were derived in accordance with his principle of least effort.^[2] At some point in all this I recalled a remark made by Professor Anderson, in a graduate school seminar in mathematical modeling, to the effect that well-developed theoretical fields tend to select fundamental dimensions and define other quantities in terms of these.

The classical example of such "dimensional analysis" is Newtonian Physics. Length is clearly a dimension, in a real sense the most basic one we have. With length, plus a few other dimensions defined in terms of length (e.g., time and mass — the distance a marker moves on a proper scale), other quantities can be defined as in the following table, the last column of which contains only s, m & t — length, mass and time:

 Variable        Symbol   Formula   Dimensions
 length, distance     s         s            s 
 area                 A        sw           s2 
 volume               V       swh           s3 
 density              D       m/V         ms-3 
 velocity             v       s/t         st-1 
 acceleration         a       v/t         st-2 
 momentum             p        mv        mst-1 
 force                F        ma        mst-2 
 work                 W        Fs       ms2t-2 
 power                P       W/t       ms2t-3 
 gravity constant     g         g         st-2   
 potential energy    PE       mgh       ms2t-2 
 kinetic energy      KE   1/2 mv2       ms2t-2 

Among other advantages, such reduction of all variables to one (or a few) dimensions makes it possible to rigorously "balance equations" dimensionally — have the same units on both sides of the equation, or specify the units for embedded "constants." Different sciences use different fundamental dimensions, e.g., Chemistry - energy; Economics - money (but see the "Labor Theory of Value" below). I recall Professor Anderson suggesting at some point in our conversations that one day, perhaps, time might prove to be a fundamental dimension in Sociology. I began to think about least-time.

Economics seems limited, as a general social science: what would its least-cost model mean in primitive societies which lack a money economy? In a similar vein, what could Geography's least-distance theories contribute to an understanding of future societies with "beam me up" technologies? I realize these are fictional, but our transportation systems make our ancestors view of distance hopelessly out of date also. Primtive societies where money is meaningless, future societies where distance is meaningless — how do you theorize about both using the same social science model? Zipf's least-effort notion was a conscious attempt to combine monetary and distance (and other) costs, but it smacked of mixing apples and oranges (dollars + miles + sweat?), and it lacked rigor: how, exactly, do you measure (quantify) "effort"?

The solution, it seems to me, is to translate these disparate variables into the single dimension of Time. If I know the average velocity of a given transportation technology, it is easy enough to translate distance into time (distance is time, multiplied by veloctiy; time = distance/velocity). Depending upon whether I use a horse or an automobile, the county seat may be either a day or a

The Labor Theory of Value We see then that that which determines the magnitude of the value of any article is the amount of labour socially necessary, or the labour-time socially necessary for its production. Each individual commodity, in this connexion, is to be considered as an average sample of its class. Commodities, therefore, in which equal quantities of labour are embodied, or which can be produced in the same time, have the same value. The value of one commodity is to the value of any other, as the labour-time necessary for the production of the one is to that necessary for the production of the other. "As values, all commodities are only definite masses of congealed labour-time." - Karl Marx, Capital I:1:i

half-hour away. Time is money, according to Franklin; money is time, multiplied by the rate at which it accumulates (time = money/rate): so much income, expenditure, rent, or interest per hour, day, month or year. Even effort, though poorly defined itself, might be translated into some expenditure of work-hours, given a level of technology (e.g., how long would it take someone to dig a given trench with no tools, with a shovel, with a backhoe?)

Pete Phillips San Francisco, June 1999

These thoughts took me back further in my education, to an undergraduate course in the history of social thought, taught by Andrew P. Phillips at what was then San Francisco State College. Along with the ideas of Comte, Durkheim and Weber, it was there that I encountered the social thought of Karl Marx and, incidentally, his labor theory of value: the value of a commodity is a direct function of the average socially necessary labor time required for its production. When I first heard of this idea I appreciated the fact that, unlike most other social science theories, it aimed at being cross-cultural, pan-historical, i.e., general — presumably applicable to all societies, even non-monetized or hyper-technological ones.

At the time I learned of these ideas of Marx, I was taking a Social Psychology course in which I had occasion to review ideas of a number of philosophers about the "essential difference" between human and animal life. The text and teacher made much of so-called "symbolic interactionism", the doctrine which asserts that the fundamental difference between humans and animals is our reliance on "meaningful symbols" (vs. mere "signs" like barks and chirps). It didn't seem to me that this was much of an improvement over the ancient metaphysical assertion that we differ by virtue of our capacity for thought (in fact, Aristotle's notion of us as the zoon politikon seems a little more sophisticated). That was back before contemporary scientists began studying animal language. Against this notion that thought is what distinguishes us from other animals, or that "symbolic interactionism" should be the (dimensional?) basis of Sociology, let me quote one of my favorite lines from Shakespeare:

But thought's the slave of life, and life time's fool;
And time, that takes survey of all the world, must have a stop.^[3]

Thought's the slave of life (all life), and life is time's fool — it could as easily have been written by Marx, or any biologist for that matter. I don't understand why Sociology (other than the classic intro text by the Lenskis) doesn't begin with Biology as Auguste Comte originally suggested. It would be a small step from there, in the study of human or non-human social systems, to see how relevant time really is as a fundamental dimension.

Just for fun, Table 9-2 contains a very small sample of Shakespeare's many other references to time. I once shut down an online literature search engine by trying to get it to list all his references to "time"; next time I tried it "time" had been removed from searches, treated as too common along with words like "the" and "is"):

Time Minimization Theory

Begin with this: Assume that social structures evolve in such a way as to minimize the time expended in their operation. Under this constraint, determine how large a county (or other territorial division) should be (in order to enforce law, maintain order, provide services, or whatever).

One way of thinking about the value assumed by any variable is to assume it to be the result of contradictory forces: those tending to increase it, and those tending to decrease it. The larger a county is, other things equal, the more people there are to pay for its operations. On the other hand, the larger it is the harder it will be for distant citizens to reach, or be reached by, the county seat of government. Originally, the "maintenance" time of a county was donated by people who volunteered their time as its officials. With the growth of county government, however, the operations of a county are carried out by paid employees. The money required to pay them comes from taxes and other sources. Ultimately, however, we may imagine the money as the result of time expenditures by those making up the county (e.g., as taxed income, itself a product of time).

Sharing the maintenance costs creates a centrifugal force, a tendency to increase the size the county, but as distance to the center increases the "interaction" time also increases, creating a centripetal force, a tendency to decrease county size. It seems reasonable to assume that actual county sizes represent a balance of these opposing tendencies, a minimizing of both "maintenance" and "interaction" times somehow taken together.

We don't know what the maintenance time actually is, only that it represents some expenditure of hours, h. We also don't know how the burden is distributed among the citizens. We can refer to counties as being more or less costly in terms of average or per-capita time expenditures,

The time expended in the operation of this social structure is the sum

total time = maintenance time + interaction time

(2)

We are trying to find an optimum value for A, the area served by the county, but A does not appear in Eq. 2, at least not directly. The definition of density is population divided by area

(3)

(4)

FIG. 8-2. CONTRADICTORY EFFECTS OF AREA ON TIME EXPENDITURE

FIG. 8-3. TOTAL TIME EXPENDITURE AS A FUNCTION OF AREA

Taking the derivative^[5] of T with respect to A gives us

(5)

(6)

(7)

(8)

(9)

Lack of Intentionality in Time Minimization

I want to emphasize something about time-minimization as I have used the term: I make no assumption about people consciously or purposely minimizing time. Least-distance and least-cost theories are often put forward as criteria for rational decision making. Zipf presents his own theory of least-effort as similarly rational, conscious, purposive. We feel comfortable enough with models which describe the way a decision maker makes rational decisions employing one or more of these criteria. And rational decision makers certainly do try to manage time expenditure (think of all the time-and-motion studies done in industry throughout this century).

In contrast, I am explicitly not trying to specify rules to be used by a rational decision maker. Nor am I in any way asserting that people are inclined toward "rational" behavior. I am only trying to account for the relationship described by Eq. 1. The relationship between size and density results from, and describes, a pattern of territorial division which is the product of countless numbers of decisions, more-or-less rational, by countless numbers of people, over very long periods of time and often huge expanses of space. The actors involved were effectively unconscious of one another and certainly unaware of the size-density relationship itself. I have no reason to assume any of them consciously tried to establish a negative two-thirds size-density relation or that they purposefully minimized time.

What I am putting forth here is pure theory, invention, fiction, imagination. I am only trying to state the simplest assumption which, if true, would account for the observed results. I make no assertion of its truth (in fact, I don't think most people do consciously minimize time, most of the time; they even "waste" it doing things they "like" - see below). It is as if I had observed the relation between temperature and pressure in a contained gas and began to speculate about perfectly elastic tiny billiard balls whizzing around in perfectly empty space, i.e., the kinetic theory of gases. In fact, the particles of a gas tend to be not-perfectly-elastic, double-atom molecules, whizzing around in not-perfectly-empty space. But it doesn't matter: the model is good enough to make sense of the results (at the level of a simple gas law). That is all I am doing with the assumption of time-minimization.

I belabor this point because I think Sociology tends to misunderstand the nature of scientific theory. Whereas most scientific explanation accounts for patterns in the recurrent behavior of very large numbers of events (atoms, molecules, cells, organisms, species, planets, galaxies), Sociology tends toward a highly individual-voluntaristic view of its "actors". When we think of crime, we usually picture a criminal rather than a crime rate; when we think of adolescence or adolescents, we seldom envision a population pyramid.

I illustrate this tendency in class lectures with reference to an exhibit of 12th century Chinese scroll paintings I saw at the DeYoung Museum in San Francisco when I was an undergraduate. The right-hand side of Fig 8-4 is a badly beat-up poster I still have from that exhibit. The two pictures on the left are details from the center-right (houses) and lower-left (some people) of that poster. Some of paintings were enormous, as much as 50 feet when fully unrolled. Ordinarily they were scrolled, a slave standing on either side, slowly revealing portions, so that the viewers, lounging before the changing scene, had a sense of taking a journey through vast regions. The painting was supposed to inspire thoughtful commentary, sometimes written on the painting itself in the form of poetry.

FIG. 8-4. EMPHASIS ON THE EXTERNAL WORLD

I was fascinated by this view of the place of humans in nature. Here are the sky, mountains, valleys, river gorges. There may be a forest or an orchard, a bridge. Often people are shown amidst the buildings or natural surroundings. Who are they? How are they grouped? The people in such a picture are subject to all sorts of laws: meteorological, biological, physical, nutritional, sociological. You don't have to enter their minds to account for their behavior any more than you have to enter the minds of molecules or monkeys to explain theirs. There is a structure in which these people, and thousands of others elsewhere and earlier, live out their lives.

The other rooms in the museum at that time were populated with European works, mostly portraits, suggested here in Fig 8-5. Though some of them had great artistic merit, their view of human nature was very different from you get in Fig. 8-4 The entire emphasis here is on the individual and his or her mood or character or claim-to-fame. About the only hint you get that they even live in a world outside themselves might be in their title (the 15th Earl implies that there were 14 before him) or the placement of a pillar in the background (his pillar in his palace in his world).

FIG. 8-5 EMPHASIS ON THE INDIVIDUAL

There is a similar contrast in point-of-view in much literature. Moby Dick begins with the highly personal

Call me Ishmael. Some years ago—never mind how long precisely—having little or no money in my purse, and nothing particular to interest me on shore, I thought I would sail about a little and see the watery part of the world.

and ends after much struggle with

Now small fowls flew screaming over the yet yawning gulf; a sullen white surf beat against its steep sides; then all collapsed, and the great shroud of the sea rolled on as it rolled five thousand years ago.

Huckleberry Finn, though less cosmic, moves similarly from the unique to the recurrent. It begins with "You don't know about me...." and concludes: "... Aunt Sally she's going to adopt me and civilize me, and I can't stand it. I been there before."

I don't see how anyone could theorize about human behavior when humans are viewed solely as willing, "motivated" individuals. (Isn't "motivation" or "motive force" what got all the pre-Galilean physicists back to Aristotle confused with regard to rest versus motion?) Each individual is just that: a genetically unique entity which, even if it were cloned a thousand times, would still be unique in terms of subsequent individual experience and personal "motivation". Predicting behavior would be a matter of getting to know that individual well enough to guess the next move correctly. The knowledge would not generalize beyond that individual psyche. Your intense love of football probably has little to say about mine for opera; your need to golf has little to do with my need to solve puzzles. If science is the study of recurrent events, then it has nothing to say about individuals and their "likes" or "wills" or "wants" or "motives". It should rather focus on the "common denominator" which by assumption underlies all recurrent actions and their patterns, something such as time-minimization.

A theory of human behavior which "makes sense" at an individual level (rational choice or libidinal repression) ought to be suspect. Theories should account for patterns which appear in macro-level data. A theory like time-minimization should not be judged in terms of intuitive "reality"; it need only account for the empirical finding better than any competitor.

Publishing the Derivation

I submitted the derivation to the American Sociological Review on April 8, 1975. The rejection arrived on June 9. One reviewer had no opinion regarding publication, but did offer some hints for tightening the presentation. The other wrote that

The basic flaw in the analysis is the reliance on the characteristics of an imaginary average region. A truly operational model would use the real average distance to regional centers...,