(2010-06-20) Collins Simmelian Numbers

Randall Collins: Simmelian Numbers. 3 -- The lowest sociological number. This is the cornerstone of Georg Simmel’s “The Significance of Numbers in Social Life” [1908]. 3 is the lowest number for social structures, because it makes possible coalitions of 2-against-1, creating a dimension which transcends the immediate 1-on-1. It also creates the sense of a group, since no one individual can destroy a group of 3 or more by leaving.

ca. 7-10 -- The maximum group size for a multi-sided conversation

Beyond size 7-10, the party breaks up into smaller groups carrying on their own conversations; or else it loses its informal character and turns into a single-speaker/audience (hub-and-spokes) structure

ca. 30 -- The dividing line between the size of an audience where it is comfortable for a speaker to give an informal, spontaneous-sounding talk, and where it feels more appropriate to read a formal, written, pre-packaged speech

ca. 75 -- The Stark-Bainbridge breaking point for cult implosion. According to the analysis of Rodney Stark and William Sims Bainbridge [The Future of Religion, 1986] a religious cult recruited by a charismatic leader’s appeal can grow to about size 75

3-6 to 1 ratio -- The most dangerous number. Photos, videos and narratives of fighting [Collins, Violence: A Micro-Sociological Theory, 2008] overwhelmingly show the pattern of a group of between 3 and 6 members beating up a single victim. This is the pattern in riots, police beatings, gang fights, bullying and other close-range violence.

The most vicious beatings, the atrocities of piling on and overkill, happen in the 3-6 to 1 ratio. 3-on-2 or even 6 or 8-on-2 are not very dangerous ratios; the minority here has the solidarity of backup, and the majority can’t get into the frenzy of emotional domination that they achieve with a sufficiently outnumbered single victim.

Group-to-group numbers

the number of groups interacting in a field

2 -- The number of mutual enemies or factions at the moment of violent confrontation

No matter how many different opposing gangs, ethnic groups, social movements, or armies there are, when it comes to actual fighting, the structure simplifies down to 2 sides

The polarization of violent conflict into 2 sides has the effect that only one line of difference can be recognized while the fight is going on; other issues get dropped or pushed into the background. This applies also to other kinds of intense conflict, such as political factions in a moment of crisis; thus skilled politicians manipulate coalitions by making some of their enemies into allies whose issues are temporarily submerged.

fighters seem to become too disoriented if they have to pay attention to more than one fight.

3-to-6 -- The Law of Small Numbers in longterm intellectual attention space. My evidence on networks of master-pupil chains of philosophers and other intellectuals [Collins, The Sociology of Philosophies, 1998] shows that if there is a period in history where new ideas are produced, there are always between 3 and 6 networks linking the creative figures of one generation to the next. There are always at least two or three major figures at the same time; if a single network dominates, it is not creative (because creativity is negating what exists, taking up an oppositional position in a field).

Some version of the Law of Small Numbers appears to exist in other fields, such as art, music and literature

Politics appears to operate more like a severe conflict field, with a tendency towards polarization into 2 factions at a moment of crisis; what happens in more routine periods of action still needs to be formulated in theoretical terms.

In economic production markets, Harrison White [Markets from Networks, 2002] argues that no single production firm can create a market without having at least one major rival to define the business they are in

These formulations of a Law of Small Number (or family of such laws) are still primitive and need to take into account time patterns

Collins’s 3-6 Law of Small Numbers for intellectual networks deals with intergenerational networks which reproduce themselves for longer than 30 years.

Political parties and social movements go through periods when there are many small contenders; how long does it take for them to winnow down to a small number? Stefan Klusemann’s research [After State Breakdown -- Dynamics of Multi-party Conflict, Violence, and Paramilitary Mobilization, UPenn Ph.D. 2010] shows how large numbers of competing revolutionary and paramilitary movements over a period of 10-15 years consolidate into dominants like the Bolsheviks and the Nazis, creating an authority structure with the ideal-typical number of state monopoly: One.

Numbers in the structure of the self:

3 -- The triadic structure of the self. George Herbert Mead formulated this as I, Me, and Generalized Other; Norbert Wiley [The Semiotic Self, 1994] uses Peirce to reshape the triad as I, You, Me, and to embed a second reflexive triad inside the primary triad of interior dialogue

The human self is reflexive because it incorporates a social viewpoint, along with its own action-viewpoint and an image of itself from outside. This underlines again Simmel’s point that 3 is the first sociological number.

0 -- As Emile Durkheim put it: “The individual, the zero of social life.”