3 -- The lowest sociological number. This is the cornerstone of Georg Simmel’s “The Significance of Numbers in Social Life” . 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. John Levi Martin [Social Structures, 2009] argues that dyads are always experienced against the background of other possible relations. Dyadic withdrawal-- such as lovers turning to each other and ignoring the rest of the world-- is special precisely because there is a rest of the world, a third party that makes an outside against their inside. Sociologically, 2 is really derivative of 3.
ca. 7-10 -- The maximum group size for a multi-sided conversation. In informal sociable situations like a party or a group gathered around a dinner table, there is often a single conversation in which everyone is paying attention. 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-- for instance when someone makes a speech or is given an award, or a guest of honor monopolizes everyone’s attention. The number zone 7-10 is the borderline between informality and formality, as we observe it micro-interactionally.
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. Below 20 or 30 people, it feels artificial to read a speech; above 30 or so, it is hard to talk casually -- what Durkheim called the demon of oratorical inspiration. The borderline is higher for professional speakers, but this means persons who have had a lot of practice and hence their speech is in fact pre-packaged. Announcers, actors, or modern-day politicians who sound comfortably casual before a large audience are putting on a frontstage performance of casualness based on a large amount of backstage training.
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; but to get beyond this size, it must delegate recruitment to disciples and their own networks. At size 75, the leader becomes smothered by the attentions of his/her followers, and is unable to spend much time adding more members. Cults of this kind implode-- they break off network connections with the surrounding society, turn in on themselves, and take their ideology to an extreme polarization between themselves and the world. This is organizational suicide for a social movement, since it can no longer grow; sometimes it literally leads to mass suicide.
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. In a riot, there may be large numbers of other people standing at a distance, but the actual beating takes place by little clusters of 3-6 against an isolated one. 1-on-1 is not a very dangerous ratio; most such confrontations are standoffs, or if there is an audience who encourages the fight, it usually is a restricted and rule-bound “fair fight.” 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.
[The main exception to the 3-6 to 1 ratio is in domestic violence, where emotional dominance is usually established by an adult male who is much bigger, stronger, and more vehement than his victims. The dangerous number-ratio holds in public violence. Why the domestic arena has a different structure needs further analysis.]
Group-to-group numbers: (Here we are not concerned with the size of a group but 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. Hobbes’ war of all-against-all is purely mythical; it has never been observed in any serious violence. [Water-splashing fights or food fights are an exception, but these are non-serious and playful in tone.] 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.
Why intense conflict polarizes to 2 has not been well-explained; but violence is tense and confusing enough, and fighters seem to become too disoriented if they have to pay attention to more than one fight. (See also Violence, chapter 7, for evidence on the one-fight-at-a-time limitation on fights in bars and entertainment venues.)
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). There is an upper limit of 6 such networks; in exceptional periods where more than 6 schools thrive at the same time, several of them fail to recruit new followers and die out in the following generation.
Some version of the Law of Small Numbers appears to exist in other fields, such as art, music and literature. But the numbers at the upper limit may differ. 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; he implies there is an upper limit in the form of a diminishing tail of market share, dropping off sharply above about 6 or 8 firms.
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. Numbers of competitors in an economic production market (such as personal computers in the 1980s) can be much larger; we need to specify the time period (which may be only a few years), and the dynamics that moves us from one time-spread to another. 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. Ogden and Richards’ The Meaning of Meaning,  argued that any unit of significance must have the triangular structure of sign, object signified, and larger context of discourse-- a formulation made by different thinkers in various terminologies. 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. One might argue that an infant-with-mother is a primal dyad; but the baby grows into a social actor and human thinker by acquiring the 3-pointed structure of the self.
0 -- As Durkheim put it: “The individual, the zero of social life.”
Explaining why such Simmelian number-patterns exist will advance us deeply in sociological theory.