Thomas J. Fararo’s (2000) review of theoretical sociology reveals extraordinary complications in theoretical sociology in the twentieth century – complications largely concerned with differences in mathematical approaches and mathematical units of analysis.
Why should this be? In the two scientific fields that have progressed the most in mathematical approaches in the last 100 or so years, physics and economics, and also in the field of mathematics itself and its branches, a notable difference from mathematical sociology and mathematical psychology is the openness of the former fields to non-data-based exploratory analysis in the preliminary phases of investigation as a general rule. Mathematical sociology and mathematical psychology take great pains to insure that nothing gets into the exploratory stage which is not thoroughly sifted and filtered as belonging to some “proper” existing theoretical school and even theoretical unit of analysis like a group or an individual respectively. This helps protect the discipline’s “integrity”, but to what extent does a discipline need to be protected beyond what science and logic already do?
My suggestion is that the most important factor in mathematical progress is not better data but better theory focused on the data that one starts with (even if the data is rather rough and small). Roughly speaking, if one wants to prove that most grass is a certain shade of green, it is more important to search for and understand theories of grass “given” the data that we have already seen (whether random or not!) than to devise ingenious ways of collecting new samples of grass. Eventually, it will be good to have new samples of grass – but it is not the most urgent initial requirement.
Since this is rather abstractly stated, I will illustrate this with two new mathematical methods, one new to sociology but used outside sociology already, and the other new to all quantitative fields. Each method will be applied to problems which have not been solved, much less attacked, with currently existing sociological methods – one or two problems for each method.
The two methods are Probable Influence (PI) and Dimensional Analysis (DA for short),the former new to all quantitative fields and the latter new to sociology but already used in engineering and physics extensively.
Probable Influence (PI) was discovered by Doctorow and Doctorow (1980-81, 1983) and applied to various fields “culminating” with theoretical physics in Doctorow (2000), although it has been developing even more rapidly since then. It is simply the probability that A influences B where A and B are set/events (although it can be extended to processes), which can be interpreted as the probability that A causes B. Symbolically, it is written P(A←B), where (A←B) is defined as an influence set which is (AB’)’ which reduces to A’ U B where A’ is the complement of A (the part of the universe outside A) and U is set union (and/or) and AB’ is the intersection (and) of A and B’, and P( ) is probability of. By the laws of elementary probability, P(A←B) reduces to 1 + P(AB) – P(A) where P(AB) is the probability of A intersect (and) B. Readers familiar with conditional probability P(B/A) or more commonly written P(B|A) will recall that conditional probability is P(AB) divided by P(A) when P(A) is not 0, so remarkably even though it was not planned like this, P(A←B) replaces division in P(B|A) by subtraction and adds 1.
Dimensional analysis was formally developed by Buckingham (1914), and is used today in almost every introductory physics and introductory engineering textbook as well as in certain branches of science for exploratory purposes including fluid physics and fluid engineering/hydrodynamics, astrophysics and cosmology, etc., and in mathematics especially in differential equations, e.g., Bluman and Kumei (1989) who generalized it to Lie group and Lie algebra point transformation analysis. Bruno, Doctorow, and Kappner (1981) applied it to population migration with special population dimensions, and De Jong applied it to economics with new dimensions such as satisfaction, labor, money.
There is little doubt that both PI and DA are not so much data-oriented as theory-oriented and very useful in exploratory analysis. For example, one reason for the importance of DA in hydrodynamics and fluid dynamics is the fact that these fields are still largely exploratory and that more data simply doesn’t do the job, and similarly in cosmology and astrophysics and population dynamics and much of economics. PI is even more obviously theory-oriented because (A←B) is the set/event analog of the logical conditional (or logical implication) a←b defined as ~(a ^ ~b) which reduces to ~a V b where ~ is negation (not), ^ is conjunction (and), V is disjunction (or). Logic is more theory-oriented than data-oriented and is fundamental to mathematics and thereby to quantitative sciences, at least in the above deductive forms with a, b propositions.
Curiously enough, conditional probability P(B/A) or P(B|A) defined by P(AB) divided by P(A) when P(A) is not 0, is largely used in a data-oriented rather than theory-oriented way, mostly because it does not have a direct connection to logic as with (A←B) and (a←b). Not being aware of P(A←B), conditional probability specialists (most of whom are called Bayesians) only discovered the data-oriented applications such as updating a sample by resampling. This is also partly due to the fact that conditional probability as above is interpreted as the probability of B given (or fixing) A, that is to say with A fixed or constant, for example the probability of some result given the data. Fortunately, P(A←B) and conditional probability P(B/A) are used in different situations largely, because when P(A) is 0, conditional probability is not defined, while in such situations P(A←B) is defined. The reason is that division by 0 is impossible in mathematics. Since P(A) = 0 occurs for very rare events A, and P(A) is near 0 for rare events (events with probability less than .05 usually), P(A←B) is the natural choice for rare events, while conditional probability is usable for other events with some success.
Let’s apply PI and DA as defined in the Introduction to the following problems.
A. Intention or decision versus cognition versus behavior or action.
B. Intention or decision versus perception versus behavior or action.
C. Interaction versus one-way communication.
None of these problems (the problem being how the variables mentioned are related) has been solved in published research, although Doctorow (2004) has outlined solutions on the internet forum sci.stat.math with copyright notices by Doctorow.
I will simply note here that Homans (1950, 1992) isolated action or activity as a key variable as well as interaction and that action and decision in statistical decision theory are directly related (e.g., Ferguson (1967)) but the form of the relationship there is too specialized to extend to sociology.
The first step is to postulate dimensions for intention or decision (I), cognition ( C ), perception (P), action (A), interaction (i), one-way communication (c), or in the last two cases a corresponding PI variable. Cognition is taken to have the dimension of Knowledge (K), corresponding to the semantic component of information or the semantic part of that whose syntactic component is information. Action (A) is taken to have the dimension of (kinetic) energy. Intention can be considered to have its own dimension (D) or it can be defined with regard to other dimensions (see below). I should add emotion or sentiment (also isolated by Homans (1950) and by de Jong as satisfaction) e and notice that emotion typically results in behavior (small scale or larger scale) and is triggered by either perception P (which will be taken as its own dimension P) or Knowledge (K) (taken to have its own dimension K) so that e can be represented dimensionally as energy (E dimensionally) per Knowledge K and per perception P so:
We’ll simplify things by noting that from (1)-(3), the dimensions of D are:
Let’s take two dimensions D and P, and three variables decision D, emotion e, and perception P. By Buckingham’s Pi Theorem (not PI of Probable Influence) (Buckingham (1914)), the number of variables minus the number of dimensions is the number of independent dimensionless products, so 3 – 2 =1, so there is one dimensionless product of the variables, and letting k be a dimensionless constant, we have the unique solution:
So intention or decision is a constant (dimensionless) times emotion e times perception P. Notice that we would want to define in addition a purely Knowledge-based decision or intention for which we could use another symbol d with dimension K. This would enable unbiased scientific decisions, whereas I turns out to be perception and emotion-based although dimensionally it is D with dimensions energy E divided by knowledge K or E times K to the –1 exponent.
By similar arguments, it is easy to derive:
Where k1 is another dimensionless constant. This doesn’t necessarily mean that intention or decision I decreases as cognition C increases since action or behavior A is not in general constant. It means that intention or decision is action per unit cognition or roughly speaking “the amount of action in each standard amount of cognition”. An interesting application of (6) is to habits in learning. If A in (6) represents a measured intensity of study habits (habits of studying) which increases the better the habits of studying, then by (6) the study habits A equal the intention times the learning or cognition amount. This corresponds to intuition also since the more the intention to study and the more the amount of study, the more we would expect the study habits to increase toward learning.
Finally, we can prove directly:
Which says that the probability of interaction equals the sums of the probabilities of one-way communication minus 1. The fastest proof is as follows. The set (A←B U B←A) has probability P(A←B U B←A) = P(A’ U B U B’ U A) = 1 since A’ U A is the universe which always ha probability 1 by elementary probability. But the probability of a union of two sets, like P(C U D) = P( C) + P(D) – P(CD) so P(A←B U B←A) = P(A←B) + P(B←A) – P((A←B)(B←A)). But P((A←B)(B←A)) = P(A→←B) since the argument (A→←B) is defined as (A←B)(B←A). Q.E.D.
Isolating dimensions in sociology corresponding to Homans’ Activity/Action, Sentiment or emotion, and a generalization of information theory’s information to (semantic) Knowledge and perception leads to solutions of problems that have not been solved before in regard to decisions, intentions, emotion, cognition.
Isolating a set/event (which can be generalized to a process) corresponding to influence which is the set/event analog of the logical conditional (or logical implication without worrying about the truth value) and then calculating the probability of this set/event which is called Probable Influence (PI) or Probable Causation solves the problem (previously unsolved) of how Homans’ Interaction is related to one-way communication.
Both of these methods, dimensional analysis (DA) and Probable Influence (PI), are theory-oriented rather than data-oriented, and fulfil a needed function in theoretical sociology with real problems.
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