An earlier version of this essay, "Behaviorism" appears in J.J. Chambliss (ed.)
Philosophy of Education: an encyclopedia New York: Garland Pub. May 1996. 52 -- 55.

The Mathematics of Behaviorism: an informal examination

Edward G. Rozycki
Widener University

edited 6/29/11

What are Behaviorists Trying to Do?

Theorists in psychology have proposed Behaviorism both as a research program and as a philosophy of psychology. As a research program Behaviorist theorists undertake to explain and predict an organism's behavior solely in terms of the effects of its present environment and the conditioning history of that individual organism. Put in simple mathematical form, it asserts that

1) B = f(E, H),

i.e. Behavior, B, is a function of Environment, E, and conditioning history, H.

In more colloquial terms what formula 1) says is that specific pairs of values, that of an environmental variable and that of a conditioning history variable, specify a unique value of a behavioral variable. (Or, in stronger causal language, a specific environmental state, given a specific conditioning history, is sufficient to cause an organism to act in a unique way.)

Thus, while a different environment may cause an organism to behave differently, or a different conditioning history may cause an organism to behave differently in the same environment, Behaviorism, as a philosophy, disallows the possibility that the organism could behave consistently differently given the same environment and no new conditioning history. For the Behaviorist, organisms do not develop independently of environmental stimuli.

As the functional formula shows, E and H are assumed to be non-interactive, because such interaction would have to be done by some theoretically disallowed internal process. Thus, any formula of the form

2) B = f(E, H, g(E, H))

where f and g are functions is disallowed. To say that a type of formula is "disallowed" is to say that it is philosophically unacceptable to the Behaviorist. This is not a decision based on empirical evidence; but is the very basis of the identity of what is called "Behaviorism."

Skinner tries to give reasons for this dismissal. According to him, the Behaviorist researcher needs to be able to assume that behavioral types, B, can be simply mapped back to environmental states, E, that presumably effect them, without worrying about physiological processes of the observed organism. Otherwise, there cannot be, on the Behaviorist's view, a science of Psychology.

To reiterate, Behaviorist doctrine presumes to say what psychology should do. It demands of that discipline that no internal states of an organism be treated as indispensible to explaining that organism's behavior. Thus , any formulation which includes internals states, e. g. X, for expectations -- as Albert Bandura does, for example --, yielding

3) B = f(E, H, X)

must be disallowed. For the Behaviorist, the organism must be "empty."

What is a Variable?

The theoretical viability of Behaviorist theory depends upon the possibility of defining such things as behavioral, environmental and conditioning history variables, B, E and H. The definitional requirements for being a variable are that a variable be a partitioned set, i.e. that it represent a class of event-types which are mutually exclusive and together exhaustive of all events in its class. This partitionability condition is a requirement of the mathematical foundations of measurement. No partition, no measurement. (But, also, in the opinion of many Behaviorists: No measurement, no science of behavior.)

A further assumption of Behaviorism is that B, E and H can be adequately and interestingly defined extensionally, that is, without reference to the beliefs or perceptions of the behaving organism -- these would be, after all, represented by additional internal variables. Thus the Behaviorist discards the generally practical distinction between "John's left biceps twitched" and "John twitched his left biceps" unless it can be made out in terms of measurable characteristics of the observed event.

Intensional definition, i.e. definition which invokes beliefs or perceptions of the subject, can be avoided, the Behaviorist hopes, by characterizing type-defining events in terms of the responses of measuring devices. So it is that behavior, or environment or conditioning history types will be identified by sets of numbers generated on certain devices. (How one would calibrate such a device from within the framework of Behaviorist theory is an interesting question. Practically, it looks like it would involve an infinite regress.)

Can a Consistent Behaviorist Finish an Experiment?

Suppose the experimenter were going to apply the same theoretical restrictions upon his own behavior as researcher, as upon his or her subject's researched behavior, i.e. be a "Consistent Behaviorist". According to Behaviorist theory, an organism can be shown to be able to discriminate between two environmental event-types, say, e1 and e2, only if it can be conditioned to respond differentially upon their presentation. If the experimenter's apparatus can discriminate n environmental-event types, it would take


experiments to map out the perceptual capabilities of the subject organism. (This is the number of combinations of pairs of contrastive events given n. If the order of presentation were crucial, then the even greater measure of the number of permutations of pairs would be the number of experiments required.)

So for a paltry 10 environmental types, it would take 45 experiments to map out the organism's respondent capabilities, assuming the organism to have only 2 response types. For m behavior-types, the stimulus-presentation set must be presented m-1 times.

It is easy to measure stimulus variation to the hundredth of a unit, and easy to measure, say, ten simultaneous potential stimulus-properties of an environment, e.g. intensity of illumination, temperature, density of magnetic flux, speed relative to Earth's surface, etc. (No a priori restrictions on possible physical variables is consistent with Behaviorist doctrine, even though, in practice, Behaviorists tend to ignore possible influencing variables that do not interest them.[1])

If we suppose that an organism is capable of ten behavioral types, then the "Consistent Behaviorist", to determine the behavioral variability of the organism, has an enormous number of experiments to perform. There are 10010 "environments" to be tested out for each of the ten response types.,

It may take, say, at least, ten (a very steep learning curve!) training trials for the organism to "learn" to differentiate its responses in a given environment. Putting all these considerations together, and given that most reported conditioning experiments were concluded within months of their initiation, one may assume that there were never any "Consistent Behaviorists" at work. The experimental results offered in evidence of Behaviorist theory must have been carried out with the surreptitious support of other theoretical scaffolding.[2]

Is the Behaviorist's Concept of Behavior Useful to Educators and Jurists?

Consider the following generally useful and possibly both educationally and legally important distinctions between emitting sound, saying , telling, informing, and surprising. Consider Harry to be the person who is our subject of investigation. From a Behaviorist perspective Harry has behaved, b1, in a way described (using adapted phonetic notation) as follows:

a. Harry emitted / 2àym+gowin hówm1 /.

But other descriptions of b1 are available. For example:

b. Harry said, "I'm goin' home."

c. Harry told John he was going home.

d. Harry informed John that he, Harry, was going home.

e. Harry surprised John with the statement that he was going home.

Does this mean that the Behaviorist has a chance at reducing descriptions b, c, d, and e to a.? Not at all. We can easily imagine a situation where all of these descriptions are true of Harry's behavior. But if Harry is a babbling idiot, a might be true and none of the rest. If Harry is reading alound a line from a script, a and b might be true and none of the rest. If John already knew that Harry was going home, a, b and c might be true but none of the rest. If John is never surprised by what Harry does, a, b, c and d might be true, but not e.

What these alternative descriptions show is the following:

1) If circumstances determine how behavior is to be described, we must take care to separate such circumstances from environmental conditions which supposedly explain that same behavior, otherwise our variables are not independent; and

2) Behavior is not, in natural language, a partitioned set of events, therefore it is not a variable in the mathematical sense. Thus, it cannot be measured, nor can it be conditioned, i.e, be subject to a procedure which requires it to be assigned a probability distribution. (Such an assignment requires a partition.)

Thus the would-be Behaviorist bears the burden of defining a behavior-partition rather than assuming that behavioral categories in our natural language, those categories with which we express our educational, legal and moral concerns, will do for a start. This burden, however, is not the Behaviorist's uniquely, but is borne by theorists of any persuasion who would attempt to analyze human behavior, as it is colloquially understood, as a causal consequence. If no behavior-partition can be constructed, no global measurement of behavior can be made; and thus, no global correlational analysis of human behavior.

Why has Behaviorism Survived in Education?

The Behaviorist theorist who has had the greatest influence on American Education is B.F. Skinner. The plausibility of the Skinnerian doctrine in education depends on the viability of the Behaviorist program. With the proliferation of devices such as hand-held calculators and computers containing internal processing devices, hidden internal processors -- anathema to Skinner -- have become a commonplace explanatory hypothesis.

Because the "behavior" of such computing devices cannot be explained and predicted solely in terms of environmental variables or their history of influence on the individual "organism", and also because the enthusiasm with with the Behaviorist program was promoted has not been correspondingly met with empirical results, the Behaviorist program has been largely abandoned in empirical psychological research -- except in cases where severely dysfunctioning individuals are concerned. Indeed, the development of computers provides a major impetus for the ascent of the Cognitivist psychological theories which have largely replaced Behaviorism in experimental psychology.

As a philosophical theory, Behaviorism is passé. However, its influence in education cannot be understood merely in terms of the viability of the scientific or philosophical standing of Behaviorism as a general theory of psychology.

Ideologically, Behaviorism serves several kinds of interests. First, those of radical educational interventionists who want to discount influences external to the school environment, e.g. genetics or parenting, in the hope that the school can be a major agent of social reconstruction.

Secondly, Behaviorist theory has provided the rationale for a variety of marketable, although short-lived teaching devices. Educational budgets have long proven to be easy prey for scientific-sounding entrepreneurs.

Finally, the idea of control within the school environment feeds the aspirations of educators to professional independence. Being able to control the in-school behavior of children without having to worry about their out-of-school circumstances is a seductive idea, that in American schools, to this day, pursues hope in the face of contrary experience.


[1] Donald S. Blough and Richard B. Millward, "Learning: Operant Conditioning and Verbal Learning," in Paul E. Farnsworth (ed.), Annual Review of Psychology, Volume 16 (Palo Alto, California: Annual Reviews, Inc., 1965) complain that operant researchers "shrug off" stimulus problems and ". . . have remained indifferent to laws that might predict what flux of energy will be a stimulus, and what manner of stimulus it will be." (p. 64)

[2] See Edward G. Rozycki, "Evaluating Theories: a first approach" on parasitical theories at


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