MATHEMATICS AND VERBAL BEHAVIOR M. JACKSON MARR GEORGIA TECH

SCOPE: SOME RELEVANT TOPICS 1. What is mathematics? 2. What’s it all about? Philosophy and mathematics 3. Platonism vs. Empiricism: Sources of mathematical concepts 4. Evolutionary Nature of mathematical concepts and methods 5. The shaping of abstraction: Example: complex variable. 6. Mathematics: Skinner’s “empirical epistemology.” 7. Contingencies, rules and relations: The mechanics of intuition 8. Coda

PLATONISM “I believe that the numbers and functions of analysis are not the arbitrary product of our spirits; I believe that they exist outside of us with the same character of necessity as the objects of objective reality; and we find or discover them and study them as do the physicists, chemists and zoologists.” --Charles Hermite “I believe that mathematical reality lies outside of us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply our notes of our observations.” --G. H. Hardy

EMPIRICISM “The mathematician is an inventor, not a discoverer.” “There correspond to our laws of logic very general facts of daily experience.” “One would like to say: the proof changes the grammar of our language, changes our concepts. It creates new connexions, and it changes the concepts of these connexions (it does not establish that they are there; they do not exist until it makes them).” --L. Wittgenstein “The verbal behavior of the mathematician, as of anyone else, is presumably a function of variables in the external environment and in other parts of his own behavior.” --B.F. Skinner “Mathematics is part of human culture and history, which are rooted in our biological nature and our physical and biological surroundings. Our mathematical ideas in general match our world for the same reason our lungs match earth’s atmosphere.” --Reuben Hersh

COMPLEX VARIABLE EVOLUTION Quadratic/cubic equations: Wessel’s complex plane: De Moivre’s theorem: Euler’s identity: Riemann’s zeta function: Contour integration: Conformal mapping: Mandelbrot set:

SHAPING OF ABSTRACTION Idealization of geometric concepts (e.g., points, lines, shapes) Simplifications and analogies in model building (“Assume a spherical cow”) Deriving the wave equation for a vibrating string assumes: 1. String is perfectly flexible, of constant density (ρ), and under constant tension, T. 2. Motion confined to X-Y plane and each point moves perpendicular to X-axis. 3. Displacement small compared to the length (L); angle with X-axis also small. 4. No external forces.

GENERALIZATION OF “DISTANCE” any triangle: n-dimensional Euclidian space: Length of a curve: Riemannian space:

Skinner’s “Empirical Epistemology” “When a speaker accurately reports, identifies, or describes a given state of affairs, he increases the likelihood that a listener will act successfully with respect to it, and when the listener looks to the speaker for an extension of his own sensory capacities, or for contact with distant events, or for an accurate characterization of a puzzling situation, the speaker’s behavior is most useful if the environmental control has not been disturbed by other variables” (VB, 1957, p. 418). “The scientific community encourages the precise stimulus control under which an object or property of an object is identified or characterized in such a way that practical action will be most effective” (VB,1957, p. 419). “One of the ultimate accomplishments of a science of verbal behavior may be an empirical logic, or a descriptive and analytic scientific epistemology, the terms and practices of which will be adapted to human behavior as a subject matter” (VB, p. 431).

VERBAL OPERANTS AND CONTROLLING RELATIONS 1.TACTS and their extensions: GENERIC, METAPHORICAL, ABSTRACT 2.INTRAVERBALS 3.AUTOCLITICS AND AUTOCLITIC FRAMES 4.RELATIONAL FRAMES 5.CONTINGENCY-CONTROLLED VERSUS RULE-GOVERNED BEHAVIOR

“Behaving intuitively in the sense of behaving as the effect of unanalyzed contingencies, is the very starting point of a behavior analysis” (Skinner, About behaviorism). “All his [Ramanujan’s] results, new or old, right or wrong, had been arrived at by a process of mingled argument, intuition, and induction of which he was entirely unable to give an account.” (G.H. Hardy, The mathematician’s apology) “The mathematician at work relies on surprisingly vague intuitions and proceeds by fits and starts with all too frequent reversals. Clearly logic as it stands fails to give a direct account of either the historical growth of mathematics or the day-to-day experience of its practitioners.” (S. Fefferman, cited in Davis and Hersh, The mathematical experience) INTUITION

Ramanujan’s Intuition

“Behaving intuitively in the sense of behaving as the effect of unanalyzed contingencies, is the very starting point of a behavior analysis” (Skinner, About behaviorism). “All his [Ramanujan’s] results, new or old, right or wrong, had been arrived at by a process of mingled argument, intuition, and induction of which he was entirely unable to give an account.” (G.H. Hardy, The mathematician’s apology) “The mathematician at work relies on surprisingly vague intuitions and proceeds by fits and starts with all too frequent reversals. Clearly logic as it stands fails to give a direct account of either the historical growth of mathematics or the day-to-day experience of its practitioners.” (S. Fefferman, cited in Davis and Hersh, The mathematical experience) INTUITION

NATURE contingencies contingency-shaped behavior Verbal behavior Intraverbals, autoclitics, relational frames, abstract tacts, etc. rule-governed behavior rules of rules, abstraction, generalization, extension, simplification, analogy, unification, formalization Laws, theories, models, theorems, conjectures Intuitions (verbal contingency- controlled behaviors) describing

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