B. Rotman: psychology + semiology of mathematics
psychology + semiology of mathematics
So I went first to linguistics and then, to what proved to be a lot more promising , to semiotics. Since then, I’ve written about mathematics from a semiotic/semiological point of view, first from the French structuralist one stemming from Ferdinand de Sauusure (the approach adopted in Signifying Nothing in relation to zero), and then the so-called Anglo-Saxon perspective of Charles Saunders Peirce (the approach adopted in Ad Infinitum in relation to infinity). From Peirce’s writings I took the notion of a thought experiment ("reflective abstraction") and elaborated it to form a semiotic model of mathematical thinking, which forms the methodological basis for the various essays in Mathematics as Sign.
That was Brian Rotman accounting his intellectual history, starting from his book on Jean Piaget, to later work on the semiotics of mathematic notations and zero.
In a talk in 2004 at Stanford, Rotman commented on the ghost in speaking, writing, and mathematics (here only the math part):
A salient feature of ghosts is their mode of embodiment and disembodiment. Their aberrant physicality, more than anything, underlies their strangeness, the spookiness of their presence, and inflects all inferences about them.
The medieval (Christian) king, according to Kantorowicz, has two bodies – a private and human one, his material body – and a divine, immortal one. The monarch’s presence and his exercise of power derived from the institutional, theological, and rhetorically assembled co-presence of these bodies.
The mathematician has three bodies, or three material arenas of operation — a mortal Person, a virtual agent, and a semiotic Subject — likewise co-assembled. The mathematical person subjectively situated in language is the one who imagines, makes judgments, tells stories, has intuitions, hunches and motives; next, the mathematical agent, imagined by the Person, is a formal construct which executes ideal actions and lacks any capacity to attach meaning to the signs which control its narratives; and between them, their interface, the mathematical subject, who embodies the materiality of the apparatus that writes and is written by mathematical thought.
Following Charles Peirce one can, as I have shown elsewhere, view mathematics as a thought-experimental process of ‘reflective observation’. According to this, the person imagines the agent performing an activity and observes the result of the activity via the symbolic mediation by the subject. The agent is a proxy or surrogate of the person, so that for the observation to be a convincing thought-experiment the agent must resemble the person. But the resemblance is necessarily partial: the agent is invoked in the first place is to execute an action – such as unlimited counting — that goes beyond the person’s temporal and/or material constraints. The agent is thus a person without a body. Or rather a person with a virtual body that has a split character. On the one hand it lacks those features of bodies that prevent the person carrying out the action. On the other, any feature not so excluded remains unaffected and available to the agent.
What is still unclear to me — which is a motivation to read his Mathematics as Signs book — is why the agent chooses to invalidate constraint X but not Y and Z. That is, principles that guide the construction of a cohesive symbol system from a multitude practical constraints.