psychology and semiology of mathematics

Some time ago it became obvious that getting rid of analytical philosophy's stranglehold on attempts to describe and theorize the subject was the first step in thinking anything new and productive about mathematics. My initial attempt at this was to go to psychology; more particularly educational psychology, since here was a subject that had been obliged to think the nature of mathematics by having to explain and deal with the incomprehension and alienation which is the common experience of most people learning it. At the time, the mid 1970s, the combination of psychology, education and mathematics spelled Jean Piaget, the Swiss psychologist whose theories of children's mental development had dominated the field for several decades. I found dozens of books and hundreds of articles detailing and expositing Piaget's work on the psychology of children but practically nothing on the philosophical ideas behind this work; ideas which Piaget had written and was continuing to write a great deal about, and which for him were the reason for his experimental investigation of children in the first place. As a result, I wrote my own account, Jean Piaget: Psychologist of the Real, a monograph that traced the intellectual genealogy of Piaget's philosophical framework and theory of mathematics, and then critically evaluated them.

Since then, psychology has moved on and away from overarching theories like Piaget's, and current work in cognitive psychology is much better placed to answer the sort of questions -- about the psychological roots of mathematical thinking -- that I would have been happy enough to get in the 1970s.

I too moved on. Of many difficulties with his ideas, my most serious was Piaget's impoverished view of language: not only did he underplay the role of natural language in thinking but, more crucially for my purposes, completely failed to understand the significance of mathematical language, of the vast and complex systems of symbols and notations that are the subject's most conspicuous feature. Why were mathematical symbols, I wanted to know, so central to the creation and the meaning, the content, of the subject?

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.