All of us are familiar with Felix Klein’s Erlanger Programm, which he published in 1872 when he became professor of mathematics at Erlangen. As with much of our history, however, we often get the details wrong. A common impression is that it was in Klein’s inaugural lecture on 7 December 1872 that he laid out the program that was to change how mathematicians looked at their discipline. In fact, Klein’s Erlanger Programm was a written paper distributed at the time of his lecture but quite separate from it. His “Erlanger Antrittsrede ” was directed to a general audience, and its theme was mathematics education (Rowe, 1983, 1985). In the lecture, Klein deplored the growing division between humanistic and scientific education. He saw the content of mathematics as outside both, even though, if a partition is made, the vital role mathematics plays in many scientific disciplines rightly puts it with the sciences. In the lecture, Klein made no reference to research in mathematics education. Later, however, after becoming a professor at Göttingen in 1886, he was active in helping get such research underway. He eventually taught courses in mathematics education, supervising the first European doctoral degree (Habilitation) in mathematics education

Abstract: Neither absolutism nor aposteriorism have questioned the progressive elements associated with the applications and the social functions of mathematical knowledge. Aporism raises this question by discussing the thesis of the formatting power of mathematics. This thesis unites linguistic relativism applied to mathematics and the idea that technology is a structuring principle in society. We are no longer surrounded by “nature”, instead we live in a techno-nature. Mathematical abstractions can be projected outside the sphere of mathematics, and in this way they modulate and eventually constitute fundamental categories of techno-nature. The Vico paradox expresses the difficulties of specifying the nature and function of technological actions: We are not even able to grasp and to understand what we have ourselves constructed. A critique cannot be guaranteed by scientific (or mathematical) thinking itself. Critique becomes a much more complex activity including reflections on technological actions. A critique includes ethical considerations, and therefore a critique of mathematicsisalsoethical.

This question (what is the philosophy of mathematics education?) provokes a number of reactions, even before one tries to answer it. Is it a philosophy of mathematics education, or is it the philosophy of mathematics education? Use of the preposition ‘a ’ suggests that what is being offered is one of several such perspectives, practices or areas of study. Use of the definite article ‘the ’ suggests to some the arrogation of definitiveness to the account given. 1 In other words, it is the dominant or otherwise unique account of philosophy of mathematics education. However, an alternative reading is that ‘the ’ refers to a definite area of enquiry, a specific domain, within which one account is offered. So the philosophy of mathematics education need not be a dominant interpretation so much as an area of study, an area of investigation, and hence something with this title can be an exploratory assay into this area. This is what I intend here. Moving beyond the first word, there is the more substantive question of the reference of the term ‘philosophy of mathematics education’. There is a narrow sense that can be applied in interpreting the words ‘philosophy ’ and ‘mathematics education’. The philosophy of some area or activity can be understood as its aims or rationale. Mathematics education understood