Consensus and Coherence in Mathematics – How Can They Be Explained in a Culturalistic View?

In the more recent philosophy of mathematics, mathematics is seen as a cultural product. In this view, it is not easy to explain the high level of coherence of mathematical theories and concepts and the wide spread consensus among mathematicians. As these phenomena are cited in orer to negate the cultural relativity of mathematics, it is worth searching for explanations. In this article, attempts at explanations are given by looking at the historical development of mathematical concepts and theories. In the last section, consequences for teaching and learning mathematics are briefly discussed. 0. Introduction: Mathematics as a cultural phenomenon “It is characteristic for mathematics as a scientific domain that it has disconnected from everyday life and the socio-cultural foundation which it has originally come from. Scarcely any other subject regards itself that definitively as being independent of time, values and culture. The exclusive reference on the formal and the abstractable [...] makes it difficult to discuss the relation between mathematics and cultural or social elements. Mathematics is [...] widely seen as the paradigm of the formal, the structural or the algorithmical and contrasted to culture – i.e. the historical, the dynamical, the informal or the intuitive or social: thus mathematics and culture are conceived to be extremes that are not reconcilable.“ (Schroeder 2000, p. 452, author’s translation) This citation provides us with a very concise characterization of mathematics as it exists in the minds of many non-mathematicians, and even mathematicians and mathematics teachers. However, in the disciplines which reflect on mathematics professionally, this characterization is questioned more and more. Rejecting this old, absolutist image of mathematics, philosophers of mathematics have established humanistic or social constructivist positions, in which mathematics is understood as a cultural product (cf. Ernest 1998, Tymoczko 1985, Restivo et al. 1993 for the social-constructivist view, or White 1993 for the humanistic position). Emphasis is put on both parts: “product“ accounts for the fact that mathematics must not only be discovered but must really be created by humans. These creations always take place in a specific cultural setting, thus it is a “cultural product“. One of the most prominent exponents of this position is Reuben Hersh. In his book “What is mathematics, really?“ (1997) he describes mathematics as a human activity: “From the viewpoint of philosophy mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context.“ (Hersh 1997, p. 11) To understand mathematics as a cultural product means to acknowledge the human influence on mathematics. Nevertheless, every individual is confronted with an apparently impartial theory. Leslie