Mathematicians’ use of external representations, such as symbols and diagrams, constitutes an important focal point in current philosophical attempts to understand mathematical practice. In this paper, we add to this understanding by presenting and analyzing how research mathematicians use and interact with external representations. The empirical basis of the article consists of a qualitative interview study we conducted with active research mathematicians. In our analysis of the empirical material, we primarily used the empirically based frameworks provided by distributed cognition and (...) cognitive semantics as well as the broader theory of cognitive integration as an analytical lens. We conclude that research mathematicians engage in generative feedback loops with material representations, that they use representations to facilitate the use of experiences of handling the physical world as a resource in mathematical work, and that their use of representations is socially sanctioned and enabled. These results verify the validity of the cognitive frameworks used as the basis for our analysis, but also show the need for augmentation and revision. Especially, we conclude that the social and cultural context cannot be excluded from cognitive analysis of mathematicians’ use of external representations. Rather, representations are socially sanctioned and enabled in an enculturation process. (shrink)

This paper investigates the notion of semiotic scaffolding in relation to mathematics by considering its influence on mathematical activities, and on the evolution of mathematics as a research field. We will do this by analyzing the role different representational forms play in mathematical cognition, and more broadly on mathematical activities. In the main part of the paper, we will present and analyze three different cases. For the first case, we investigate the semiotic scaffolding involved in pencil and paper multiplication. For (...) the second case, we investigate how the development of new representational forms influenced the development of the theory of exponentiation. For the third case, we analyze the connection between the development of commutative diagrams and the development of both algebraic topology and category theory. Our main conclusions are that semiotic scaffolding indeed plays a role in both mathematical cognition and in the development of mathematics itself, but mathematical cognition cannot itself be reduced to the use of semiotic scaffolding. (shrink)

The production of mathematical formalism on state of the art computers is quite different than by pen and paper. In this paper, I examine the question of how different media influence mathematical writing. The examination is based on an investigation of professional mathematicians' use of various media for their writing. A model for describing mathematical writing through turn-takings is proposed. The model is applied to the ways mathematicians use computers for writing, and especially it is used to understand how interaction (...) with the computer system LaTeX is different in the case of formulas than in the case of diagrams. (shrink)

Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to be surveyed (...) by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions. (shrink)

In this paper, we wish to explore the role that textual representations play in the creation of new mathematical objects. We do so by analyzing texts by Joseph-Louis Lagrange (1736–1813) and Évariste Galois (1811–1832), which are seen as central to the historical development of the mathematical concept of groups. In our analysis, we consider how the material features of representations relate to the changes in conceptualization that we see in the texts.Against this backdrop, we discuss the idea that new mathematical (...) concepts, in general, are increasingly abstract in the sense of being detached from material configurations. Our analysis supports the opposite view. We suggest that changes in the material aspects of textual representations (i.e., the actual graphic inscriptions) play an active and crucial role in conceptual change.We employ an analytical framework adapted from Bruno Latour’s 1999 account of intertwined material and representational practices in the empirical sciences. This approach facilitates a foregrounding of the interconnection between the conceptual development of mathematics, and the construction, (re-)configuration, and manipulation of the materiality of representations. Our analysis suggests that, in mathematical practice, distinctions between the material and structural features of representations are not permanent and absolute. This problematizes the appropriateness of the distinction between concrete inscriptions and abstract relations in understanding the development of mathematical concepts. (shrink)