Mikkel Willum Johansen

An Empirical Approach to the Mathematical Values of Problem Choice and Argumentation

In this paper we describe and discuss how mathematical values influence researchers’ choices when practicing mathematics. Our paper is based on a qualitative investigation of mathematicians’ practices, and its goal is to gain an empirically grounded understanding of mathematical values. More specifically, we will analyze the values connected to mathematicians’ choice of problems and their choice of argumentative style when communicating their results. We suggest that these two situations can be understood as relating to the three mathematical values: recognizability, formalizability and believability. Furthermore, we discuss three meta-issues concerning the general nature of mathematical values, namely (1) the origin of mathematical values, (2) the extent to which different values change over time and (3) the situatedness of mathematical values; that is the extent to which mathematical values depend on the specific context in which you are located. We conclude the chapter by recommending a methodological pluralism in future investigations of mathematical values.

Semiotic Scaffolding in Mathematics

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.

What’s in a Diagram?

In this paper I analyze the cognitive function of symbols, figures and diagrams. The analysis shows that although all three representational forms serve to externalize mental content, they do so in radically different ways, and consequently they have qualitatively different functions in mathematical cognition. Symbols represent by convention and allow mental computations to be replaced by epistemic actions. Figures and diagrams both serve as material anchors for conceptual structures. However, figures do so by having a direct likeness to the objects they represent, whereas diagrams have a metaphorical likeness. Thus, I claim that diagrams can be seen as material anchors for conceptual mappings. This classification of diagrams is of theoretical importance as it sheds light on the functional role played by conceptual mappings in the production of new mathematical knowledge.

Inquiry-Reflective Learning Environments and the Use of the History of Artifacts as a Resource in Mathematics Education

In this paper we explore the possibility of using the historical development of cognitive artifacts as a resource in mathematics education. We present three examples where the introduction of new artifacts has played a role in the development of a mathematical theory. Furthermore, we present a methodological approach for using original sources in the classroom. The creation of an inquiry-reflective learning environment in mathematics is a significant element of this methodology. It functions as a mediating link between the theoretical analysis of sources from the past and a classroom practice where the students are invited into the workplace of past mathematicians through history. We illustrate our methodology by applying it to the use of artifacts in original sources, hereby introducing a first version of such an inquiry-reflective learning environment in mathematics through history.

A Typology of Mathematical Diagrams

In this paper, we develop and discuss a classification scheme that allows us to distinguish between the types of diagrams used in mathematical research based on the cognitive support offered by diagrams. By cognitive support, we refer to the gain that research mathematicians get from using diagrams. This support transcends the specific mathematical topic and diagram type involved and arises from the cognitive strategies mathematicians tend to use. The overall goal of this classification scheme is to facilitate a large-scale quantitative investigation of the norms and values governing the publication style of mathematical research, as well as trends in the kinds of cognitive support used in mathematics. This paper, however, focuses only on the development of the classification scheme.

Science Fiction at the Far Side of Technology: Vernor Vinge’s Singularity Thesis Versus the Limits of AI-Research

The notion of a coming technological singularity is a key concept in contemporary science fiction, futurism and popular science. It is the central theme in block-buster movies such as The Matrix (1999) and The Terminator (1984) and a number of science fiction novels, such as William Gibson’s Neuromancer (1984) and William Thomas Quick’s trilogy Dreams of Flesh and Sand (1988), Dreams of Gods and Men (1989), and Singularities (1990) (see Esterbrook 2012, for further references and analysis of the science fiction literature). Furthermore, there are dozens of popular science books, hundreds of academic papers and even a congressional report on the singularity. There is a Singularity University and an annual singularity conference, where top academics and intellectuals including Douglas Hofstadter, Rodney Brooks, David Chalmers and Stephen Wolfram have given talks. So, yes, the singularity is a concept to be taken seriously.

Computers as a Source of A Posteriori Knowledge in Mathematics

ABSTRACT 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.