Over the last century, there have been considerable variations in the frequency of use and types of diagrams used in mathematical publications. In order to track these changes, we developed a method enabling large-scale quantitative analysis of mathematical publications to investigate the number and types of diagrams published in three leading mathematical journals in the period from 1885 to 2015. The results show that diagrams were relatively common at the beginning of the period under investigation. However, beginning in 1910, they (...) were almost completely unused for about four decades before reappearing in the 1950s. The diagrams from the 1950s, however, were of a different type than those used earlier in the century. We see this change in publication practice as a clear indication that the formalist ideology has influenced mathematicians’ choice of representations. Although this could be seen as a minor stylistic aspect of mathematics, we argue that mathematicians’ representational practice is deeply connected to their cognitive practice and to the contentual development of the discipline. These changes in publication style therefore indicate more fundamental changes in the period under consideration. (shrink)

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the (...) effectiveness of mathematics in natural science. (shrink)

Recent philosophical analyses of the epistemic dimension of images in the sciences show a certain trend in acknowledging potential roles of these images beyond their merely decorative or pedagogical functions. We argue, however, that this new debate has yet paid little attention to a special type of pictures, we call ‘visual metaphor’, and its versatile heuristic potential in organizing data, supporting communication, and guiding research, modeling, and theory formation. Based on a case study of Conrad Hal Waddington’s epigenetic landscape images (...) in biology, we develop a descriptive framework applicable to heuristic roles of various visual metaphors in the sciences. (shrink)

Recent philosophical analyses of the epistemic dimension of images in the sciences show a certain trend in acknowledging potential roles of these images beyond their merely decorative or pedagogical functions. We argue, however, that this new debate has yet paid little attention to a special type of pictures, we call ‘visual metaphor’, and its versatile heuristic potential in organizing data, supporting communication, and guiding research, modeling, and theory formation. Based on a case study of Conrad Hal Waddington’s epigenetic landscape images (...) in biology, we develop a descriptive framework applicable to heuristic roles of various visual metaphors in the sciences. (shrink)

We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.

In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the (...) distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education. (shrink)