A Subjective Comparison Between a Historical and a Contemporary Textbook on Geometry

In order to investigate how a 19th century mathematical textbook (in contrast to a contemporary one) would be experienced by a novice reader, we embarked on the following project: In the summer of 2013, a student with no previous training in college-level mathematics (the first author) set out to learn projective geometry from Pasch\u27s 1882 textbook Lectures on Modern Geometry. Afterwards, he studied the same material from Coxeter\u27s 1994 popular undergraduate textbook Projective Geometry. We report here some of his experiences and impressions contextualizing them along the way

Extended mathematical cognition: external representations with non-derived content

Funder: Social Sciences and Humanities Research Council of Canada; doi: http://dx.doi.org/10.13039/501100000155Abstract: Vehicle externalism maintains that the vehicles of our mental representations can be located outside of the head, that is, they need not be instantiated by neurons located inside the brain of the cogniser. But some disagree, insisting that ‘non-derived’, or ‘original’, content is the mark of the cognitive and that only biologically instantiated representational vehicles can have non-derived content, while the contents of all extra-neural representational vehicles are derived and thus lie outside the scope of the cognitive. In this paper we develop one aspect of Menary’s vehicle externalist theory of cognitive integration—the process of enculturation—to respond to this longstanding objection. We offer examples of how expert mathematicians introduce new symbols to represent new mathematical possibilities that are not yet understood, and we argue that these new symbols have genuine non-derived content, that is, content that is not dependent on an act of interpretation by a cognitive agent and that does not derive from conventional associations, as many linguistic representations do

From a Doodle to a Theorem: A Case Study in Mathematical Discovery

We present some aspects of the genesis of a geometric construction, which can be carried out with compass and straightedge, from the original idea to the published version (Fernández González 2016). The Midpoint Path Construction makes it possible to multiply the length of a line segment by a rational number between 0 and 1 by constructing only midpoints and a straight line. In the form of an interview, we explore the context and narrative behind the discovery, with first-hand insights by its author. Finally, we discuss some general aspects of this case study in the context of philosophy of mathematical practice

Peano on Symbolization, Design Principles for Notations, and the Dot Notation

Peano was one of the driving forces behind the development of the current mathematical formalism. In this paper, we study his particular approach to notational design and present some original features of his notations. To explain the motivations underlying Peano’s approach, we first present his view of logic as a method of analysis and his desire for a rigorous and concise symbolism to represent mathematical ideas. On the basis of both his practice and his explicit reflections on notations, we discuss the principles that guided Peano’s introduction of new symbols, the choice of characters, and the layout of formulas. Finally, we take a closer look, from a systematic and historical perspective, at one of Peano’s most striking innovations, his use of dots for the grouping of subformulas.Peano a été l’une des forces motrices dans le développement du formalisme mathématique actuel. Dans cet article, nous étudions son approche particulière de la conception notationnelle et présentons quelques caractéristiques originales de ses notations. Pour motiver l’approche de Peano, nous présentons d’abord sa vision de la logique comme méthode d’analyse et son désir d’un symbolisme rigoureux et concis pour représenter les idées mathématiques. Sur la base à la fois de sa pratique et de ses réflexions explicites sur les notations, nous discutons des principes qui ont guidé Peano dans l’introduction de nouveaux symboles, le choix des caractères et la mise en forme des formules. Enfin, nous examinons de plus près, d’un point de vue systématique et historique, l’une des innovations les plus marquantes de Peano, à savoir l’usage de points pour regrouper des sous-formules

The cultural challenge in mathematical cognition

In their recent paper on “Challenges in mathematical cognition”, Alcock and colleagues (Alcock et al. [2016]. Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20-41) defined a research agenda through 26 specific research questions. An important dimension of mathematical cognition almost completely absent from their discussion is the cultural constitution of mathematical cognition. Spanning work from a broad range of disciplines – including anthropology, archaeology, cognitive science, history of science, linguistics, philosophy, and psychology – we argue that for any research agenda on mathematical cognition the cultural dimension is indispensable, and we propose a set of exemplary research questions related to it

Learning from the existence of models. On psychic machines, tortoises, and computer simulations

Using four examples of models and computer simulations from the history of psychology, I discuss some of the methodological aspects involved in their construction and use, and I illustrate how the existence of a model can demonstrate the viability of a hypothesis that had previously been deemed impossible on a priori grounds. This shows a new way in which scientists can learn from models that extends the analysis of Morgan (1999), who has identified the construction and manipulation of models as those phases in which learning from models takes place