The symbolic model for algebra : functions and mechanisms

The symbolic mode of reasoning in algebra, as it emerged during the sixteenth century, can be considered as a form of model-based reasoning. In this paper we will discuss the functions and mechanisms of this model and show how the model relates to its arithmetical basis. We will argue that the symbolic model was made possible by the epistemic justification of the basic operations of algebra as practiced within the abbaco tradition. We will also show that this form of model-based reasoning facilitated the expansion of the number concept from Renaissance interpretations of number to the full notion of algebraic numbers

Identifying adequate models in physico-mathematics: Descartes' analysis of the rainbow

The physico-mathematics that emerged at the beginning of the seventeenth century entailed the quantitative analysis of the physical nature with optics, meteorology and hydrostatics as its main subjects. Rather than considering physico-mathematics as the mathematization of natural philosophy, it can be characterized it as the physicalization of mathematics, in particular the subordinate mixed mathematics. Such transformation of mixed mathematics was a process in which physico-mathematics became liberated from Aristotelian constraints. This new approach to natural philosophy was strongly influenced by Jesuit writings and experimental practices. In this paper we will look at the strategies in which models were selected from the mixed sciences, engineering and technology adequate for an analysis of the specific phenomena under investigation. We will discuss Descartes’ analysis of the rainbow in the eight discourse of his Meteorology as an example of carefully selected models for physico-mathematical reasoning. We will further demonstrate that these models were readily available from Jesuit education and literature

Health warning: might contain multiple personalities - the problem of homonyms in Thomson Reuters Essential Science Indicators

Author name ambiguity is a crucial problem in any type of bibliometric analysis. It arises when several authors share the same name, but also when one author expresses their name in different ways. This article focuses on the former, also called the “namesake” problem. In particular, we assess the extent to which this compromises the Thomson Reuters Essential Science Indicators (ESI) ranking of the top 1% most cited authors worldwide. We show that three demographic characteristics that should be unrelated to research productivity – name origin, uniqueness of one’s family name, and the number of initials used in publishing – in fact have a very strong influence on it. In contrast to what could be expected from Web of Science publication data, researchers with Asian names – and in particular Chinese and Korean names – appear to be far more productive than researchers with Western names. Furthermore, for any country, academics with common names and fewer initials also appear to be more productive than their more uniquely named counterparts. However, this appearance of high productivity is caused purely by the fact that these “academic superstars” are in fact composites of many individual academics with the same name. We thus argue that it is high time that Thomson Reuters starts taking name disambiguation in general, and non-Anglophone names in particular, more seriously

Why 'scaffolding' is the wrong metaphor : the cognitive usefulness of mathematical representations.

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.Peer reviewe

Counting on the mental number line to make a move: sensorimotor ('pen') control and numerical processing

Mathematics is often conducted with a writing implement. But is there a relationship between numerical processing and sensorimotor ‘pen’ control? We asked participants to move a stylus so it crossed an unmarked line at a location specified by a symbolic number (1–9), where number colour indicated whether the line ran left–right (‘normal’) or vice versa (‘reversed’). The task could be simplified through the use of a ‘mental number line’ (MNL). Many modern societies use number lines in mathematical education and the brain’s representation of number appears to follow a culturally determined spatial organisation (so better task performance is associated with this culturally normal orientation—the MNL effect). Participants (counter-balanced) completed two consistent blocks of trials, ‘normal’ and ‘reversed’, followed by a mixed block where line direction varied randomly. Experiment 1 established that the MNL effect was robust, and showed that the cognitive load associated with reversing the MNL not only affected response selection but also the actual movement execution (indexed by duration) within the mixed trials. Experiment 2 showed that an individual’s motor abilities predicted performance in the difficult (mixed) condition but not the easier blocks. These results suggest that numerical processing is not isolated from motor capabilities—a finding with applied consequences

Adaptive Logics as a Necessary Tool for Relative Rationality: Including a Section on Logical Pluralism

In this paper, I show that adaptive logics are required by my epistemological stand. While doing so, I defy the reader to cope with the problems I am able to cope with. The last section of the paper contains a defense of a specific form of logical pluralism. Although this section is an integral part of the paper, it may be read separately