Thinking through drawingDiagram constructions as epistemic mediators in geometrical discovery

That is a kind of definition that prescribes ‘what you are to do in order to gain perceptual acquaintance with the object of the world’ (Peirce 1931–1958: 2.330).

The analysis of the role of diagrams and visualizations in mathematical reasoning has been recently promoted by philosophers and historians of science (cf. e.g. the seminal studies by Brown, 1997, 1999 and Giaquinto, 1992, 1994, 2007). At present, research on visual and diagrammatic cognition is a common topic in various areas of cognitive science, logic (Allwein & Barwise, 1996), and computer science (Glasgow et al., 1995; Anderson et al., 2000). In this tradition, the role of visualizations and diagrams as external tools that can be described in the framework of distributed and embodied cognition has often been disregarded; however, an increasing attention is currently devoted to these aspects. A fruitful feedback from cognitive science has favored new research in the area of mathematical reasoning (cf. e.g. the recent special issue of Educational Studies in Mathematics, 2009 (70), ‘Gestures and Multimodality in the Construction of Mathematical Meaning,’ edited by L. Edwards, L. Radford, and F. Arzarello), which contains various studies on the role of embodiment (gestures), distributed cognition (artifacts, external representations, etc.), and multimodality. The approach I present in this paper aims at deepening the role of externalities in mathematical diagrammatization, also taking advantage of the clarifying role of the basic concept of abduction, which is a synthetical cognitive and epistemological tool able to unify various aspects of hypothetical cognition.

On the philosophical, computational, and cognitive aspects of the relationships between geometry and space, also from a historical point of view, cf. Magnani (2001b).

A discussion concerning the ignorance-preserving vs. knowledge-enhancing character of abduction is illustrated in Magnani (2013).

On the epistemological and eco-cognitive aspects of abductive cognition, cf. my recent book (Magnani, 2009b).

Of course, in the case, we are using diagrams to demonstrate already known theorems (for instance, in didactic settings), the strategy of manipulations is already available and the result is not new. Further details on this issue are illustrated in the study by Magnani (2009b: Ch. 3).

The names galea and batello referred to a boat, which the outline of the work was thought to correspond. The process that underlies the galley division method is illustrated in Smith ([1925] 1958: 137–144, vol. II).

This method of visualization was invented by Stroyan (2005), and improved by Tall (2001).

I maintain that, in general, spatial transformations are represented by a visual component and a spatial component (Glasgow & Papadias, 1992).

Usually, scientists try to determine identity, when they make a comparison to determine the individuality of one of the objects; alignment, when they are trying to determine an estimation of fit of one representation to another (e.g. visually inspecting the fit of a rough mental triangular shape to an external constructed triangle); and feature comparison, when they compare two things in terms of their relative features and measures (size, shape, color, etc.; cf. Trafton et al., 2005).

Further details on model-based abduction are illustrated in Magnani (2009b: Ch. 1).

On multimodal abduction, cf. Magnani (2009b: Ch. 4).

The role of affordances in abductive cognition is illustrated in Magnani and Bardone (2008).

Magnani and Dossena (2005) and Dossena and Magnani (2007) illustrate that external representations such as the ones I call unveiling diagrams can not only enhance the consistency of a cognitive process but also provide more radically creative suggestions for new useful information and discoveries.

Also called limit sphere or orisphere.

Lobachevsky called the new theory ‘imaginary geometry,’ and also ‘pangeometry.’

Given that Lobachevsky designates the size of a line by a letter with an accent added, for example, x′, to indicate this has a relation to that of another line, which is represented by the same letter without the accent x, ‘which relation is given by the equation $$\[--><$>{\rm{II}}(x)\, + \,{\rm{II}}(x^{\prime})\, = \,\frac{1}{2}{\rm{\pi }} <$><!--$$’ (Proposition 35).

In other problems-solving cases, the end-product of perception directly picked up is the end-product of the whole problem-solving process.

On the limitations of the Lobachevskyan perspective, cf. Torretti (1978) and Rosenfeld (1988).

We have seen how Lobachevsky did this by using Figure 9.

This approach in computer science, involving the use of diagram manipulations as forms of acceptable methods of reasoning, was opened by Gelernter's Geometry Machine (Gelertner, 1963), but the diagrams played a very secondary role.