I made a lot of use of this in my MRes work, but never wrote up any notes on it, so here goes.

As Mancosu (2008, p. 2) summarizes, ‘‘the epistemology of mathematics needs to be

extended well beyond its present confines to address epistemological issues having to

do with fruitfulness, evidence, visualization, diagrammatic reasoning, understanding,

explanation and other aspects of mathematical epistemology which are orthogonal to

the problem of access to ‘abstract objects’.’’ p. 829A figure per se does not have a meaning; in order to become

meaningful, it has to be considered inside a particular context of use, and therefore

interpreted in such a context. It is only when the intention behind the figure is

recognized that the figure is seen as a representation and as a consequence it becomes

an illustration or a diagram. By illustration, we mean a static representation, which

can be useful by conveying information in a single display, but where modifications

are not well-defined. By diagram, we mean a dynamic representation, on which we

can perform moves that can count as inferential procedures. Diagrams are dynamic

inferential tools that are modified and reproduced by the experts for various epistemic

purposes. They do not only represent strategies to solve problems but also give

evidence for their solutions. p. 830Knot diagrams are in a sense privileged points of view on knots and knot types: They

display only a certain number of properties by selecting the relevant ones. If illustrations

are analogous to pictures, diagrams are like maps. In order to draw a map, it is required

to define conventions that would make it legible. p. 833Moreover, there are particular geometric properties, such as the

presence of symmetries, which can be used to infer properties of the represented knot

type. In the same example, the diagrams in Fig. 3a and in Fig. 3b, which are the same

diagram also if considered geometrically, display a clear symmetry of order four,

while the diagram in Fig. 3c does not. p. 834Experts perform

actions on diagrams by re-drawing them in appropriate ways, according to the way

they interpret them. For this reason, novices need to train their imagination in order to

recognize the various possible moves on diagrams, and then be able to effectively use

them. Moreover, these manipulations are similar to the manipulations we can perform

on concrete objects, but instead of having a pragmatic aim, they have an epistemic one

(Kirsh and Maglio 1994). The use of diagrams triggers a form of manipulative

imagination that gets enhanced by the practice. p. 836

This is something that Giardino expanded on in her talk at Stanford – this manipulative imagination and some way of defining that idea. It was a very interesting talk, making use of Walton’s theory of art and make-believe alongside some embodied mind ideas.

The meaning of a knot diagram is fixed by its context of use: Diagrams are the

results of the interpretation of a figure, depending on the moves that are allowed on

them and at the same time on the space in which they are embedded. Once we

establish the appropriate moves, we fix the ambient space, thus determining the

different equivalence relations. The context of use does not have to be pre-defined.

This is not a ‘‘damaging ambiguity’’11; on the contrary, it expresses the richness ofthis notation and explains why it is effective in promoting inference. The

indetermination of meaning makes different interpretations available, and therefore

allows attending to various properties and moves. In order to ‘see’ a diagram in a

figure we have to recognize the relevant information and be aware of the possible

modifications. For this reason, the effectiveness of a diagram increases with

expertise: Only experts are able to fully exploit the richness of the different possible

meanings that a figure can acquire. p. 838-839The knot group can be used to distinguish knots and

in particular to give an alternative proof of the existence of non-trivial knots.

Moreover, it shows that diagrammatic and algebraic reasoning can be related since

knot diagrams can be interpreted algebraically. This reveals that these diagrams can

be used as syntactic devices. p. 841