This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact mathematical (...) constructions. Therefore, they are legitimate components of the theory (combinatorial and geometric group theory) and the pictures of Cayley graphs are a part of practical mathematical procedures. (shrink)

Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant des représentations visuelles, peut conduire à de nouveaux résultats mathématiques. L'argumentation est basée sur l'étude du cas d'un domaine des mathématiques relativement récent et prometteur: la théorie géométrique des groupes. L'article discute comment la représentation des groupes par les graphes de Cayley rendit possible la découverte de nouvelles propriétés géométriques de groupes.

Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant des représentations visuelles, peut conduire à de nouveaux résultats mathématiques. L'argumentation est basée sur l'étude du cas d'un domaine des mathématiques relativement récent et prometteur: la théorie géométrique des groupes. L'article discute comment la représentation des groupes par les graphes de Cayley rendit possible la découverte de nouvelles propriétés géométriques de groupes.

Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to their visualisations? Using an example from graph theory, this paper argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians’ sensitivity to aesthetics of the abstract.

This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact mathematical (...) constructions. Therefore, they are legitimate components of the theory and the pictures of Cayley graphs are a part of practical mathematical procedures. (shrink)

This paper concerns the role of intuitions in mathematics, where intuitions are meant in the Kantian sense, i.e. the “seeing” of mathematical ideas by means of pictures, diagrams, thought experiments, etc.. The main problem discussed here is whether Platonistic argumentation, according to which some pictures can be considered as proofs (or parts of proofs) of some mathematical facts, is convincing and consistent. As a starting point, I discuss James Robert Brown’s recent book Philosophy of Mathematics, in particular, his primarily examples (...) and analogies. I then consider some alternatives and counterarguments, namely John Norton’s opposite view, that intuitions are just pictorially represented logical arguments and are superfluous; and the Kantian transcendental theory of construction in imagination, as it is developed in the works of Marcus Giaquinto and Michael Friedman. Although I support the claim that some intuitions are essential in mathematical justification, I argue that a Platonistic approach to intuitions is partial and one should go further than a Platonist in explaining how some intuitions can deliver new mathematical knowledge. (shrink)

As in science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that are unavailable in a given practice, e.g., Euclidean geometry. Thought experiments invite us to perform an imaginary scenario involving counterfactual, deductive and sensory elements. This chapter aims to pinpoint the beneficial peculiarities of thought experiments in mathematics in comparison with inferences, diagrams and calculative procedures. Reflection about thought experiments assists us to realize both the limits and opportunities in mathematical thinking. Henceforth, the analysis (...) of examples suggests a broader understanding of mathematical, thought and experiment. (shrink)