What are mathematical diagrams?

Abstract

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term "mathematical diagram" is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance for a taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations (or both). I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain (away) certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading.

Visual Proofs as Counterexamples to the Standard View of Informal Mathematical Proofs?

“…While adapting Hamami's account to diagrams from homological-algebraic and category-theoretical proof practice, such as commutative diagrams and the accompanying method of "diagram-chasing," appears to be relatively unproblematic, since these can be expressed rather straightforwardly with the help of sequences of equations-which is also mentioned by Avigad (cf. [3, p. 7380])-more work seems to be necessary concerning what Silvia De Toffoli calls "geometric-topological diagrams," such as knot diagrams [18]. A promising first step of how one might try to adapt the standard view is by distinguishing between the criterion of informal rigor itself and criteria of acceptability for rigorous proofs as suggested by her in [17] which appears to fit nicely with Avigad's augmentation of Hamami's model of informal rigor which I have briefly mentioned at the end of Sect.…”

Tables as Powerful Representational Tools

“…Early examples of the arrangement of data into tables are ledgers, e. g., to record the sales in a store [8, p. 125-146]. 3 Historically, such ledgers often consisted of sheets that could be spread on a table (their modern cousins, called 'spreadsheets', are discussed below, in Sect. 6).…”

“…If diagrams are understood from a semiotic point of view as structured signs, then tables are simply a particular, well-defined category of diagrams. However, diagrams that are frequently discussed in the literature (e. g., [3,6,15,20]) are often more general than tables (pace Stenning, who considers directly interpreted diagrams to be subject to more constraints than tables [23, p. 45]), since they do not require any vertical or horizontal arrangement of their elements, and as more expressive, since they typically use lines or arrows to represent relations between elements. Nevertheless, it also seems that, whenever possible, such diagrams are presented in such a way that they look very much like tables!…”

“…We then turn to studying the relation of these features to the, so to speak, passive uses of tables: for the general presentation of information, information retrieval, and the visualization of relational structure and patterns (Sects. [3][4][5]. The latter is particularly important, because it shows how tables can contribute to the detection of new patterns and generation of new knowledge.…”