The role of diagram materiality in mathematics

Cognitive Semiotics 2016; 9(2): 183–201 Anna Kiel Steensen* and Mikkel Willum Johansen The role of diagram materiality in mathematics DOI 10.1515/cogsem-2016-0008 Abstract: Based on semiotic analyses of examples from the history of mathematics, we claim that the influence of the material aspects of diagram tokens is anything but trivial. We offer an interpretation of examples of diagrammatic reasoning processes in mathematics according to which the mathematical ideas, arguments, and concepts in question are shaped by the physical features of the chosen diagram tokens. Keywords: diagrammatic reasoning, mathematical diagrammatic representations, C.S Peirce reasoning, diagrams, 1 Introduction The use of representations in scientific reasoning is currently a major area of focus in semiotics (Stjernfelt 2007), the philosophy of science (Dogan and Nersessian 2012; Nersessian 2012), and science education research (De Freitas and Sinclair 2014). Although the topic has been approached using several different theoretical frameworks, such as distributed cognition and cognitive semantics, Peirce’s theory of diagrammatic signs and diagrammatic reasoning remains a vital source of inspiration for an attempt to understand how representations arise and the roles that representations play in human thinking. In Peirce’s semiotics, a sign can relate to its object by direct connection, by habit (such as convention, social or natural law, or instinctual behavior), or by similarity, and is called an index, a symbol, or an icon, respectively. The diagrammatic sign belongs to the latter category; in Peirce’s own words: “[…] it is as an Icon that the Diagram represents the definite Form of intelligible relation which constitutes its Object, that is, that it represents that Form by a more or less vague resemblance thereto.” (Peirce 1906/1976: 316). *Corresponding author: Anna Kiel Steensen, Independent Scholar, Teglværksgade 16 3th, DK-2100, Copenhagen Ø, Denmark, E-mail: [email protected] Mikkel Willum Johansen, Department of Science Education, University of Copenhagen, Øster Voldgade 3, DK-1350, Copenhagen K, Denmark, E-mail: [email protected] Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM 184 Anna Kiel Steensen and Mikkel Willum Johansen Peirce specifies the diagram to be an icon that is similar to its object by representing a structure of rational relations involving the object: To begin with, then, a Diagram is an Icon of a set of rationally related objects. By rationally related, I mean that there is between them, not merely one of those relations which we know by experience, but know not how to comprehend, but one of those relations which anybody who reasons at all must have an inward acquaintance with. [… The] Diagram not only represents the related correlates, but also, and much more definitely represents the relations between them, as so many objects of the Icon. (Peirce 1906/1976: 316–317) As Stjernfelt points out (2007: 90), this relational similarity between icon and object amounts to an operational criterion of iconicity, according to which the iconicity of a sign is the possibility to derive new information – that is, information not used to construct or recognize the icon as such – about the object by manipulating and observing the sign (c. f., Peirce 1960a: 158 [CP 2.279]1). The performative aspect of iconicity is essential; that is, when a sign is manipulated to obtain information, then the sign is an icon of the object to which the information is thought to apply. Moreover, for a sign to be a diagram, its iconicity should be rational (in the sense expressed by Peirce in the previous quote), meaning that the represented relations and the operations applied to the sign must all be rational. According to Peirce, the derivation of new information is an inherently experimental process that involves elements of action and perception. In a diagrammatic reasoning process, something must be done with the diagram; a diagram token must be operated and the effects observed: “Thinking in general terms is not enough. It is necessary that something should be DONE. In geometry, subsidiary lines are drawn. In algebra, permissible transformations are made. Thereupon the faculty of observation is called into play.” (Peirce 1960b: 194 [CP 4.233]). It should be noted, however, that a diagram is not necessarily an external representation. The diagram token can be a concrete physical object, or it can be something we imagine and experiment with in our imagination. However, in Peirce’s semiotics, the only semiotic difference between the external and “internal” diagrammatic representation seems to be simply that one has a materiality that the other lacks. As a realist,2 Peirce can dismiss such material differences as accidental to the diagrammatic reasoning, since the reality of the represented relational structure controls the reasoning process. In either case, 1 For references to the Collected Papers of Charles Sanders Peirce, we will supplement the page reference with a reference to the volume and paragraph. 2 At its most basic, realism is the idea that there is something that does not depend ontologically on whether we observe or perceive it. In particular, semiotic realism holds that signs are signs of such real objects and relations, and that we can obtain actual knowledge about these objects and relations through signs. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM Diagram materiality in mathematics 185 the token embodies the represented relational structure, and thus the token is placed at center stage as the object that is perceived and acted on (Peirce 1906/ 1976: 315). The fact that a relational structure can be diagrammatized using materially different diagrams will be central to our discussion. A simple example is a drawing of a circle of radius 1 with the center in the origin of a coordinate system which can be used as a diagram of the same relational structure as the algebraic equation x2 + y2 = 1. Moreover, if it is possible to derive the same information about an object by different diagrams, then the diagrams are equal modulo their different tokens, from an operational point of view.3 Tokens may have completely incomparable material qualities (such as typography, notation, and spatial arrangement) and still serve in diagrammatizations of the same relations. Thus, as noted by Stjernfelt (2011: 400), the operational criterion cannot capture the semiotic difference between materially different diagram types, if such differences exist. As we see it, this inattentiveness to the materiality of diagrams constitutes a major dilemma in Peirce’s theory of diagrammatic reasoning: to Peirce, diagrammatic reasoning on the one hand consists of experiments on concrete diagram tokens, but on the other hand, the specific material properties of the tokens are presumed not to influence the outcome of the experiments in any substantial way. In other words, if contingent facts are disregarded, such as the instrumental needs and interests of the reasoner and the constitution of human cognitive and perceptual systems, it does not matter whether the reasoner uses this or that type of diagram, as long as the diagram types are equally iconic with respect to the operational criterion. We consider this view on the materiality of diagrams to be questionable, and in this paper we will investigate the dilemma by analyzing the roles played by different diagram tokens in actual mathematical reasoning processes. The choice of mathematical reasoning is natural, as Peirce considered mathematical reasoning to be diagrammatic reasoning per se, and Peirce’s own work, as well as the discussions of it, is full of mathematical examples. The examples, however, are primarily basic textbook mathematics that only covers a limited scope of mathematical reasoning. With this paper, we wish to enrich the discussion by 3 In this paper, two (mathematical) diagrammatic representations are said to be equal if they are used as equal in mathematical practice. Indeed, we can never know the full operational potential of a diagram, so we settle with this more pragmatic definition. This definition is different from (but not incompatible with) both the more technical definition of equality found for example in (Barwise and Etchemendy 1996), and mathematical “equality” understood as an isomorphism between the represented mathematical structures. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM 186 Anna Kiel Steensen and Mikkel Willum Johansen introducing examples that do better justice to the creative and explorative parts of mathematical reasoning; these are some of the kinds of reasoning and semiotic processes that we find in mathematics research. Due to the hierarchical nature of mathematics, however, it is difficult for outsiders to follow examples from the current research fronts. Therefore, we have chosen three historical cases that are sufficiently advanced to reflect the creative and explorative parts of mathematics while simultaneously not requiring a serious mathematical background to follow. The cases show three ways in which materiality of diagrams can interfere with the reasoning process, namely conceptual development, domain expansion, and tool/object-conversion. 2 Conceptual development As our first case, we will analyze an example of a diagrammatic representation of the natural numbers. In modern mathematics, natural numbers are usually represented with Hindu-Arabic numerals or algebraic symbols. In Peircean terms, these representations are symbols, since they represent by way of convention, but these are symbols that enter into diagrams; for instance, when we use the numerals as the basis for pen-and-paper multiplication. Although modern notation may to us seem inseparable from the natural number system, the natural numbers have historically been represented in several qualitatively different ways, and the aim of this section is to compare one of them, Pythagorean figural numbers, with modern notation. In Pythagorean figural number notation, numbers are represented using collections of dots or pebbles in a straightforward way; “•” represents the number one, “••” represents the number two, etc. All four basic arithmetic operations can easily be performed by arranging and counting dots following simple algorithms; for example, the product of two natural numbers n and m can be found by constructing an n-by-m rectangular collection of dots and counting the total number of dots in it. Consequently, and in this context importantly, this Pythagorean dot representation supports the same basic diagrammatic reasoning with natural numbers as a representation using the Hindu-Arabic numerals. The Pythagoreans also used dot notation to investigate general properties of the natural number system. As part of this investigation, they made numerous conceptual innovations. For example, they introduced a distinction between even and odd based on whether the dot collection representing a number could be broken into two new collections of equal size or not (Heath 1921: 70), Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM Diagram materiality in mathematics 187 and they introduced figural numbers, such as triangular, oblong, and square numbers, based on the shape of possible outlines of dot collections (ibid., 76). Equipped with these second-order concepts, the Pythagoreans used dot notation to establish several non-trivial properties of the natural numbers. To take two examples, dot notation makes it clear that any square number is the sum of two consecutive triangular numbers and vice versa (Figure 1, Heath 1921: 83) and that any partial sum of the series of consecutive odd numbers 1 + 3 + 5 + … will be a square number (Heath 1921: 77) (Figure 2). These same properties can be shown through experiments with a diagram that uses an algebraic token. First, in algebraic notation, for a natural number n, triangular numbers can be defined as numbers of the form 21
n
ðn + 1Þ, oblong numbers as n
ðn − 1Þ, and square numbers as n2 . These were later followed by pentagonal numbers 21
n
ð3n + 1Þ, hexagonal numbers 21
n
ð2n − 1Þ, and more. A pair of consecutive triangular numbers can be represented as 21
n
ðn + 1Þ and 1 2
ðn + 1Þ
ðn + 2Þ, and by standard algebraic manipulations it can easily be seen that this pair sums to the square ðn + 1Þ2 . There are equally simple proofs of the other property. The typographical differences between this algebraic diagram Figure 1: The sum of two triangular numbers is a square and vice versa. Figure 2: Any partial sum of the series 1 + 3 + 5 + … will form a square. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM 188 Anna Kiel Steensen and Mikkel Willum Johansen and the diagram constructed with Pythagorean dot notation evidently do not affect the possibility of representing these specific rational relations or justifying their validity. Although the diagrams appear to be different, it may be reasonable to regard them as instances of the same diagrammatic representation of this relation, since they allow the same information about it to be derived. Here we have two types of diagram that are extremely different from a typographical point of view, and yet the diagrams can hold the same content and facilitate the same derivations and computations. In other words, the materiality of the diagrams seems completely accidental. The case so far seems to corroborate Peirce’s understanding of diagrammatic reasoning; different notations may in fact support the very same reasoning processes. In mathematics, it is a common method to shift between different types of diagrams during the reasoning process. These shifts may be motivated by cognitive convenience, cultural and instrumental demands, or other factors, but we now see why the shifts are justified. The diagrams are considered to hold the same mathematical content despite their material differences, and as long as two representations diagrammatize the same rational relations, they can be freely substituted. In the case at hand, the same results will be reached regardless of whether the relation is diagrammatized using algebraic notation or Pythagorean dot notation, and the same idea can be extended to other representational types. This is, therefore, a specific instance of a general idea with huge methodological and practical significance. Peirce’s theory captures and articulates an insight that is fundamental to an important part of mathematical practice. And yet, the case above also reveals a significant shortcoming of the theory. The two representational types may well be interchangeable when it comes to mathematical deduction, but if we move to mathematical ideation and creativity, differences begin to become apparent. As noted above, the dot diagrams led to the development of many new concepts. Some of them, such as odd, even, and square numbers, have simple algebraic representations, while others do not. Triangular, oblong, pentagram, and hexagram numbers may spring out naturally in dot diagrams, whereas they seem less distinctive in their algebraic definitions. Most of these less-distinctive concepts have vanished from active use along with the research programs they might have inspired. This shows how the choice in diagram type matters; the different material properties of different diagram types may inspire different conceptual developments. Even some of the concepts still in use have a clear, non-trivial indication of their origin in dot notation. As a striking example, triangular numbers have evolved from being mathematical entities associated directly with a certain arrangement of dots to being considered mathematical properties that are enjoyed by certain natural numbers – although these are properties with a strong geometric connotation. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM Diagram materiality in mathematics 189 Here, the dot diagram catalyzed the formation of a concept robust enough to transcend and outlive the dot diagram. The realist might object that the reality of the diagrammatized mathematical relation, not the diagram typography, controls the diagrammatic reasoning. All the concepts above reflect real relational properties of the natural numbers, and concept formation is a discovery made by diagrammatic experiments. Consequently, by this argument, the semiotic aspect of a choice between diagram typographies amounts to cognitive convenience. But this objection fails to acknowledge how semiotically complicated it is to establish second-order concepts when we have no direct perceptual access to the objects we investigate, such as in mathematics. When a new second-order concept is introduced, it may influence the directions and methods of further investigations. It may even interfere with the ontology of mathematics: In what sense did pentagonal and hexagonal numbers exist before they were defined using dot diagrams? 3 Domain expansion As our second case, we look at how typographically different diagrams have been used to solve quadratic equations. In Babylonian and Greek mathematics, it was common to represent quantities as line segments and to use these representations to construct solutions to certain classes of problems (e. g., Katz 1998: 36). Part of this body of work, in particular book II of Euclid’s The Elements as well as parts of book I and VI, was considered under the heading “geometrical algebra” in the nineteenth and twentieth century, because the problem types and solution procedures seemed to translate directly and naturally into modern algebraic language. In standard translations such as Heath (1908/2006), the geometric constructions are accompanied by and explained with algebraic notation. Today, historians of mathematics have abandoned the term “geometrical algebra” because it lacks sensitivity to the historical context (e. g., Unguru 1975; Waerden 1976). We will give an example that questions the interchangeability of the two notations from a semiotic point of view. Both Babylonian and Greek mathematicians knew of an algorithm that allowed them to construct solutions to problems of the form: “Given a line segment, break it into two parts such that the area of the rectangle they enclose is equal to a given quantity.” In The Elements (book VI, proposition 28), we find the following version of the algorithm (here simplified): “Construct a square on half of the given line segment. Subtract the given area from the square and rearrange the remaining area such that it forms a new square. The problem can Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM 190 Anna Kiel Steensen and Mikkel Willum Johansen now be solved by breaking the given line in two such that one part equals half of the given line plus the root of the new square and the other part equals half of the given line minus the root of the new square.” The problem can be understood algebraically as two equations with two unknowns: if A is the length of the given line and B is the given area, we have to find two numbers x and y such that x + y = A and x
y = B. By eliminating y, the problem translates into the quadratic equation − x2 + Ax − B = 0, which can be solved by applying the standard solution algorithm. The problem can be diagrammatized with a geometric or an algebraic representation and solved by reasoning with diagrams of either type. Moreover, both representations support a general solution algorithm. So, although these two diagrams are typographically different, they denote the same objects and allow for the same diagrammatic reasoning. However, the solutions of the algebraic diagram actually differ radically from those of the geometric diagram. This famously became apparent in Cardano’s Ars Magna, where Cardano considers the problem of dividing a line of length 10 into two parts such that the product of the lengths of the parts is 40 (Cardano 2007: 219). Cardano seemingly sets out to solve the problem by applying the geometric algorithm; he constructs a square on half the length of the given line. The next step in the algorithm requires him to subtract 40 (the given area) from 25 (the area of the square). The algorithm then breaks down, since a geometric shape with a negative area cannot be constructed. This limitation in the algorithm was well-known, but rather than rejecting the problem as impossible to solve, Cardano translates it into a semi-algebraic form and carries out the algorithm as algebraic manipulations. This gives him the (complex number) pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi solutions 5 + − 15 and 5 − − 15, which can neither be obtained by the geometric algorithm nor expressed with the geometric diagram. Cardano did not consider these to be proper solutions (he likens the whole maneuver to a sophistry that is “as refined as it is useless” [Cardano 2007: 220]), but mathematicians gradually came to accept such complex solutions to quadratic equations (as well as the equally geometrically impossible negative solutions) (c. f., Johansen and Kjeldsen 2015). This might not seem in conflict with a Peircean conception of diagrams. Cardano and his followers simply discovered that algebraic notation is more general, in the sense that it allows us to handle problems that are impossible in geometric notation. There may be an overlap between the relational structures represented by the geometric and the algebraic diagrams, but the overlap is not complete (because the geometric diagrams implicitly add the relation “quantities as line segments” to anything they represent). The extra solutions obtained by the algebraic diagram do not come from the algebraic typography but from the Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM Diagram materiality in mathematics 191 diagrammatized relational structure itself. The different outcomes of the diagrammatic reasoning processes simply show that the represented relational structures are not identical, and the domain extension is new information about the relations represented by the algebraic diagram. In this interpretation of the case, the domain extension represents a relational property of the algebraic diagram. So the case is completely in line with the operational criterion, and the diagram typographies are still as accidental to the diagrammatic reasoning process as Peirce suggested. This interpretation, however, takes the historical context too lightly. The two diagrams are not operationally equal to us, but in a mathematical environment where there only exist non-negative real-number solutions to the problem, the representations support the same allowed diagrammatic reasoning and deliver the same allowed solutions. To the mathematicians before Cardano – and to Cardano himself – the representations were interchangeable, because they differed only by typographical features. There is only one relational structure in play, and the extra solutions are not new information about that relational structure; they are nothing but meaningless attachments to the algebraic diagram. The gap between the representations only came to be because the typographical features of the algebraic representation suggested an expansion of the domain of solutions, an extension impossible even to describe with the geometric representation. This does not imply that the domain extension begun as an accidental typographical aspect of the algebraic diagram. What it means, however, is that there was a significant semiotic potential in the typography of the algebraic diagram. This illustrates a semiotic potential in the typographical difference between operationally equal diagrams, and the potential extends beyond typography, because the diagrams can suggest different domains for the relations they are meant to handle. As mentioned in Section 1, the performative aspect is decisive; in this case the diagrams were used interchangeably. The solutions obtained by the algebraic diagram are incompatible with the geometric diagram, and were not acknowledged as parts of the represented relational structure. This suggests that it is by virtue of material properties that the algebraic diagram enabled Cardano and his followers to interpret these extra solutions as an extension of the domain. 4 When the accidental becomes essential As our last case, we will investigate an important episode in the development of graph theory, which has been established during the last 50 years as a prolific Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM 192 Anna Kiel Steensen and Mikkel Willum Johansen and vibrant sub-area in mathematics. Several journals are dedicated to graph theory, and more than 1,000 papers classified within the area are published every year (Gross et al. 2014 : xiii). The introduction of graph notation inspired a conceptual development, just as the Pythagorean dot representations did. Today, a graph, informally speaking, consists of a number of vertices that are joined by edges,4 and we have a rich vocabulary to describe different types of graphs and their properties. Graphs can be connected and have different colorings, they can have leaves and walks, and they can be trees, forests, etc. (e. g., Gross et al. 2014: ch. 1). But while the Pythagoreans never lost sight of the natural numbers as their objects of study, the development of graph theory happened as graphs evolved from a notational device used to study other mathematical structures into mathematical objects in their own right. This can be illustrated by looking at one of the seminal papers in graph theory, Julius Petersen’s 1891 paper Die Theorie der regulären Graphs (Petersen 1891). Graph notation has its origin in the treatment of questions of obvious topological character, and during the nineteenth century, graphs turned out to be useful when dealing with problems that were more abstract (e. g., Cayley 1857). In his 1891 paper, Petersen sets out to solve a purely algebraic problem in so-called invariant theory by using graphs. Invariant theory investigates products of the form: ðx1 − x2 Þα
ðx1 − x3 Þβ
ðx2 − x3 Þγ


ðxn − 1 − xn Þ [1] where x1 , x2 , . . . , xn are given variables, and the exponents α, β, etc. are natural numbers. A product is required to be of the same degree5 in each variable. It was known at the time that all such products can be generated from a finite selection of the products (Mulder 1992: 159), and Petersen sets out to explicitly determine these generators. Invariant theory is an algebraic theory; both Petersen’s problem and the known proof of the finiteness of the set of generators were stated in algebraic symbols and treated using algebraic techniques. Petersen, however, opens his paper by explaining how the problem can be represented using graph notation (an idea probably suggested to him by Sylvester [Mulder 1992: 158–159]): 4 Formally, a graph G is defined as a pair of sets V, E for which every element in E has either one or two of the elements of V associated to it. The elements of V are the vertices, and the elements of E the edges. The vertex or vertices associated to an edge is called the endpoint(s) of the edge and the endpoints are said to be joined by this edge (Gross et al. 2014: 2). 5 The degree of a variable is the value of its largest exponent. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM Diagram materiality in mathematics 193 One can give the problem a geometrical form by representing x1 , x2 , . . . , xn as arbitrary vertices in the plane and indicating each factor xm − xp by means of an edge joining xm and xp . A product thus gives rise to a figure made up of n vertices joined in such a way that each vertex is incident with the same number of edges. The same two vertices may be joined by several edges. For example, consider Figure (3) which represents the product ðx1 − x2 Þ2
ðx3 − x4 Þ2
ðx1 − x3 Þðx2 − x4 Þðx1 − x4 Þðx2 − x3 Þ (Petersen 1891: 194, in the translation of Biggs et al. 1976: 190–191) Figure 3: Graph representation of a product. Each vertex represents one of the unknowns x1 to   x4 , and each edge represents one of the factors xi − xj of the product. Redrawn from Petersen (1891: 201). This token-token translation of products into graphs is accompanied by an operation-operation translation: The concatenation of products corresponds to superimposing graphs, and, conversely, the factorization of a product corresponds to splitting the graph into parts. With this translation scheme, Petersen moves freely between the graph representation and the algebraic representation. In Peircean terms, the two representations function as typographically different diagrammatic signs of the same relational structure. Since graph theory as such was not an established mathematical field at the time, what is happening in the paper would not and should not be regarded as a translation between two independent mathematical fields. From a mathematical point of view, it is especially noticeable that Petersen does not justify his use of graphs in more detail; instead, he presupposes that reasoning with graphs produces results applicable to invariant theory (as long as the translation scheme is not violated). The simple structural correspondence contained in the translation scheme seems sufficient to convince Petersen, which indicates that he does not consider graphs to be mathematical objects in their own right. Otherwise, he would have needed to justify that graphs were mathematically compatible with invariant theory, and stating a translation key between the notations would not have been sufficient. As other authors have noticed (Mulder 1992: 162) – and which his quote suggests – Petersen indeed seems to use graphs as a pictorial representation of the algebra, not as a translation of the algebraic problem into another mathematical field. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM 194 Anna Kiel Steensen and Mikkel Willum Johansen If Petersen’s initial intention was to use graph notation as a tool to investigate an algebraic structure, then the content of his paper to some extent contradicts that intention. The paper is organized as a series of investigations of graphs, and the results lead step by step to a partial solution to the problem that he set out to solve, stated in graph theoretical form. Petersen interprets this partial solution in terms of the (algebraic) invariant theory, but he then returns to graphs and proceeds with the solution. By the end of the paper, he has extended the partial solution, but he never explicitly returns to invariant theory. The final result is left untranslated, and there is no attempt to present an invariant theoretical meaning of the intermediary investigations of graphs. Each step in the series of results is stated without reference to invariant theory, and together they form a de facto investigation of graphs. In effect, the paper results in a study in graph theory as much as a study in invariant theory. In other words, the tool has become the object of investigation. This shift from regarding graphs as a tool to treating them as an object of investigation was gradually mirrored by the general society of mathematicians. Today, Petersen’s 1891 paper is considered to be one of the first papers containing significant results that are explicitly in the theoretical investigation of graphs (Mulder 1992: 158). What we see in the paper and in the attitude toward graphs among mathematicians can be interpreted as an instance of a more general phenomenon wherein epistemic tools and techniques are gradually transformed into epistemic objects (Rheinberger 1997; Epple 2004; Kjeldsen 2009). Although the discovery of such a notion of tool/object-conversions is fairly new, several important occurrences have already been identified in the history of mathematics, such as in the development of category theory (Johansen and Misfeldt 2015) and the emergence of convex bodies in Minkowski’s practice (Kjeldsen 2009). Clearly, graph theory is not a singular case, but rather illustrates a common and important force in the development and expansion of mathematics. As we have seen, the graphs were used as tokens to diagrammatize the same relations as the initial algebraic diagram. When Petersen shifts to the graph diagram, the choice between the two is largely a matter of cognitive convenience, and the typographical differences are considered to be mathematically insignificant. With the tool/object-conversion, the once accidental typographical features of graphs become diagrammatic representations of relations formed in their image (those denoted as leaves, forests, connectedness, and so on). These relations are the subject matter of modern graph theory, the objects we investigate and about which we prove theorems. What began as a simple choice between typographically different instantiations of the same diagram had semiotic consequences far beyond the design of the token. We do not claim that there are no micro-sociological or cognitive aspects in the tool/object-conversion of Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM Diagram materiality in mathematics 195 graphs, but such effects cannot account for the full semiotics of the conversion. This case shows that diagram materiality can play an active role in conversions from tool to object and consequently in the formation of mathematical objects and relations, so tool/object-conversion as a semiotic phenomenon calls on us to add the semiotics of the material to Peirce’s theory of diagrams. 5 Discussion We will now return to the dilemma in Peirce’s theory that motivated our paper: Peirce recognized that diagrammatic thinking means thinking with tokens, but there is no semiotic potential in diagram typography from the perspective of his theory, because the typographical properties of the diagram are regarded as accidental to the diagrammatic reasoning process. According to Peirce, any conclusion obtained by diagrammatic reasoning is necessarily true about the diagrammatized relational structure. In particular, when a diagrammatic reasoning process shows that the diagram has some property, then it is the relational structure that has this property, no matter what typographical qualities the diagram token has. In each of the three cases, we see this idea in action as shifts between different representation types were followed by ascriping the results of the diagrammatic experiments to one common relational structure. This also works in reverse, as it becomes possible to relate different mathematical concepts if shared relational properties can be identified and diagrammatized. The Peircean diagram completely captures both of these essential roles played by representations in contemporary mathematical practice. While Peirce’s theory enables a well-articulated description of these shifts between representations, it is silent when it comes to the observed influence of diagram typographies on mathematical reasoning. And, as the cases show, this influence can have far-reaching consequences (even ontologically) and crucially affect the development of mathematics. A theory of diagrammatic reasoning should account for that without rejecting the possibility of seeing typographically different diagrams as diagrams of the same relational structure. We open the discussion of the dilemma by examining the notion of optimal iconicity posed by Stjernfelt (2011). An icon is said to be optimal when every relation involved in the represented relational structure is displayed by the icon token: “A diagram ought to be as iconic as possible, that is, it should represent relations by visible relations analogous to them.” (Peirce 1960b: 348 [CP 4.433]). Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM 196 Anna Kiel Steensen and Mikkel Willum Johansen We should clearly not understand this idea of analogy between visible relations and represented relations in the naïve sense of the word; of all the relations we are able to diagrammatize, many do not have an obvious or unique visible form. Moreover, to determine and agree on the precise criteria for analogy would in itself be a delicate task. In any case, optimal iconicity addresses the visual aspect of the diagram and thus draws attention to the specific material properties of that diagram. This perspective might be what we need in order to add a more nuanced view on the materiality of diagrams to Peirce’s semiotics. We must nevertheless be aware of the ontological difficulty stemming from the idea that we can measure some kind of similarity between diagram and object. This idea amounts to: “[…] the attempt at making the graphs match general ontological structure to as large a degree as possible […]” (Stjernfelt 2011: 413). Now, to avoid psychologism and contingency, the use of a similarity criterion requires us to insist upon the reality of the represented relations as well as the reality of the similarity itself, a reality that can secure an absolute standard by which to measure. Before we can use optimal iconicity to describe the degree of iconicity of one diagram relative to another, we have to consider what it means to evaluate this proclaimed similarity between a diagram and its object. For example, in the Petersen case, the graph diagram and the algebraic diagram have typographically different tokens but are used as mathematical equivalents; that is, it is as if the choice between them were only a matter of contingent, heuristic factors, such as which representation happens to be most intuitive to work with. So, as diagrammatic representations, they are equal with respect to operational iconicity. In a product given in algebraic form, such as [1] above, the number of occurrences of, say, x1 − x2 equals the exponent of x1 − x2 , but by looking at the corresponding graph, we can literally see each occurrence as an edge connecting the vertices associated with x1 and x2 . In the graph diagram, the degree of the product is defined by the number of edges from any vertex; but to obtain this degree from the algebraic expression, we must first choose a variable and then add all the exponents of that variable. However, to say that a variable occurs a given number of times in the product or that a vertex in the graph is incident with a number of edges is to verbally represent mathematical properties, properties that are not perceptible unless they are diagrammatized. The mathematical object that is represented by the vertex is not “incident” with anything, nor does it “occur” a certain number of times. So how can we refer directly to mathematical relations? The graph at first seems highly iconic, because each minus sign is displayed as a typographical reality. In the algebraic expression, this typographical reality is replaced by a symbol, the exponent of x1 , which by convention represents the same property as the number of edges and the number of minus signs. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM Diagram materiality in mathematics 197 Such replacement decreases the iconicity of the sign (roughly speaking, fewer symbols amount to higher iconicity), but this merely compares one representation to another; it does not compare a representation directly to the relational structure. The latter comparison can only be made once we have direct access to the mathematical relation. The problem is that we do not have perceptual access to the represented relational structure without the diagram, so how can we determine whether the diagram resembles the structure? Can we represent a mathematical object without constructing a less than optimal icon of it? Perhaps, but can we conceive of a mathematical object in a way that is concise, robust, and definite enough to be comparable to some diagram of the object without already having constructed an actual representation of it? In short, we need to perceive the mathematical object in order to evaluate how iconic a given representation of it is. But how can we perceive mathematical objects as pre-represented? The twofold problem with optimal iconicity when it comes to describing the semiosis of representations in mathematics thus seems to be that the concept 1) has to do with how the represented object is, because it is based on how the object looks and requires a representation of the object and 2) commits to a strong form of realism with respect to mathematics, a realism that insists that it is possible, in principle, to represent any mathematical relation. If we already have an “inward acquaintance” (Peirce 1906/1976: 316) with the relation, then this relation does not need to have a certain form that is comparable to other physical forms in such a way that we are in accordance as to which form best matches the relation. Also, to claim that we already have the relation is problematic; as we have seen, physical forms can provide form to mathematical relations not previously given. As long as we have a pre-established ontology, these assumptions are perhaps reasonable enough, but they seem dubious in the creative and explorative parts of mathematics. Within the confinements of a pre-established ontology and a pre-established theory, it seems indeed possible to diagrammatize the same content using qualitatively different representations. In this sense, Peirce offers an insightful justification for the practice of substituting one representation with another. Yet the choice of representation can have far-reaching consequences for the theories and ontologies we are building (provided we understand mathematics as a creative and growing field of human knowledge). As we saw in the Cardano case, the typography of the diagram can extend the relational structure of the diagram beyond the pre-established ontology. In the dot representation case, we found a conceptualization grounded in specific typographical properties of a diagrammatization. This conclusion is largely independent of a basic ontological attitude toward mathematics; even a realist will have to admit that the Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM 198 Anna Kiel Steensen and Mikkel Willum Johansen primordial mathematical reality must be conceptualized in one way or another. But the theoretical outset and conceptual system, which any organized exploration must have, will guide which aspects of the mathematical reality to explore and how to do it. With this in mind, what we have seen in this paper is that the typography of diagrams plays an important role beyond the instrumental and cognitive aspects of the diagrammatic reasoning process, and it is a role that is not mutually exclusive with realism. This conclusion brings a general message to the discussion of diagrammatic reasoning. The majority of interest has been aimed at understanding the expressive power of different diagrammatizations, such as establishing whether Venn diagrams or Peirce’s existential graphs are logically equivalent to first-order logic (Shin 1994). Such work is clearly important, but it only considers the synchronic dimension: It seeks to understand how well different representational systems function within a pre-established ontology. The three cases show that there is a diachronic dimension as well, spanned at least in part by the material properties of diagrammatizations. Finally, we ask whether our results are necessarily confined to diagrammatic reasoning and what the prospects of further diachronic investigations of diagrams could be. In mathematics, we can only perceive the relations we reason about through diagrammatic representations. This situation is not universal, but it is also not unique. In the social sciences, we encounter many relations that are only accessible through representations; relations such as the unemployment rate cannot be perceived directly but need to be represented by the Phillips curve (for instance). In the natural sciences, we often reason about relations that are not observable but may be the result of chains of representations from one system to another (e. g., Latour 1999). These representations are performative, in the sense that they are an active part of our perception, conceptualization, and ontological conception of the field. The diachronic semiotic effects of representations in other sciences are not of less interest than these effects are in mathematics. If the materiality of representations plays an important role in mathematics, this may well also be the case in other sciences. The relevance of studying the diachronic dimension of mathematical diagrams need not stop here. Due to the precision of mathematics, we can determine situations where two representations diagrammatize the same relations with the same degree of iconicity, and we can isolate the effect of the material aspects of the various diagrammatizations. This opens the possibility of describing the effects qualitatively and independent of the mathematical framework. Equipped with such metatheoretical descriptions, we can perhaps obtain a better understanding of the diachronic dimension of the use of representations in human reasoning across specific sciences. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM Diagram materiality in mathematics 199 6 Conclusion In our interpretation, the cases confirm the general idea that the material difference between different diagram types can result in profound semiotic differences, and the choice of diagram type can have real consequences for the development of mathematics. Pierce presupposed that the role played by diagram materiality in diagrammatic reasoning processes is trivial; this view has been passed on with his diagram notion, and research in the use of diagrammatic representations is to some extend still shaped by it. This view and its consequences (such as the focus on the synchronic dimension of diagrammatic representations) should be revised in the light of our results. We have argued that even though realism is intertwined with the idea that diagram materiality is accidental, it is possible to simultaneously maintain realism with respect to rational relations while also acknowledging the semiotic potential of material properties. However, in our opinion, realism cannot explain all semiotic aspects of diagram materiality. We have seen that the material properties of a diagram constitute an important part of the diachronic dimension of the diagram, and also that the diachronic dimension is relevant to the semiotic process related to the use of a particular representational system. This semiotic process can catalyze changes in theoretical perspective and ontology. We have pointed out three such changes that can happen caused by changes of representational system, but there may, of course, be many more. These results suggest the diachronic dimension as a new area of research in the use of representations in scientific reasoning. References Barwise, J. & J. Etchemendy. 1996. Heterogeneous logic. In G. Allwein & J. Barwise (eds.), Logical reasoning with diagrams, 179–200. New York: Oxford University Press. Biggs, N. L., E. K. Lloyd & R. J. Wilson. 1976. Graph theory, 1736–1936. Oxford: Clarendon Press. Cardano, G. 1545/2007. The rules of Algebra (Ars Magna). T. R. Witmer, trans. 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Towards a history of epistemic things: Synthesizing proteins in the test tube. Stanford: Stanford University Press. Shin, S. J. 1994. The logical status of diagrams. Cambridge: Cambridge University Press. Stjernfelt, F. 2007. Diagrammatology, an investigation on the borderlines of phenomenology, ontology, and semiotics, Synthese Library v. 336. Dordrecht London: Springer. Stjernfelt, F. 2011. On operational and optimal iconicity in Peirce’s diagrammatology. Semiotica 186. 395–419. Unguru, S. 1975. On the need to rewrite the history of Greek mathematics. Archive for History of Exact Sciences 15(1). 67–114. van der Waerden, B. L. 1976. Defence of a “shocking” point of view. Archive for History of Exact Sciences 15(3). 199–210. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM Diagram materiality in mathematics 201 Bionotes Anna Kiel Steensen Anna Kiel Steensen was born in 1987 in Copenhagen. She holds an MSc in mathematics from University of Copenhagen, 2015, and is a Ph.D. student in the philosophy of mathematics at ETH Zürich as of February, 2017. Mikkel Willum Johansen Mikkel Willum Johansen was born in 1973 in Copenhagen. He holds a Master’s Degree in philosophy with a minor in mathematics. Beginning with his Ph.D. dissertation in 2011, Naturalism in the Philosophy of Mathematics, he has investigated the philosophy of mathematics, primarily from naturalized and practice oriented perspectives. He is a member of the Association for the Philosophy of Mathematical Practice and is an associate professor in philosophy of science at University of Copenhagen. Brought to you by | University of Toronto-Ocul Authenticated Download Date | 11/7/16 10:03 AM